Chapter 9 slides, 3rdedition - University of North Floridawkloster/crypto/CryptographyNetSecurity...

Cryptography and Network Security 1

Cryptography and Network Security

© by Xiang-Yang Li

Department of Computer Science, IIT


Notice©This lecture note (Cryptography and Network Security) is

prepared by Xiang-Yang Li. This lecture note has benefited from numerous textbooks and online materials. Especially the “Cryptography and Network Security” 2nd edition by William Stallings and the “Cryptography: Theory and Practice” by Douglas Stinson.

You may not modify, publish, or sell, reproduce, create derivative works from, distribute, perform, display, or in any way exploit any of the content, in whole or in part, except as otherwise expressly permitted by the author.

The author has used his best efforts in preparing this lecture note. The author makes no warranty of any kind, expressed or implied, with regard to the programs, protocols contained in this lecture note. The author shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these.


About Instructor Assistant Professor IIT from 2000 PhD/MS UIUC 1997-2000 BS, BE Tsinghua University

Research Interests: Algorithm design and analysis Wireless networks Game theory Computational geometry

Contact Information Phone 022-27891275, 27408884 Email: [email protected]


Office and Office hours Office

SB 237D Office hour

Monday 3PM – 4PM. Wednesday 3PM– 4PM. Or by contact: email [email protected], phone 312 567 5207


About This CourseTextbook

Cryptography: Theory and Practice by Douglas R. Stinson CRC press

Cryptography and Network Security: Principles and Practice; By William Stallings Prentice Hall

Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, CRC Press I have electronic version!

http://www.cacr.math.uwaterloo.ca/~dstinson/gifs/cover.gif

http://www.amazon.com/gp/product/images/0138690170/ref=dp_primary-product-display_0/104-8278000-3955142?%5Fencoding=UTF8&n=283155&s=books

http://www.cacr.math.uwaterloo.ca/hac/about/cover.gif


Grading and Others Grading

Homework 30% Mid Term 20% Report 20% (select your own topic),

15 pages report Final exam 30% (closed book)

Policy Do it yourself Can use library, Internet and so on


Homeworks Do it independently

No discussion No copy Can use reference books

Staple your solution Write your name also,

For report, you should NOT discuss

with classmates then write your own report (about 10 pages for the topic you selected)

HW1 (Due 2/7/05) HW2 (Due 3/7/05) HW3 (Due 4/04/05)

Report (Due 05/07/05)

Type your solution!

And print it then submit


Topics Introduction Number Theory Traditional Methods: secret key system Public Key System Digital Signature Other topics:

secret sharing, zero-knowledge proof, bit commitment, oblivious transfer,…


OrganizationChapters

Introduction Number Theory Conventional Encryption Block Ciphers Public Key System Key Management Hash Function and Digital Signature Identification Secret Sharing Pseudo-random number Generation Email Security IP Security Others



IntroductionXiang-Yang Li


Introduction

The art of war teaches us not on the likelihood of the enemy’s not coming, but on our own readiness to receive him; not on the chance of his not attacking, but rather on the fact that we have made our position unassailable. --The art of War, Sun Tzu


Cryptography Cryptography (from Greek kryptós, "hidden",

and gráphein, "to write") is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — the art of encryption.

Past: Cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats.

In recent decades, cryptography has expanded its remit in two ways mechanisms for more than just keeping secrets: schemes like digital

signatures and digital cash, for example. in widespread use by many civilians, and users are not aware of it.


Crypto-graphy, -analysis, -logy The study of how to circumvent the use of cryptography

is called cryptanalysis, or codebreaking. Cryptography and cryptanalysis are sometimes grouped

together under the umbrella term cryptology, encompassing the entire subject.

In practice, "cryptography" is also often used to refer to the field as a whole; crypto is an informal abbreviation.

Cryptography is an interdisciplinary subject, linguistics Mathematics: number theory, information theory, computational

complexity, statistics and combinatorics engineering


Close, but different fields Steganography

the study of hiding the very existence of a message, and not necessarily the contents of the message itself (for example, microdots, or invisible ink)

Traffic analysis which is the analysis of patterns of communication in order

to learn secret information.


Network Security Model

Trusted Third Party

principal principal

Security transformation

Security transformation

attacker


Attacks, Services and Mechanisms Security Attacks

Action compromises the information security Security Services

Enhances the security of data processing and transferring

Security mechanism Detect, prevent and recover from a security attack


Attacks Passive attacks

Interception Release of message contents Traffic analysis

Active attacks Interruption, modification, fabrication

Masquerade Replay Modification Denial of service


Information Transferring


Attack: Interruption

Cut wire lines,Jam wireless

signals,Drop packets,


Attack: Interception

Wiring, eavesdrop


Attack: Modification

intercept Replaced info


Attack: Fabrication

Also called impersonation


Important Features of Security Confidentiality, also known as secrecy:

only an authorized recipient should be able to extract the contents of the message from its encrypted form. Otherwise, it should not be possible to obtain any significant information about the message contents.

Integrity: the recipient should be able to determine if the message has

been altered during transmission. Authentication:

the recipient should be able to identify the sender, and verify that the purported sender actually did send the message.

Non-repudiation: the sender should not be able to deny sending the message.


Cryptography Cryptography is the study of

Secret (crypto-) writing (-graphy) Concerned with developing algorithms:

Conceal the context of some message from all except the sender and recipient (privacy or secrecy), and/or

Verify the correctness of a message to the recipient (authentication)

Form the basis of many technological solutions to computer and communications security problems


Basic Concepts Cryptography

encompassing the principles and methods of transforming an intelligible message into one that is unintelligible, and then retransforming that message back to its original form

Plaintext The original intelligible message

Ciphertext The transformed message

Message Is treated as a non-negative integer hereafter


Basic Concepts Cipher

An algorithm for transforming an intelligible message into unintelligible by transposition and/or substitution

Key Some critical information used by the cipher, known

only to the sender & receiver Encipher (encode)

The process of converting plaintext to ciphertext Decipher (decode)

The process of converting ciphertext back into plaintext


Basic Concepts cipher

an algorithm for encryption and decryption. The exact operation of ciphers is normally controlled by a key — some secret piece of information that customizes how the ciphertext is produced

Protocols specify the details of how ciphers (and other cryptographic

primitives) are to be used to achieve specific tasks. A suite of protocols, ciphers, key management, user-

prescribed actions implemented together as a system constitute a cryptosystem;

this is what an end-user interacts with, e.g. PGP


Encryption and Decryption

Plaintext ciphertext

Encipher C = E(K1)(P)

Decipher P = D(K2)(C)

K1, K2: from keyspace


Security Two fundamentally different securities

Unconditional security No matter how much computer power is

available, the cipher cannot be broken Using Shannon’s information theory

Computational security Given limited computing resources (e.G time

needed for calculations is greater than age of universe), the cipher cannot be broken

Proved by some complexity equivalence approach



Elementary Number TheoryXiang-Yang Li


Number theory Elementary number theory

Main topic of this course divisibility, the Euclidean algorithm to compute

greatest common divisors, factorization Fermat's little theorem and Euler's theorem, the Chinese

remainder theorem and Euler's φ function are investigated;

Analytic number theory Algebraic number theory Geometric number theory Computational number theory


GCD Greatest common divisor gcd(a,b)

The largest number that divides both a and b Euclid's algorithm

Find the GCD of two numbers a and b, a<b Use fact if a and b have divisor d so

does a-b, a-2b …d m a n b

d a b

d a b

d a b

d a qb

2

3


Cont. GCD (a,b) is given by:

let g0=b g1=a gi+1 = gi-1 mod gi when gi =0 then gcd(a,b) = gi-1

The algorithm terminates in O(log b) rounds Why? Every round, the total number of bits of a and b is decreased by at

least one

What is a more precise complexity bound?


Properties For any two integers a and b

Exist integers m and n: gcd(a,b) =ma+bn Example:

a=2, b=3; we choose m=-1, n=1 so –2+3=1 a=6, b=11; we choose m=2, n=-1 so 2*6-11=1

Simple proof? Integer n can be factored as

n=p1a1 p2

a2 p3a3…. pn

an where pi is prime number


Extended Euclidean Algorithm input are two integers a and b,

computes their greatest common divisor (gcd) as well as integers x and y such that ax + by = gcd(a, b).

It later can also be used to compute the inverse of an integer

a n 1 m od


Proof Assume we compute gcd(x0,y0), x0>y0

Let Xi=(xi,yi); 0xi-qi+1yi+1<|yi| Then Xi=MiXi-1, where Mi=(0,1; 1,-qi) Assume the gcd algorithm terminates in n steps We have MnMn-1

…M1X0=(gcd(x0,y0), 0)T

Assume MnMn-1…M1=( )

Then ax0+by0=gcd(x0,y0) The above algorithm is to keep track of a,b,c,d, and xi,yi

values.

a bc d


Modular Arithmetic Congruence

a b mod n says when divided by n that a and b have the same remainder

It defines a relationship between all integers a a a b then b a a b, b c then a c


Cont. addition

(a+b) mod n (a mod n) + (b mod n) subtraction

a-b mod n a+(-b) mod n multiplication

a b mod n derived from repeated addition Possible: a*b 0 where neither a, b 0 mod n

Example: 2*3 =0 mod 6


Addition and Multiplication Integers modulo n with addition and

multiplication form a commutative ring with the laws of Associativity

(a+b)+c a+(b+c) mod n Commutativity

a+b b+a mod n Distributivity

(a+b)*c (a*c)+(b*c) mod n


Cont. Division

b/a mod n multiplied by inverse of a: b/a = b*a-1 mod n a-1*a 1 mod n 3-1 7 mod 10 because 3*7 1 mod 10 Inverse does not always exist!

Only when gcd(a,n)=1


Euclid's Extended GCD Routine If (a,n)=1 then the inverse always exists Can extend Euclid's algorithm to find

inverse by keeping track of gi = ui.n + vi.a Extended Euclid's (or binary GCD)

algorithm to find inverse of a number a mod n (where (a,n)=1) is:


Inverse Inverse(a,n) is given by:

X=(x1,x2,x3)=(1,0,n); Y=(y1,y2,y3)=(0,1,a) If y3=0 return x3=gcd(a,n); no inverse If y3=1 return y3=gcd(a,n); y2=a-1 mod n Q=[x3/y3] T=X-Q*Y X=Y; Y=T Goto 2nd step


When inverse exists If gcd(a,n)=1 inverse exists

We can find x, y such that ax+ny=1 Then x= a-1 mod n

If inverse exists gcd(a,n)=1 Let x be the inverse of a, i.e., ax=1 mod n Then x a=1+q n for some integer q Let gcd(a,n)=d. Then d | (x a-q n ) Obviously d=1 since x a-q n =1


Galois Field If n is constrained to be a prime number

p then this forms a Galois field modulo p denoted GF(p) and all the normal laws associated with integer arithmetic work

Exponentiation b = ae mod p

Discrete Logarithms find x where ax = b mod p


Relative primes Two numbers a and n are relative primes if

gcd(a,n)=1 Consider all integers 0<a <n

How many are relative prime to n? Equivalently, how many a such that a-1 mod n exists

Typically Zn={0,1,2,….,n-1} : all integers 0<= a < n Zn

*={a| 0<= a < n, gcd(a,n)=1} All integers in Zn that are co-prime with n Also called reduced residue set mod n


Euler Totient Function If consider arithmetic modulo n, then a

reduced set of residues is a subset of the complete set of residues modulo n which are relatively prime to n eg for n=10, the complete set of residues is {0,1,2,3,4,5,6,7,8,9} the reduced set of residues is {1,3,7,9}

The number of elements in the reduced set of residues is called the Euler Totient function (n)


cont Compute (n)

If factoring of n is known (n)=n (1-1/pi) where pi is its prime factor

Otherwise It is expensive! But not proved yet

computing (n) when knowing fact n =pq but not the number p and q Conjectured to be a hard question But not proved yet. Equivalent to find p and q


cont Equivalency: finding p,q computing (n)

ProofIf we found p and q, then (n)=(p-1)(q-1)if we found (n), then solve p, q from equations

n p qn p q

( ) ( )( )1 1


Euler's Theorem Let gcd(a,n)=1 then

a(n) mod n = 1 Proof:

consider all reduced residues xi in Zn*={x| 0<= x < n,

gcd(x,n)=1} Then axi,1<=i <= (n) also form reduced residues set Using axi = xi mod n

Using Zn* and aZn

* are same sets! We have a(n) xi = xi mod n Thus, a(n) =1 mod n

Using the fact that xi has inverse


Fermat's Little Theorem Let p be a prime and gcd(a,p)=1 then

ap-1 mod p = 1 Proof: similar to the proof of Euler’s theorem But consider all integers in Zp

Generally, for any prime number p ap mod p = a (true for any number a)

Generally, for any number n=pq a(n) mod n = a (true for any number a)

Need to prove for the case gcd(a,n)>1Do it

yourself


Efficient computing of exponential Compute ab mod n efficiently when b, n

large? Example: compute a1024 mod 21024 +1 Simple approach: repetitively time a 1024 times? Efficient computation:

Write number b in binary format as xkxk-1xk-2….x2x1x0

Let t1=a mod n. Then compute ti+1= ti * ti mod n for i<k

Then

a n a n

a n

t n

b x x x x x x

x

i k

ix

i k

k k k

ii

i

m od m od

[ ] m od

m od

....

( )

1 2 2 1 0

2

0

0

Time complexity?


Chinese Remainder Theorem By Qin Jiushao Let m1,m2,….mk be pair-wise relative prime numbers Assume integer x= ai mod mi for 1<= I <= k Then x= ai ei mod M

Where M= mi; Mi=M/ mi

ei= Mi * (Mi-1 mod mi)

Proof For each i, the integers mi and M/mi are coprime, and using the extended Euclidean

algorithm we can find integers r and s such that r mi + s M/mi = 1. If we set ei = s M/mi, then we have

ei =1 mod mi and ei =1 mod mj for j<>i.


General CRT Sometimes, the simultaneous

congruences can be solved even if the mi's are not pairwise coprime. a solution x exists if and only if ai ≡ aj (mod gcd(ni, nj))

for all i and j. All solutions x are congruent modulo the least common

multiple of the ni. Methods: successive substitution


Example consider the simultaneous congruences

x ≡ 3 (mod 4) x ≡ 5 (mod 6)

Can be transformed to x ≡ 3 (mod 4) x ≡ 5 (mod 2) x ≡ 1 (mod 2) x ≡ 5 (mod 3)

Then transformed to x ≡ 3 (mod 4) x ≡ 2 (mod 3)

Using CRT X=11 (mod 12)


Primality Testing To check if exists integer a such that a|n

Primary school method Test a=2,3,4,5,6,….,n-1 Test a=2,3,4,5,…, n0.5

Test a=2,3,5,7,11,…., p, where prime number p<=n0.5

Two slow! Check almost n numbers Check n0.5 numbers At least around (n/ln n)0.5 numbers need be checked

Example Number n~21024, then (n/ln n)0.5~(21024 /1024) 0.5 ~ 2507

Assume 230 numbers per second, takes about 2507-30*16 = 227 days Any improvement?


Simple Fact Equation x21 mod p has only solutions

1,-1 If p is prime number Simple proof: (x+1)(x-1) 0 mod p

So if we find another solution, then p can not be prime number! Miller and Rabin 1975,1980

Randomly chosen integer a If a21 mod p then p is not prime number

Integer a is called the witness Otherwise p maybe, or maybe not a prime number


Witness AlgorithmWitness(a,n)

Let bkbk-1…b1b0 be the binary code of n-1 Let d=1 For i=k downto 0 x=d; d=d*d mod n If d=1 and x1, and x n-1 return TRUE If bi=1 then d=d*a mod n Endfor If d 1 then return TRUE Return FALSE


Facts Analysis the result of witness

If returns TRUE, then n is not prime number Find other solutions for x21 mod n

Otherwise, n maybe prime number Given odd n and random a

Witness fails with probability less than 0.5 Run witness algorithm s times

If one time, it is TRUE Then n is not prime number

Otherwise, Pr(n is prime)>1-2-s


Randomized Methods Las Vegas Method

Always produces correct results Runs in expected polynomial time

Monte Carlo Method Runs in polynomial time May produce incorrect results with bounded probability No-Biased Monte Carlo Method

Answer yes is always correct, but the answer no may be wrong

Yes-biased Monte Carlo Method Answer no is always correct, but the answer yes may

be wrong


Witness Algorithm Witness Algorithm is based on Monte

Carlo Method It actually test compositeness, not primality

When it reports yes, the number is always composite

When it reports no, input may be composite, prime

Probability Result Pr(input=composite | ans=composite)= 1 Pr(ans=no | input=composite)<1/2 Pr(input=composite | ans=no) 1/4


Time Complexity Each round of witness cost O(log n)

Unit: integer multiplication and modular arithmetic

So the primality testing cost O(s log n) The confidence is 1-2-s if report prime The confidence is 1 if report non-prime


Primitive Root Order of integer ordn(a)

The order of a modulo n is the smallest positive k such that ak1 mod n

Primitive Root Integer a is a primitive root of n if the order of a

modulo n is (n) Not all integers have primitive root

Example n=pq for primes p and q Prime p has (p-1) primitive roots


cont When primitive root exists

Number n in format of p, 2p, pk, 2pk for some integer k and prime number p

Otherwise the primitive root does not exist Find a PR for p such that

Let a=2, i=1 If i>k, a is a PR, otherwise go to step 3 If let i=i+1 and go to step 2;

otherwise let i=1, and a=a+1 and repeat this step 3.

p q qak

a k 1 11 . . . .

a pp q i( ) / m od 1 1


Some “hard” questions Some questions that are assumed to be

hard, will be used as bases for cryptography Integer factorization

Given n, find all its prime factors Discrete logarithm

Given g, y, and p, find x such that gxy mod p Square root

Given b, find x such that x2b mod n. Here n is not a prime number


Integer Factorization write an integer as product of prime numbers.

For example, given the number 45, the prime factorization would be 32·5. The factorization is always unique, according to the fundamental theorem of

arithmetic Given two large prime numbers, it is easy to multiply them. However, given

their product, it appears to be difficult to find the factors. This is relevant for many modern systems in cryptography. If a fast method

were found for solving the integer factorization problem, then several important cryptographic systems would be broken.

Although fast factoring is one way to break these systems, there may be other ways to break them that don't involve factoring. So it is possible that the integer factorization problem is truly hard, yet these systems can still be broken quickly.

A rare exception is the BBS generator. It has been proved to be exactly as hard as integer factorization: if you can break the generator in polynomial time then you can factorize integers in polynomial time, and vice versa


Current state of the art If a large, n-bit number is the product of

two primes that are roughly the same size, no polynomial time factoring algorithm is known the best known algorithms are sub-exponential, but

super-polynomial: asymptotic running time by the general number field sieve (GNFS) algorithm, is

Polynomial methods known for quantum computer!


Factoring algorithms Special purpose

its running time depends on the properties of unknown factors: size, special form, etc.

Examples Trial division, Pollard's rho algorithm, Pollard's p-1

algorithm, Lenstra elliptic curve factorization, Congruence of squares, Special number field sieve

General purpose running time depends solely on the size of the integer to be

factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose algorithms are based on the congruence of squares method.

Examples: Quadratic sieve, General number field sieve


Discrete Logarithms Y gx mod p

Given y, g, and p, compute x as logg(y) Time complexity O(e(ln p)1/3(ln ln p)2/3)

Best known until now In other words, if p is large, then it is very hard to solve the discrete logarithm problem

Several protocols are based on this ElGamal discrete log cryptosystem, Diffie-Hellman key exchange and the Digital

Signature Algorithm. Current methods:

the Pohlig-Hellman algorithm if p-1 is a product of small primes, so this should be avoided in those applications


Quadratic Residue Quadratic Residue

Integer b is a quadratic residue of modulo integer n if and only if x2 b mod n has a solution for x

Number x is called the square root of b Otherwise b is called quadratic nonresidue

Given odd prime p, b is quadratic residue, iff b(p-1)/2 1 mod p b is quadratic nonresidue, iff b(p-1)/2 -1 mod p These facts can be used to test primes with probability


Computing Square root mod p Given number a, find number x, x2 =a

mod p If p=3 mod 4, then x=a(p+1)/4 mod p is a solution. If p=5 mod 8, a(p-1)/4 =1 mod p then x= a(p+3)/8 mod p If p=5 mod 8, a(p-1)/4 =-1 mod p then x= 2a(4a)(p-5)/8 mod p If p=1 mod 8, x a N

hs k

12

p hk 1 2 Here h is an odd number


Compute square-root mod p Find a solution to x2 =a mod p if exists

Let r=0, s=p-1; while s even, {r=r+1; s=s/2;} Choose random n such that Let z=ns mod p; x=a(s+1)/2 mod p; b=as mod p; If b=1, return x as a solution Let m=1, y=b2 mod p; while y<>1 {y= y2 mod p; m=m+1;} If r=m then a is Quadratic non-residue; exit; Let x=xz2r-m-1 mod p and b=bz2r-m mod p and z=z2r-m mod p Go to step 4

The expected running time is O(log4 p)

np 1


Complexity Theory The input length of a problem is the number n of

symbols used to characterize it Complexity of a method

Function f(n) is order O(g(n)) if f(n)<=c*|g(n)|, for all n>=N0, for some c

Function f(n) is order (g(n)) if f(n)>=c*|g(n)|, for all n>=N0, for some c

Function f(n) is order (g(n)) if c1*|g(n)|<=f(n)<=c2*|g(n)|, for all n>=N0, for some c1 and c2

Polynomial time algorithm (P) solves any instance of a particular problem with input length n in time

O(p(n)), where p is a polynomial


Cont. Non-deterministic polynomial time

algorithm (NP) is one for which any guess at the solution of an instance of

the problem may be checked for validity in polynomial time.

NP-complete problems are a subclass of NP problems for which it is known that if

any such problem has a polynomial time solution, then all NP problems have polynomial solutions.

Co-NP: the complements of NP problems.



Conventional MethodsXiang-Yang Li


Roadmap of Cryptography classical cryptography (--- 1920s)

secret writing required only pen and paper Mostly: transposition, substitution ciphers Easily broken by statistics analysis (e.g., frequency)

mechanical devices invented for encryption Rotor machines (e.g. Enigma cipher) 1930s-1950s featured in films, such as in the James Bond adventure From Russia

with Love specification of DES and the invention of RSA

(1970s) --- modern ciphers Public key system, most notably

Quantum Cryptography (future?)


Quantum Cryptography Quantum cryptography currently has two aspects.

quantum key exchange (also known as quantum key distribution), a method for secure communications based on quantum mechanics

conjectured effect of quantum computing on cryptanalysis, although it is currently, like quantum computing itself, only a theoretical concept.

Basic idea of quantum key exchange is to use the "noisy" properties of light to render incoherent an image that acts to complement a secret key. This image can be represented in a number of ways, but the ability to decode

that image rests upon an understanding of how it was made. No way to intercept the transmission without changing it is possible, so key information can be exchanged with great confidence it has been transmitted secretly.

quantum computing will considerably extend the reach of cryptanalysis, making brute force key space searches much more effective -- if such computers ever become possible in actual practice


History Ancient ciphers

Have a history of at least 4000 years Ancient Egyptians enciphered some of their

hieroglyphic writing on monuments Ancient Hebrews enciphered certain words in the

scriptures 2000 years ago Julius Caesar used a simple substitution

cipher, now known as the Caesar cipher Roger bacon described several methods in 1200s


History Ancient ciphers

Geoffrey Chaucer included several ciphers in his works Leon Alberti devised a cipher wheel, and described the

principles of frequency analysis in the 1460s Blaise de Vigenère published a book on cryptology in

1585, & described the polyalphabetic substitution cipher

Increasing use, esp in diplomacy & war over centuries


Classical Cryptographic Techniques Two basic components of classical ciphers:

Substitution: letters are replaced by other letters Transposition: letters are arranged in a different order

These ciphers may be: Monoalphabetic: only one substitution/ transposition is

used, or Polyalphabetic:where several substitutions/ transpositions

are used Product cipher:

several ciphers concatenated together


Encryption and Decryption

Plaintextciphertext

Encipher C = E(K)(P) Decipher P = D(K)(C)

Key source


Key Management Using secret channel Encrypt the key Third trusted party The sender and the receiver generate key

The key must be same We will talk more about how we can generate keys for

two parties who are “unknown” of each other before, and want secure communication


Attacks Recover the message Recover the secret key

Thus also the message Thus the number of keys possible must

be large!


Possible AttacksCiphertext only

Algorithm, ciphertext

Known plaintext Algorithm, ciphertext, plaintext-ciphertext pair

Chosen plaintext Algorithm, ciphertext, chosen plaintext and its ciphertext

Chosen ciphertext Algorithm, ciphertext, chosen ciphertext and its plaintext

Chosen text Algorithm, ciphertext, chosen plaintext and ciphertext


Steganography Conceal the existence of message

Character marking Invisible ink Pin punctures Typewriter correction ribbon

Cryptography renders message unintelligible!


Contemporary Equiv. Least significant bits of picture frames

2048x3072 pixels with 24-bits RGB info Able to hide 2.3M message

Drawbacks Large overhead Virtually useless if system is known


Caesar CipherReplace each letter of message

by a letter a fixed distance away Reputedly used by Julius Caesar

Example: L FDPH L VDZ L FRQTXHUHG I CAME I SAW I CONGUEREDThe mapping is ABCDEFGHIJKLMNOPQRSTUVWXYZ

DEFGHIJKLMNOPQRSTUVWXYZABC


Mathematical Model Description Encryption E(k) : i i + k mod 26 Decryption D(k) : i i - k mod 26


Cryptanalysis: Caesar CipherKey space: 26

Exhaustive key search Example

GDUCUGQFRMPCNJYACJCRRCPQ HEVDVHRGSNQDOKZBDKDSSDQR

Plaintext: JGXFXJTIUPSFQMBDFMFUUFSTKHYGYKUJVGRNCEGNGVVGTU

Ciphertext: LIZHZLVKWRUHSODFHOHWWHUVMJAIAMWXSVITPEGIPIXXIVW


Character Frequencies In most languages letters are not

equally common in English e is by far the most common letter

Have tables of single, double & triple letter frequencies

Use these tables to compare with letter frequencies in ciphertext, a monoalphabetic substitution does not change relative

letter frequencies do need a moderate amount of ciphertext (100+ letters)


Letter Frequency Analysis Single Letter

A,B,C,D,E,….. Double Letter

TH,HE,IN,ER,RE,ON,AN,EN,…. Triple Letter

THE,AND,TIO,ATI,FOR,THA,TER,RES,…


Modular Arithmetic Cipher Use a more complex equation to

calculate the ciphertext letter for each plaintext letter

E(a,b) : i ai + b mod 26 Need gcd(a,26) = 1 Otherwise, not reversible So, a2, 13, 26 Caesar cipher: a=1


Cryptanalysis Key space:23*26

Brute force search Use letter frequency counts to guess a

couple of possible letter mappings frequency pattern not produced just by a shift use these mappings to solve 2 simultaneous equations

to derive above parameters


Playfair Cipher

s i/j m p le a b c df g h k no q r t uv w x y z

Key: simple

Used in WWI and WWII


Playfair Cipher Use filler letter to separate repeated

letters Encrypt two letters together

Same row– followed letters ac--bd

Same column– letters under qw--wi

Otherwise—square’s corner at same row ar--bq


Analysis Size of diagrams: 25!

But the actual different diagrams are not 25! Two diagrams are the same if they derive the same

encryption and decryption method Then what is the number of difference diagrams in

playfair cipher? 25!/25=24!

Difficult using frequency analysis But it still reveals the frequency information


Hill Cipher Encryption

Assign each letter an index C=KP mod 26 Matrix K is the key

Decryption P=K-1C mod 26 Thus, we can decrypt iff gcd(det(K), 26) =1.


How to Decrypt? Compute K-1

Compute det(K) Check if gcd(det(K), 26) =1 If not, then K-1 do not exist Else K-1 is

1 1

1 1

1 11 1

11

11

2

1

K K

K KK

nn

nn

nn n

, ,

, ,

d et( )


cont

K

k k k k

k k k kk k k k

k k k k

i j

j j n

i i j i j i n

i i i i

n n j n j n n

,

, , , ,

, , , ,

, , , ,

, , , ,

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1


Analysis Difficult to use frequency analysis But vulnerable to known-plaintext

attack Give simple method to attack hill cipher under the

known-plaintext assumption? How to attack under the chosen plaintext assumption?


Polyalphabetic Substitution Use more than one substitution

alphabet Makes cryptanalysis harder

since have more alphabets to guess and flattens frequency distribution

same plaintext letter gets replaced by several ciphertext letter, depending on which alphabet is used


Vigenère Cipher Basically multiple Caesar ciphers key is multiple letters long

K = k1 k2 ... kd ith letter specifies ith alphabet to use use each alphabet in turn, repeating from start after d

letters in message Plaintext THISPROCESSCANALSOBEEXPRESSED

Keyword CIPHERCIPHERCIPHERCIPHERCIPHE Ciphertext VPXZTIQKTZWTCVPSWFDMTETIGAHLH


Enigma Machine Enigma was a portable cipher machine

used to encrypt and decrypt secret messages. a family of related electro-mechanical rotor machines

German military

Japan commercial


Enigma MachineEnigma encryption for two consecutive letters — current is passed into set of rotors, around the reflector, and back out through the rotors again. Letter A encrypts differently with consecutive key presses, first to G, and then to C. This is because the right hand rotor has stepped, sending the signal on a completely different route.


Enigma the actual encipherment of a letter is

performed electrically. When a key is pressed, the circuit is completed; current flows

through the various components and ultimately lights one of many lamps, indicating the output letter.

Current flows from a battery through the switch controlled by the depressed key into a fixed entry wheel. This leads into the rotor assembly (or scrambler), where the complex internal wiring of each rotor results in the current passing from one rotor to the next along a convoluted path. After passing through all the rotors, current enters the reflector, which relays the signal back out again through the rotors and the entry wheel — this time via a different path — and, finally, to one of the lamps (the earliest Enigma models do not have the reflector).


Rotors performs a very simple type of

encryption a simple substitution cipher


World War II Era Encryption Devices A few here

Sigaba (United States) Typex (Britain) Lorenz cipher (Germany) Geheimfernschreiber (Germany)

For more, see http://w1tp.com/enigma/


One-time Pad theoretically unbreakable (Claude Shannon)

the plaintext is combined with a random "pad" the same length as the plaintext.

Patent by Gilbert Vernam (AT&T) and Joseph Mauborgne

Encryption C=PK

Decryption P=CK

Claude Shannon's work can be interpreted as that any information-theoretically secure cipher will be effectively equivalent

to the one-time pad algorithm. Hence one-time pads offer the best possible mathematical security of any encryption scheme, anywhere and anytime.


One-time pad--cont Drawbacks

it requires secure exchange of the one-time pad material, which must be as long as the message

pad disposed of correctly and never reused In practice

Generate a large number of random bits, Exchange the key material securely between the users before sending

a one-time enciphered message, Keep both copies of the key material for each message securely until

they are used, and Securely dispose of the key material after use, thereby ensuring the

key material is never reused.

It requires a perfect random numbers as key We will learn how to generate pseudo-random numbers


Random numbers needed If the key material is generated by a

deterministic program then it is not actually random should never be used in a one-time pad cipher. If so used, the method becomes a stream cipher; these

usually employ a short key that is used to generate a long pseudorandom stream, which is then combined with the message using some such mechanism as those used in one-time pads. Stream ciphers can be secure in practice, but they cannot be absolutely secure in the same provable sense as the one-time pad


Stream ciphers Stream ciphers

The most famous: Vernam cipher Invented by Vernam, ( AT&T, in 1917) Process the message bit by bit (as a stream) different from the one-time pad– some call same Simply add bits of message to random key bits

Examples A well-known stream cipher is RC4; others include: A5/1, A5/2, Chameleon, FISH, Helix. ISAAC, Panama, Pike, SEAL, SOBER,

SOBER-128 and WAKE. Usage

Stream ciphers are used in applications where plaintext comes in quantities of unknowable length - for example, a secure wireless connection


Simplest Stream Cipher

Plaintext

Key

Ciphertext Ciphertext

Key

Plaintext


Pros and Cons Drawbacks

Need as many key bits as message, difficult in practice (ie distribute on a mag-tape or CDROM)

Strength Is unconditionally secure provided key is truly random


Key Generation Why not to generate keystream from a

smaller (base) key?Use some pseudo-random function to do this Although this looks very attractive, it proves to be very

very difficult in practice to find a good pseudo-random function that is cryptographically strong

This is still an area of much research


Transposition Methods Permutation of plaintext Example

Write in a square in row, then read in column order specified by the key

Enhance: double or triple transposition Can reapply the encryption on ciphertext



Block CiphersXiang-Yang Li


Block Ciphers The message is broken into blocks,

Each of which is then encrypted (Like a substitution on very big characters - 64-bits or

more)


Substitution and Permutation In his 1949 paper Shannon also

introduced the idea of substitution-permutation (S-P) networks, which now form the basis of modern block ciphers An S-P network is the modern form of a substitution-

transposition product cipher S-P networks are based on the two primitive

cryptographic operations we have seen before


Substitution A binary word is replaced by some other

binary word The whole substitution function forms

the key If use n bit words,

The key space is 2n! Can also think of this as a large lookup

table, with n address lines (hence 2n addresses), each n bits wide being the output value

Will call them s-boxes


Cont.


Permutation A binary word has its bits reordered

(permuted) The re-ordering forms the key If use n bit words,

The key space is n! (Less secure than substitution) This is equivalent to a wire-crossing in

practice (Though is much harder to do in software)

Will call these p-boxes


Cont.


Substitution-permutation Network Shannon combined these two primitives He called these mixing

transformations A special form of product ciphers where

S-boxes Provide confusion of input bits

P-boxes Provide diffusion across s-box inputs


Confusion and Diffusion Confusion

A technique that seeks to make the relationship between the statistics of the ciphertext and the value of the encryption keys as complex as possible. Cipher uses key and plaintext.

Diffusion A technique that seeks to obscure the statistical

structure of the plaintext by spreading out the influence of each individual plaintext digit over many ciphertext digits.


Desired Effect Avalanche effect

A characteristic of an encryption algorithm in which a small change in the plaintext gives rise to a large change in the ciphertext

Best: changing one input bit results in changes of approx half the output bits

Completeness effect where each output bit is a complex function of all the

input bits


Practical Substitution-permutation Networks In practice we need to be able to

decrypt messages, as well as to encrypt them, hence either: Have to define inverses for each of our S & P-boxes,

but this doubles the code/hardware needed, or Define a structure that is easy to reverse, so can use

basically the same code or hardware for both encryption and decryption


Feistel Cipher Invented by Horst Feistel,

working at IBM Thomas J Watson research labs in early 70's,

The idea is to partition the input block into two halves, l(i-1) and r(i-1), use only r(i-1) in each round i (part) of the cipher

The function g incorporates one stage of the S-P network, controlled by part of the key k(i) known as the ith subkey


Cont.


Cont. This can be described functionally as:

L(i) = R(i-1) R(i) = L(i-1) g(k(i), R(i-1))

This can easily be reversed as seen in the above diagram, working backwards through the rounds

In practice link a number of these stages together (typically 16 rounds) to form the full cipher


Data Encryption Standard Adopted in 1977 by the National Bureau

of Standards, now the National Institute of Standards and Technology

Data are encrypted in 64-bit blocks using a 56-bit key

The same algorithm is used for decryption.

Subject to much controversy


History IBM LUCIFER 60’s

Uses 128 bits key Proposal for NBS, 1973 Adopted by NBS, 1977

Uses only 56 bits key Possible brute force attack

Design of S-boxes was classified Hidden weak points in in S-Boxes?

Wiener (93) claim to be able to build a machine at $100,00 and break DES in 1.5 days


DES DES encrypts 64-bit blocks of data,

using a 56-bit key the basic process consists of:

an initial permutation (IP) 16 rounds of a complex key dependent calculation f a final permutation, being the inverse of IP Function f can be described as L(i) = R(i-1) R(i) = L(i-1) P(S( E(R(i-1)) K(i) ))


DES


Initial and Final Permutations Inverse Permutations

40 8 48 16 56 24 64 3239 7 47 15 55 23 63 3138 6 46 14 54 22 62 3037 5 45 13 53 21 61 2936 4 44 12 52 20 60 2835 3 43 11 51 19 59 2734 2 42 10 50 18 58 2633 1 41 9 49 17 57 25


Function f


Expansion Table Expands the 32 bit data to 48 bits

Result(i)=input( array(i))

32 1 2 3 4 54 5 6 7 8 98 9 10 11 12 1312 13 14 15 16 1716 17 18 19 20 2120 21 22 23 24 2524 25 26 27 28 2928 29 30 31 32 1


S-Boxes S-Box is a fixed 4 by 16 array Given 6-bits B=b1b2b3b4b5b6,

Row r=b1b6

Column c=b2b3b4b5

S(B)=S(r,c) written in binary of length 4


Example S-Box S1

14

4 13

1 2 15

11

8 3 10

6 12

5 9 0 7

0 15

7 4 14

2 13

1 10

6 12

11

9 5 3 8

4 1 14

8 13

6 2 11

15

12

9 7 3 10

5 0

15

12

8 2 4 9 1 7 5 11

3 14

10

0 6 13


Permutation Table The permutation after each round

16 7 20 2129 12 28 171 15 23 265 18 31 102 8 24 1432 27 3 919 13 30 622 11 4 25


Subkey Generation Given a 64 bits key (with parity-check

bit) Discard the parity-check bits Permute the remaining bits using fixed table P1 Let C0D0 be the result (total 56 bits)

Let Ci =Shifti(Ci-1); Di =Shifti(Di-1) and Ki be another permutation P2 of CiDi (total 56 bits) Where cyclic shift one position left if i=1,2,9,16 Else cyclic shift two positions left


Permutation Tables

57

49

41

33

25

17

9

1 58

50

42

34

26

18

10

2 59

51

43

35

27

19

11

3 60

52

44

36

63

55

47

39

31

23

15

7 62

54

47

38

30

22

14

6 61

53

45

37

29

21

13

5 28

20

12

4

14

17

11

24

1 5

3 28

15

6 21

10

23

19

12

4 26

8

16

7 27

20

13

2

41

52

31

37

47

55

30

40

51

45

33

48

44

49

39

56

34

53

46

42

50

36

29

32

Permutation table P1 Permutation table P2


DES in Practice DEC (Digital Equipment Corp. 1992)

built a chip with 50k transistors Encrypt at the rate of 1G/second Clock rate 250 Mhz Cost about $300

Applications ATM transactions (encrypting PIN and so on)


Model Mode of use

The way we use a block cipher Four have been defined for the DES by ANSI in the

standard: ANSI X3.106-1983 modes of use) Block modes

Splits messages in blocks (ECB, CBC) Stream modes

On bit stream messages (CFB, OFB)


Block Modes Electronic Codebook Book (ECB)

where the message is broken into independent 64-bit blocks which are encrypted

Ci = DESK1 (Pi) Cipher Block Chaining (CBC)

again the message is broken into 64-bit blocks, but they are linked together in the encryption operation with an IV

Ci = DESK1 (PiCi-1) C-1=IV (initial value)


Stream Model Cipher FeedBack (CFB)

where the message is treated as a stream of bits, added to the output of the DES, with the result being feed back for the next stage

Ci = Pi DESK1 (Ci-1) C-1=IV (initial value)


Cont. Output FeedBack (OFB)

where the message is treated as a stream of bits, added to the message, but with the feedback being independent of the message

Ci = Pi Oi Oi = DESK1 (Oi-1) O-1=IV (initial value)


DES Weak Keys With many block ciphers there are some

keys that should be avoided, because of reduced cipher complexity

These keys are such that the same sub-key is generated in more than one round, and they include:


Cont. Weak keys

The same sub-key is generated for every round DES has 4 weak keys

Semi-weak keys Only two sub-keys are generated on alternate rounds DES has 12 of these (in 6 pairs)

Demi-semi weak keys Have four sub-keys generated


Cont. None of these causes a problem since

they are a tiny fraction of all available keys

However they MUST be avoided by any key generation program


Possible Techniques for Improving DES Multiple enciphering with DES Extending DES to 128-bit data paths

and 112-bit keys Extending the key expansion calculation


Double DES? Using two encryption stages and two

keys C=Ek2(Ek1(P)) P=Dk1(Dk2(C))

It is proved that there is no key k3 such that C=Ek2(Ek1(P))=Ek3(P)

But Meet-in-the-middle attack


Meet-in-the-Middle Attack Assume C=Ek2(Ek1(P)) Given the plaintext P and ciphertext C Encrypt P using all possible keys k1

Decrypt C using all possible keys k2 Check the result with the encrypted plaintext lists If found match, they test the found keys again for

another plaintext and ciphertext pair If it turns correct, then find the keys Otherwise keep decrypting C


Triple DES DES variant Standardized in ANSI X9.17 & ISO 8732

and in PEM for key management Proposed for general EFT standard by

ANSI X9 Backwards compatible with many DES

schemes Uses 2 or 3 keys


Cont. No known practical attacks Brute force search impossible (very

hard) Meet-in-the-middle attacks need 256

Plaintext-Ciphertext pairs per key Popular current alternative


IDEA: Developed by James Massey & Xuejia

Lai at ETH originally in Zurich in 1990, then called IPES: X Lai, J L Massey, "A Proposal for a New Block

Encryption Standard" in Advances in Cryptology - Eurocrypt '90, Lecture

Notes in Computer Science, vol 473, pp 389-404, X Lai, J L Massey, S Murphy, "Markov Ciphers and

Differential Cryptanalysis" in Advances in Cryptology - Eurocrypt '91, Lecture

Notes in Computer Science, vol 547, pp 17-38, name changed to IDEA in 1992


Basic Features Encrypts 64-bit blocks using a 128-bit key Based on mixing operations from different

(incompatible) algebraic groups XOR, + mod 2^(16) , X mod 2^(16) +1) On 16-bit sub-blocks, with no permutations used

IDEA is patented in Europe & US, however non-commercial use is freely permitted used in the public domain PGP (with agreement) currently no attack against IDEA is known

Seem secure against differential cryptanalysis, brute force


Operations Operations

XOR, Addition mod 216, multiplication mod 216 +1 Why these special mod for addition, multiplication

They do not satisfy the distributive law They do not satisfy the associative law


MA: multiplication/addition Multiplication/addition

Basic block to provide diffusion Input of MA

Two sub-blocks derived from 4 input sub-blocks, 4 sub-keys

Two other sub-keys Output

Two sub-blocks Needs four operations

Four operations are the minimum to provide full diffusion


Overview


Cont. IDEA encryption works as follows:

Use 8-rounds The 64-bit data is divided into: X1 , X2 , X3 , X4

Each round The sub-blocks are added (2,3), multiplied (1,4) with

sub-keys The results are XORed [1,3] and [2,4] to 2 sub-blocks The XOR results set as input of MA structure,

It outputs two subblocksResults are then XORed with 2,4 and 1,3 subblocks respectively

The second and third sub-blocks are swapped Finally new sub-keys are combined with the sub-blocks


Sub-Keys Total need 52=6*8+4 sub-keys

First are directly from key in order Left shift of 25 bits, and then next 8 sub-keys Each sub-key is a sub-block of the original key

Decryption Much more complicated It needs the inverse of the encryption key

For addition, multiplication


Decryption The process of decryption is essentially

the same as encryption But with different selection of sub-keys Basic Operations

K1.1^(-1 ) is the multiplicative inverse mod 2^(16) +1

-K1.2 is the additive inverse mod 2^(16) The original operations are:

(+) bit-by-bit XOR + additional mod 2^(16) of 16-bit integers * multiplication mod 2^(16) +1 (where 0 means 2^(16) )


Decryption Sub-Keys

Round Encryption Keys Decryption Keys 1 K1.1 K1.2 K1.3 K1.4 K1.5 K1.6 K9.1-1 -K9.2 -K9.3 K9.4-1

K8.5 K8.6 2 K2.1 K2.2 K2.3 K2.4 K2.5 K2.6 K8.1-1 -K8.3 -K8.2 K8.4-1

K7.5 K7.6 3 K3.1 K3.2 K3.3 K3.4 K3.5 K3.6 K7.1-1 -K7.3 -K7.2 K7.4-1

K6.5 K6.6 4 K4.1 K4.2 K4.3 K4.4 K4.5 K4.6 K6.1-1 -K6.3 -K6.2 K6.4-1

K5.5 K5.6 5 K5.1 K5.2 K5.3 K5.4 K5.5 K5.6 K5.1-1 -K5.3 -K5.2 K5.4-1

K4.5 K4.6 6 K6.1 K6.2 K6.3 K6.4 K6.5 K6.6 K4.1-1 -K4.3 -K4.2 K4.4-1

K3.5 K3.6 7 K7.1 K7.2 K7.3 K7.4 K7.5 K7.6 K3.1-1 -K3.3 -K3.2 K3.4-1

K2.5 K2.6 8 K8.1 K8.2 K8.3 K8.4 K8.5 K8.6 K2.1-1 -K2.3 -K2.2 K2.4-1

K1.5 K1.6 Output K9.1 K9.2 K9.3 K9.4 K1.1-1 -K1.2 -K1.3 K1.4-1


Important Feature The size of the sub-block

Need 216+1 be prime number To compute the inverse for each possible subkey

So sub-block size 8 is also possible 28+1=257 is prime number


CAST-128 By Carlisle Adams, Stafford Tavares

Defined in RFC 2144 Use key size varying from 40 to 128 bits Structure of Feistel network 16 rounds on 64-bits data block Four primitive operations

Addition, substration (mod 232) Bitwise exclusive-OR Left-circular rotation


Skipjack and Clipper Skipjack

used in Clipper escrowed encryption scheme(US govt) Skipjack is a block cipher, 64-bit data hardware only implementation 80-bit key (escrowed in 2 halves) 32 round all design details and descriptions are classified has been very considerable debate over its use attack by Matt Blaze (ATT) on the LEAF component of

the Clipper protocol for secure phone communications


Blowfish Scheme Developed by Bruce Schneier

Fast, compact, simple and variably secure Two basic operations: addition, XOR Key ranges from 32 bits to 448 bits Similar to Feistel scheme The sub-key and s-boxes are complicated So not suitable when key changes often Function g is very simple, unlike DES


RC5 Developed by R. Rivest

Suitable for hardware or software Fast, simple, low memory, data-dependent rotations Adaptable to processors of different word length

A family of algorithms determined by word length, number of rounds, size of secret key

Decryption and encryption are not the same With little variations

Primitive operations Addition, XOR, left circular rotation


Characteristics Key features of advanced sym block

cipher Variable key length Mixed operators Data dependent rotation Key dependent rotation Key dependent S-boxes Lengthy key schedule algorithm Variable function F Variable of number of rounds Operation on both halved data each round


AES Advanced Encryption Standard (Rijndael)

key size and the block size may be chosen to be any of 128, 192, or 256 bits (later only key, block fixed 128)

Rijndael has a variable number of rounds. Not counting an extra round performed at the end of encipherment with one step omitted, the number of rounds in Rijndael is: 9 if both the block and the key are 128 bits long. 11 if either the block or the key is 192 bits long, and

neither of them is longer than that. 13 if either the block or the key is 256 bits long.

Three big blocks first perform an Add Round Key step (XORing a subkey with

the block) by itself, then regular rounds noted above, the final round with the Mix Column step


AES Each regular round

involves four steps. Byte Sub step,

each byte of the block is replaced by its substitute (inverse) in an S-box.

Shift Row step cyclically shifts the bytes in

each row by a certain offset

Mix Column step Add Round Key: XORs the

subkey for the current round


S-Box S-box calculation:

the first step was to replace each byte with its reciprocal in the same GF(2^8), except that 0, which has no reciprocal, is replaced by itself,

then a bitwise modulo-2 matrix multiply was used, and finally the hexadecimal number 63 is XORed with the

result.


Shift Step Considering the block to be made up of

bytes 1 to 16, these bytes are arranged in a rectangle, and shifted as follows:

from to 1 5 9 13 1 5 9 13 2 6 10 14 6 10 14 2 3 7 11 15 11 15 3 7 4 8 12 16 16 4 8 12 Blocks that are 192 and 256 bits long

are shifted also follow a rule.


Structure view



Public key systemXiang-Yang Li


Public Key Encryption Two difficult problems

Key distribution under conventional encryption Digital signature

Diffie and Hellman, 1976 Astonishing breakthrough One key for encryption and the other related key for

decryption It is computationally infeasible to determine the

decryption key using only the encryption key and the algorithm


Public Key Cryptosystem Essential steps of public key

cryptosystem Each end generates a pair of keys

One for encryption and one for decryption Each system publishes one key, called public key, and

the companion key is kept secret It A wants to send message to B

Encrypt it using B’s public key When B receives the encrypted message

It decrypt it using its own private key


Applications of PKC Encryption/Decryption

The sender encrypts the message using the receiver’s public key Q: Why not use the sender’s secret key?

Digital signature The sender signs a message by encrypt the message or

a transformation of the message using its own private key

Key exchange Two sides cooperate to exchange a session key,

typically for conventional encryption


Conditions of PKC Computationally easy

To generate public and private key pair To encrypt the message using encryption key To decrypt the message using decryption key

Computational infeasible To compute the private key using public key To recover the plaintext using ciphertext and public key

The encryption and decryption can be applied in either order


One Way Function PKC boils down to one way function

Maps a domain into a range with unique inverse The calculation of the function is easy The calculation of the inverse is infeasible

Easy The problem can be solved in polynomial time

Infeasible The effort to solve it grows faster than polynomial time For example: 2n

It requires infeasible for all inputs, not just worst case


Trapdoor One-way Function Trapdoor one way function

Maps a domain into a range with unique inverse Y=fk(X)

The calculation of the function is easy The calculation of the inverse is infeasible if the key is

not known The calculation of the inverse is easy if the key is

known


Possible Attacks Brute force

Use large keys Trade-off: speed (not linearly depend on key size) Confined to small data encryption: signature, key

management Compute the private key from public key

Not proven that is not feasible for most protocols! Probable message attack

Encrypt all possible messages using encryption key Compare with the ciphertext to find the matched one! If data is small, feasible, regardless of key size of PKC


History http://www.research.att.com/~smb/

nsam-160/ British National Security Action

Memorandum 160 Kennedy Nuclear Weapon http://www.research.att.com/~smb/nsam-160/pg1.html


RSA Algorithm R. Rivest, A. Shamir, L. Adleman (1977)

James Ellis came up with the idea in 1970, and proved that it was theoretically possible. In 1973, Clifford Cocks a British mathematician invented a variant on RSA; a few months later, Malcom Williamson invented a Diffie-Hellman analog

Only revealed till 1997 Patent expired on September 20, 2000. Block cipher using integers 0~n-1

Thus block size k is less than log2n Algorithm:

Encryption: C=Me mod n Decryption: M=Cd mod n

Both sender and the receiver know n


Requirements Possible to find e and d such that

M=Mde mod n for all message M Easy to conduct encryption and

decryption Infeasible to compute d

Given n and e


RSA Example1. Select primes: p=17 & q=112. Compute n = pq =17×11=1873. Compute ø(n)=(p–1)(q-1)=16×10=1604. Select e : gcd(e,160)=1; choose e=75. Determine d: de=1 mod 160 and d < 160

Value is d=23 since 23×7=161= 10×160+16. Publish public key KU={7,187}7. Keep secret private key KR={23,17,11}


RSA Example cont sample RSA encryption/decryption is: given message M = 88 (nb. 88<187) encryption:

C = 887 mod 187 = 11 decryption:

M = 1123 mod 187 = 88


Key Generation Recall Euler Theorem

a(n)+1 =a mod n for all a Then ed=1 mod (n) is sufficient to make algorithm

correct RSA chooses the following

Integer n=pq for two primes p and q Select e, such that gcd(e, (n))=1 Compute the inverse of e mod (n)

The result is set as d


Key Generation The prime numbers p and q must be

sufficiently large They are chosen by applying primality testing of

randomly chosen large numbers About n/ln n prime numbers less than n

Implies needs to check about 2ln n random numbers to find 2 primes numbers around n

Compute n=pq, keep p and q secret! Select random number e

Test gcd(e, (n))=1, and get d if equation holds


Exponentiation can use the Square and Multiply Algorithm a fast, efficient algorithm for exponentiation concept is based on repeatedly squaring base and multiplying in the ones that are needed to

compute the result look at binary representation of exponent only takes O(log2 n) multiples for number n

eg. 75 = 74.71 = 3.7 = 10 mod 11 eg. 3129 = 3128.31 = 5.3 = 4 mod 11


Exponentiation


More on Exponention (PGP) To compute md mod n, we compute md mod p and md mod q Remember that the receiver could keep

p,q Then Chinese Remainder Theorem to

find md mod n


Security of RSA Brute force: try all possible private keys Factoring integer n, then know (n)

Not proven to be NPC Determine (n) directly without factoring

Equivalent to factoring! (1996) Determine d directly without knowing (n) Currently appears as hard as factoring

But not proven, so it may be easier!


Practical Considerations Testing p, q using probability first, then deterministic methods A good random number generator is needed for p,q

'random' and 'unpredictable' Primes p and q should be in similar scale Both p-1 and q-1 should have large prime factor The gcd(p-1,q-1) should be small The encryption key e = 2 should not be used The decryption key d should larger then n1/4

RSA is much slower than symmetric cryptosystems. In practice, typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short)

symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice.


Fixed point of RSA How many m such that

me=m mod n assume that gcd(m,n)=1 It is same as me-1=1 mod n Thus, me-1=1 mod p and me-1=1 mod q Solutions gcd(e-1,p-1)*gcd(e-1,q-1)


Cyclic Attack Compute me mod n, me2 mod n, me3 mod

n…till it reaches some message readable.

Need period large Let r be the largest prime of p-1, l be

the largest prime of r-1 Then period is at least l with high

probability.


How to deal with p, q Delete them securely Or used for speed-up calculation from

CRT Compute Me mod p and Me mod q Then find using Me mod n CRT


Timing Attacks Keep track of how long a computer

takes to decrypt a message! Paul Kocher, 1995, Dec-7 Stunning attack strategy and cipher only attack! Guessing the key bit by bit

Countermeasures (Rivest 11 Dec 1995) Constant exponentiation time Random delay Blinding (add a random number for encryption and

decryption)


Chosen Ciphertext Collect ciphertext c (send to Alice), want to

find m=cd mod n Attacker chooses random r Compute x= re mod n; y=xc mod n; and t=

r-1 mod n Attacker gets Alice to sign y with private

key using RSA: yd mod n Alice sends u= yd mod n to Attacker Attacker then computes tu mod nm


Other attacks on RSA Comprised decryption key

If the private key d (for decryption of received ciphertext) of a user is comprised, then the user has to reselect n and e and d

It cannot use the old number n! Otherwise attacker already can factor n almost surely!

The number n can only be used by one person If two user uses the same n, even they do not know the

factoring of n, they still could figure out the factoring of n with probability almost one.


Bit security of RSA Given ciphertext C,

We may want to find the last bit of M, denoted by parity(C)

We may want to find if M>n/2, denoted by half(C) We may want to find all bits of M

The above three attacks are the same! If we can solve one, we can solve the other two!


Other Public Key Systems Rabin Cryptosystem

Decryption is not unique Elgamal Cryptosystem

Expansion of the plaintext (double) Knapsack System

Already broken Elliptic Curve System

If directly implement Elgamal on elliptic curve Expansion of plaintext by 4; Restricted plaintext

Menezes-Vanston system is more efficient


Rabin Cryptosystem Procedure

Let n=pq and p=3 mod 4, q=3 mod 4 Publish n, and a number b<n For message m

C=m(m+b) mod n The receiver decrypts ciphertext C

(b2/4+C)1/2-b/2


Analysis For receiver, need solve equation

x2+xb=C mod n Let x1=x+b/2, c=b2/4+C, then need

Solve x12 =c mod n

Chinese Remainder Theorem implies that x1

2 =c mod p x1

2 =c mod q When p=3 and q=3 mod 4

Solution x1=c(p+1)/4 mod p and x1=c(q+1)/4 mod q Then Chinese Remainder Theorem again to

combine solution


Security Secure against

Chosen plaintext attack Not secure against

Chosen ciphertext attack


ElGamal Cryptosystem Based on Discrete Logarithm

Find unique integer a such that gx=y mod p Here x is a primitive element in Zp, p is prime

Procedure Make p, g, y public, keep x secret Encryption:

Ek(m)=(gk mod p, m y k mod p) Decryption

Dk(y1,y2)=y2(y1x)-1 mod p


Security of ElGamal ElGamal is a simple example of a semantically

secure asymmetric key encryption algorithm (under reasonable assumptions).

ElGamal's security rests, in part, on the difficulty of solving the discrete logarithm problem in G. Specifically, if the discrete logarithm problem could be solved

efficiently, then ElGamal would be broken. However, the security of ElGamal actually relies on the so-called Decisional Diffie-Hellman (DDH) assumption. This assumption is often stronger than the discrete log assumption, but is still believed to be true for many classes of groups.


Bit security of Discrete Log Given gx=y mod p

We may want to find the value of x Find some bits of x

Assume that p-1 = 2st We can find the last s bits of x for sure But to find the other bits of x is same as to find all bits

of x! Example, the last bit of x is

0 y is QR iff y(p-1)/2=1 mod p 1 y is NQR iff y(p-1)/2=-1 mod p


DH Assumption Consider a cyclic group G of order q. The DDH

assumption states that, given (g,ga,gb) for a randomly-chosen generator g and random ,

the value gab "looks like" a perfectly random element of G. This intuitive notion is formally stated by

saying that the following two ensembles are computationally indistinguishable: (g,ga,gb,gab), where g,a,b are chosen at random as

described above (this input is called a "DDH tuple"); (g,ga,gb,gc), where g,a,b are chosen at random and c

is chosen at random. Diffie-Hellman problem

computing gab from (g,ga,gb)


Knapsack Cryptosystem Based on subset sum problem

Given a set, find a subset with half summation value It is NPC problem generally

Superincreasing set if si>j<isj The subset problem over

superincreasing set can be solved in polynomial time!

Been broken by Shamir, 1984 Using integer programming tech by Lenstra


Solve Subset Problem Let T be the half summation, t=T; For i=n downto 1 do

If tsi then t=t-si

Set xi=1 Else xi=0

If xisi=T then (x1, x2,… xn) is the solution Else, there is no solution


Knapsack System Procedure

Select a superincreasing set s Let p be prime larger than set summation of s, Select integer a, keep s, a, p secret Make t=(as1, as2,…asn) mod p public Encryption

E(x1,x2,…xn)=xiti

Decryption Solve the subset summation problem (s, a-1C mod

p)


Elliptic Curve Cryptography majority of public-key crypto (RSA, D-H)

use either integer or polynomial arithmetic with very large numbers/polynomials

imposes a significant load in storing and processing keys and messages

an alternative is to use elliptic curves offers same security with smaller bit

sizes


Real Elliptic Curves an elliptic curve is defined by an equation

in two variables x & y, with coefficients consider a cubic elliptic curve of form

y2 = x3 + ax + b where x,y,a,b are all real numbers also define zero point O

have addition operation for elliptic curve geometrically sum of Q+R is reflection of intersection R


Real Elliptic Curve Example


Finite Elliptic Curves Elliptic curve cryptography uses curves

whose variables & coefficients are finite have two families commonly used:

prime curves Ep(a,b) defined over Zp use integers modulo a prime p best in software

binary curves E2m(a,b) defined over GF(2n) use polynomials with binary coefficients best in hardware


Elliptic Curve Cryptography ECC addition is analog of modulo multiply ECC repeated addition is analog of modulo

exponentiation need “hard” problem equiv to discrete log

Q=kP, where Q,P belong to a prime curve is “easy” to compute Q given k,P but “hard” to find k given Q,P known as the elliptic curve logarithm problem

Certicom example: E23(9,17)


ECC Diffie-Hellman can do key exchange analogous to D-H users select a suitable curve Ep(a,b) select base point G=(x1,y1) with large

order n s.t. n*G=O A & B select private keys nA<n, nB<n compute public keys: PA=nA×G, PB=nB×G compute shared key: K=nA×PB, K=nB×PA

same since K=nA×nB×G


ECC Encryption/Decryption several alternatives, will consider simplest must first encode any message M as a

point on the elliptic curve Pm select suitable curve & point G as in D-H each user chooses private key nA<n and computes public key PA=nA×G to encrypt Pm : Cm={kG, Pm+k PA}, k

random decrypt Cm compute:

Pm+kPA–nA(kG) = Pm+k(nAG)–nA(kG) = Pm


ECC Security relies on elliptic curve logarithm

problem fastest method is “Pollard rho method” compared to factoring, can use much

smaller key sizes than with RSA etc for equivalent key lengths computations

are roughly equivalent hence for similar security ECC offers

significant computational advantages


Cryptography and Network

Key Management and generation

Xiang-Yang Li


Key Exchange Public key systems are much slower

than private key system Public key system is then often for short data

Signature, key distribution Key distribution

One party chooses the key and transmits it to other user Key agreement

Protocol such two parties jointly establish secret key over public communication channel

Key is the function of inputs of two users


Distribution of Public Keys can be considered as using one of:

Public announcement Publicly available directory Public-key authority Public-key certificates


Public Key Management Simple one: publish the public key

Such as newsgroups, yellow-book, etc. But it is not secure, although it is convenient

Anyone can forge such a announcement Ex: user B pretends to be A, and publish a key for

A Then all messages sent to A, readable by B!

Let trusted authority maintain the keys Need to verify the identity, when register keys User can replace old keys, or void old keys


Possible Attacks Observe all messages over the channel

So assume that all plaintext messages are available to all

Save messages for reuse later So have to avoid replay attack

Masquerade various users in the network So have to be able to verify the source of the message


Public Announcement users distribute public keys to recipients

or broadcast to community at large eg. append PGP keys to email messages or post to news

groups or email list major weakness is forgery

anyone can create a key claiming to be someone else and broadcast it

until forgery is discovered can masquerade as claimed user


Publicly Available Directory can obtain greater security by

registering keys with a public directory directory must be trusted with

properties: contains {name,public-key} entries participants register securely with directory participants can replace key at any time directory is periodically published directory can be accessed electronically

still vulnerable to tampering or forgery


Public-Key Authority improve security by tightening control

over distribution of keys from directory has properties of directory and requires users to know public key

for the directory then users interact with directory to

obtain any desired public key securely does require real-time access to directory when keys

are needed


Public-Key Authority


Cont. More advanced distribution

A sends request-for-key(B) to authority with time-stamp, that is, Ida|Idb|Time

Authority replies with key(B) (encrypted by its private key), that is EKTta(KUb| Ida|Idb|Time)

A initiates a message to B, including a random number Na, its IDA

B then ask authority to get key(A) B sends A (encrypted by A’s public key) Na and Nb

A then replies B Nb encrypted by B’s public key


Cont. In above scheme, the authority is

bottleneck New approach: certificate

Any user can read certificate, determine name and public key of the certificate’s owner

Any user can verify the authority of certificate Only the authority can create and update certificate Any user can verify the time-stamp of certificate

The certificate is CA=EKRauth[T,IDA, KUA] Time-stamp is to avoid reuse of voided key


Public-Key Certificates certificates allow key exchange without

real-time access to public-key authority a certificate binds identity to public

key usually with other info such as period of validity, rights

of use etc with all contents signed by a trusted

Public-Key or Certificate Authority (CA) can be verified by anyone who knows

the public-key authorities public-key


Public-Key Certificates


Public-Key Distribution of Secret Keys use previous methods to obtain public-

key can use for secrecy or authentication but public-key algorithms are slow so usually want to use private-key

encryption to protect message contents hence need a session key have several alternatives for

negotiating a suitable session


Simple Secret Key Distribution proposed by Merkle in 1979

A generates a new temporary public key pair A sends B the public key and their identity B generates a session key K sends it to A encrypted

using the supplied public key A decrypts the session key and both use

problem is that an opponent can intercept and impersonate both halves of protocol


Secret key Distribution Simple secret key distribution

A generates KUA and KRA, sends KUA to B B generates a secret key ks

B sends ks to A using A’s public key KUA

A decrypts the message to get the secret key ks

To get more security, the public/private keys can be regenerated when needed

But vulnerable to the active attack! Attacker E can compromise the communication

between A and B as follows


Cont. Attacking

A generates KUA and KRA, sends IDA, KUA to B E intercepts the message, transmits IDA, KUE to B B generates a secret key ks

B sends ks to A using A’s “public key” KUE

E intercepts the message, decrypt it and get ks

E sends A the message Ks, encrypted by KUA

A decrypts the message to get the secret key ks

Now E knows Ks, but A, B are unaware of it


Secret Key Distribution So need confidentiality and

authentication A and B need to use a secure method to exchange their

public keys Schemes

A initiates a message to B, EKUB(Na,IDa) B replies it with EKUA(Na,Nb) A then replies it with EKUB(Nb) A sends B the message EKUB (EKRA(Ks))

Security The first 3 steps are used to assure that A is A, B is B


Public-Key Distribution of Secret Keys if have securely exchanged public-keys:


Key Predistribution Trusted Authority (TA) generates keys

for all pair of users and transmits to them Large overhead (for TA and user)

Blom Scheme Keys are chosen from a finite field Zp

P is public prime number TA transmits k+1 elements of Zp to each user over

secure channel Secure condition: any set of at most k users (not U,V)

can not determine any information about Ku,v


Blom Scheme Scheme (when k=1)

Each user u has distinct element ru from Zp

TA choose a,b,c and defines f(x,y)=a+b(x+y)+cxy mod p

For each u, TA computes gu(x)=f(x, ru) mod p

TA transmits gu(x) to user u Two users u and v compute the common key

f(ru, rv)= a+b(ru + rv)+c ru rv mod p Here f(ru, rv)= gv(ru)= gu(rv)


Security of Blom Scheme Less than k users can not determine

keys However, more than k users can

compute any keys Solving equations to get a,b,c for k=1

Generally Function f(x,y)=ai,jxiyi mod p Here ai,j=aj,i


Diffie-Hellman Key Predist. Computationally secure

if discrete logarithm is intractable Scheme

Assume prime number p public and an integer c public Each user u has secret component au

User u computes bu=c au mod p

TA certifies it by computing (ID(u), bu, sigTA(ID(u), bu))

The common key of two users u and v is K=c

au av mod p


Diffie Hellman Around September 1974, Diffie (Graduate

student) had been traveling USA with his wife, Mary, discussing cryptography with anyone who was available. At the time, there was very little published material

about modern methods and much was classified. Very few people were interested in the topic and Marty Hellman even says that many of his colleagues felt that it was "born classified," like secrets about the atomic bomb, because it was so important to national security.

John Gill gave the idea of exponential


Diffie-Hellman Problem Diffie-Hellman problem definition

Given bu=gau mod p, bv=gav mod p, how to compute gavau mod p? Here g is a primitive element of mod p

The problem is not harder than the discrete log-arithmetic problem, because the later one can always be used to solve it

It can be proved that it has the same difficulty as the ElGamal encryption system


Diffie-Hellman Key Exchange Computationally secure

if discrete logarithm is intractable Scheme

Assume prime number p public and an integer c public Each user u chooses a secret component au (new!) User u computes bu=c

au mod p User v computes bv=c

av mod p The common key of two users u and v is

K=c au av mod p


Middle Attack Intruder w intercept the

communications Intruder w communications with u Intruder w communications with v The key computed by u is

K=c au av’ mod p

u w vc

au c au’

c av’ c

av


Authenticated Key Agreement Introducing the identification scheme

before key exchange does not help The attacker remains inactive until identification done

Simplified station to station protocol Key agreement protocol itself authenticates the user’s

identity at the same time the key being defined


Station-to-station ProtocolScheme

Each user has a certificate C(v)=(Idv,verv,sigTA(Idv,verv))

User u selects au and computes bu=c au mod p

User v selects av and computes Value bv=c

av mod p Key K=c

au av mod p Signature yv=sigv(bu,bv)

User v sends (C(V), bv, yv) to U User u computes K=c

au av mod p, verifies yv, and C(V) User u computes yu=sigu(bu,bv), sends (C(u),yu) to V User v verifies yu, and C(u)


MTI Agreement Protocol Scheme

Assume prime number p public and an integer c public Each user has certificate c(u)=(Idu,bu, sigTA(Idu,bu))

Here bu= c au mod p

Each user u chooses a secret component ru (new!) User u computes su=c

ru mod p, sends (c(u),su) User v computes sv=c

rv mod p, sends (c(v),sv) The common key of two users u and v is

K=c rvau+ ru av mod p= sv

aubv ru mod p= su avbu rv mod p



AuthenticationXiang-Yang Li


Message Authentication Digital Signature Authentication

Authentication requirements Authentication functions

Mechanisms MAC: message authentication code Hash functions, security in hash functions Hash and MAC algorithms

MD5, SHA, RIPEMD-160, HMAC Digital signatures


Message Attacks Possible attacks

Disclosure Traffic analysis Masquerade Content modification Sequence modification Time modification Repudiation

Denial of the receipt of message by the destination or

Denial of the transmitting by the source


Authentication Enables receiver to verify message

authenticity Using some lower level functions as primitive

Three types of functions Message encryption Message authentication code (MAC) Hash function


AuthenticationGoal: Bob wants Alice to “prove” her

identity to himProtocol ap1.0: Alice says “I am Alice”

Failure scenario??“I am Alice”


AuthenticationGoal: Bob wants Alice to “prove” her

identity to himProtocol ap1.0: Alice says “I am Alice”

in a network,Bob can not “see”

Alice, so Trudy simply declares

herself to be Alice“I am Alice”


Authentication: another tryProtocol ap2.0: Alice says “I am Alice” in an IP packet

containing her source IP address

Failure scenario??“I am Alice”Alice’s

IP address


Authentication: another tryProtocol ap2.0: Alice says “I am Alice” in an IP packet

containing her source IP address

Trudy can createa packet

“spoofing”Alice’s address“I am Alice”Alice’s

IP address


Authentication: another tryProtocol ap3.0: Alice says “I am Alice” and sends her

secret password to “prove” it.

Failure scenario??

“I’m Alice”Alice’s IP addr

Alice’s password

OKAlice’s IP addr



secret password to “prove” it.

playback attack: Trudy records Alice’s

packetand later

plays it back to Bob


Alice’s password

OKAlice’s IP addr


Alice’s password


Authentication: yet another tryProtocol ap3.1: Alice says “I am Alice” and sends her

encrypted secret password to “prove” it.

Failure scenario??


encrypted password

OKAlice’s IP addr



encrypted secret password to “prove” it.

recordand

playbackstill works!


encryptedpassword

OKAlice’s IP addr


encryptedpassword


Authentication: yet another tryGoal: avoid playback attack

Failures, drawbacks?

Nonce: number (R) used only once –in-a-lifetimeap4.0: to prove Alice “live”, Bob sends Alice nonce, R.

Alicemust return R, encrypted with shared secret key

“I am Alice”

RK (R)A-B

Alice is live, and only Alice knows key to encrypt

nonce, so it must be Alice!


Authentication: ap5.0ap4.0 requires shared symmetric key can we authenticate using public key techniques?ap5.0: use nonce, public key cryptography

“I am Alice”R

Bob computes

K (R)A-

“send me your public key”K A

+

(K (R)) = RA-K A

+

and knows only Alice could have the

private key, that encrypted R such that

(K (R)) = RA-K A

+


ap5.0: security holeMan (woman) in the middle attack: Trudy poses

as Alice (to Bob) and as Bob (to Alice)

I am Alice I am AliceR

TK (R)-

Send me your public key

TK +AK (R)-

Send me your public key

AK +

TK (m)+

Tm = K (K (m))+T

-Trudy gets

sends m to Alice encrypted

with Alice’s public key

AK (m)+

Am = K (K (m))+A

-

R


ap5.0: security holeMan (woman) in the middle attack: Trudy poses

as Alice (to Bob) and as Bob (to Alice)

Difficult to detect: Bob receives everything that Alice sends, and vice versa. (e.g., so Bob, Alice can meet one week later and recall conversation) problem is that Trudy receives all messages as well!


Message Encryption Conventional Encryption

Authentication provided due to the secret key But the message need to be meaningful

What happened it message is not readable? How to determine intelligible automatically?

Approach Checksum or frame check sequence(FCS) to message Encrypt the message and the appending FCS Receiver decrypt the ciphertext Computes FCS of message, compare with received one


Public Key Encryption Direct encryption by receiver’s public

key Only confidentiality, no authentication

For authentication Encrypt using sender’s private key Assume the message is intelligible No confidentiality: everyone can decrypt

Confidentiality and authentication Encrypt by sender’s, then receiver’s public key But too time-consuming: 4 rounds RSA on large data


Message Authentication Code Assume both uses share secret key k Procedure

Sender computes MAC=Ck(M) for M Sent M and MAC of it to receiver Receiver computes the MAC on received M Compare it with received MAC If match, then accepts the message

MAC is similar to encryption, but not need be reversible!


MAC with Confidentiality Two options

Using another key to encrypt M and MAC Using another key to encrypt M only

Requirements of MAC Size of MAC: n Size of key: k Need 2n computations of MAC and n/k pairs of Mi and

MACi


Why not Conventional Encrypt Possible situations

Broadcast a message (one destination can verify) Authentication is done selectively Authentication of computer program Authentication may be important than secrecy Architecture flexibility Authentication lasts longer than secret protection


MAC Requirements Computationally infeasible to construct

M’ such that Ck(M’)=Ck(M) Ck(M) uniformly distributed


Data Authentication Algorithm ANSI standard X9.17 Based on DES Using Cipher Block Chaining mode

Data is grouped into 64 bits blocks Padding 0’s if necessary

Outputi=Ek(DiOutputi-1) 0<i, and Output0=0’s

The data authentication code DAC consists of the leftmost m bits of the last output, m16


Authentication Protocols Central issues

Confidentiality: prevent masqueraded and compromised

Timeliness: prevent replay attacks Simple replay, repetition within timestamp, replay

arrives but not the true messages,backward replay attack to the sender

Mutual authentication One-way authentication


Coping with Replay Time stamps

Party A accepts a message only if has valid timestamp within a valid time

Need synchronized clock How to set the synchronized clock?

Network delay consideration? Challenge/response

Party A, (receiver), sends B a nonce (challenge) and requires the subsequent message contains it


Challenge-Response To ensure a password is never sent in

the clear. Given a client and a server share a key server sends a random challenge vector client encrypts it with private key and returns this server verifies response with copy of private key can repeat protocol in other direction to authenticate

server to client (2-way authentication) Secret key management

physically distributed before secure communications keys are stored in a central trusted key server


Weakness Step 4 and 5 prevent the replay of step

3 Assume that Ks is not compromised

If Ks is compromised Vulnerable to replay attack Attacker can replay step 3 Unless B remembers all previous session keys with A,

it can not tell that it is a replay!


Denning Protocol Denning Protocol

Party A KDC Ida|Idb KDCA Eka(Ks|Idb|T|Ekb(Ks|Ida|T)) AB Ekb(Ks|Ida|T) BA Eks(Nb) AB Eks(f(Nb))

Here T is timestamp assures the freshness of the key Ks Rely on synchronized clock


Public-key Encryption App. The simple one proposed by Denning

AS: authentication server AAS Ida|Idb ASA Ekras

(KUa|Ida|T)|Ekras(Kub|Idb|T)

AB Ekras(KUa|Ida|T)|Ekras

(Kub|Idb|T)|

Ekub(Ekra(Ks|T)) It needs clock synchronization


Cont. Protocol by Woo and Lam, using nonce

AKDC Ida|Idb KDCA EKRau(Idb|KUb) AB EKUb(Na|Ida) BKDC Idb|Ida|EKUau(Na) KDCB EKRau(Ida|KUa)|EKUb(EkRau(Na|Ks|Ida|Idb)) BA EKUa(EkRau(Na|Ks|Ida|Idb) | Nb) AB Eks(Nb)


One-way Authentication Using Public Key approach

If confidentiality is main concern AB: EKUb(Ks) | Eks(M)

If authentication is main concern AB: M|EKRa(H(M)) This can not avoid the interception and replay

attack Sign the message then

EKUb(M|EKRa(H(M)) ) Or EKUb(Ks) | Eks(M|EKRa(H(M)) ) Also A can sends the digital certificate EKRau(T|Ida|

KUa)


Authentication Applications will consider authentication functions developed to support application-level

authentication & digital signatures will consider Kerberos – a private-key

authentication service then X.509 directory authentication

service


Kerberos Trusted key server system developed

by MIT Provides centralized third-party authentication in a

distributed network access control may be provided for

each computing resource in either a local or remote network (realm)

A Key Distribution Centre (KDC), containing database: principles (customers and services) encryption keys

KDC provides non-corruptible authentication credentials (tickets or tokens)


Kerberos Two Kerberos versions

4 : restricted to a single realm 5 : allows inter-realm authentication, in beta test Kerberos v5 is an Internet standard specified in RFC1510

To use Kerberos need to have a KDC on your network need to have Kerberised applications running on all participating

systems US export restrictions

Cannot be directly distributed outside US in source format Crypto libraries must be re-implemented locally


Kerberos Requirements first published report identified its

requirements as: security reliability transparency scalability

implemented using an authentication protocol based on Needham-Schroeder


Kerberos 4 Overview a basic third-party authentication

scheme have an Authentication Server (AS)

users initially negotiate with AS to identify self AS provides a non-corruptible authentication credential

(ticket granting ticket TGT) have a Ticket Granting server (TGS)

users subsequently request access to other services from TGS on basis of users TGT


Kerberos 4 Overview


Kerberos Realms a Kerberos environment consists of:

a Kerberos server a number of clients, all registered with server application servers, sharing keys with server

this is termed a realm typically a single administrative domain

if have multiple realms, their Kerberos servers must share keys and trust


Kerberos Version 5 developed in mid 1990’s provides improvements over v4

addresses environmental shortcomings encryption alg, network protocol, byte order,

ticket lifetime, authentication forwarding, interrealm auth

and technical deficiencies double encryption, non-std mode of use, session

keys, password attacks specified as Internet standard RFC 1510


Authentication Protocols used to convince parties of each others

identity and to exchange session keys may be one-way or mutual key issues are

confidentiality – to protect session keys timeliness – to prevent replay attacks


Replay Attacks where a valid signed message is copied

and later resent simple replay repetition that can be logged repetition that cannot be detected backward replay without modification

countermeasures include use of sequence numbers (generally impractical) timestamps (needs synchronized clocks) challenge/response (using unique nonce)


Using Symmetric Encryption as discussed previously can use a two-

level hierarchy of keys usually with a trusted Key Distribution

Center (KDC) each party shares own master key with KDC KDC generates session keys used for connections

between parties master keys used to distribute these to them


Needham-Schroeder Protocol original third-party key distribution

protocol for session between A B mediated by KDC protocol overview is:

1. A→KDC: IDA || IDB || N1

2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ]

3. A→B: EKb[Ks||IDA]

4. B→A: EKs[N2]

5. A→B: EKs[f(N2)]


Needham-Schroeder Protocol used to securely distribute a new

session key for communications between A & B

but is vulnerable to a replay attack if an old session key has been compromised then message 3 can be resent convincing B that is

communicating with A modifications to address this require:

timestamps (Denning 81) using an extra nonce (Neuman 93)


Using Public-Key Encryption have a range of approaches based on

the use of public-key encryption need to ensure have correct public keys

for other parties using a central Authentication Server

(AS) various protocols exist using

timestamps or nonces


Denning AS Protocol Denning 81 presented the following:

1. A→AS: IDA || IDB

2. AS→A: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T]

3. A→B: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T] || EKUb[EKRas[Ks||T]]

note session key is chosen by A, hence AS need not be trusted to protect it

timestamps prevent replay but require synchronized clocks


One-Way Authentication required when sender & receiver are

not in communications at same time (eg. email)

have header in clear so can be delivered by email system

may want contents of body protected & sender authenticated


Using Symmetric Encryption can refine use of KDC but can’t have

final exchange of nonces, vis:1. A→KDC: IDA || IDB || N1

2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ]

3. A→B: EKb[Ks||IDA] || EKs[M] does not protect against replays

could rely on timestamp in message, though email delays make this problematic


Public-Key Approaches have seen some public-key approaches if confidentiality is major concern, can use:

A→B: EKUb[Ks] || EKs[M] has encrypted session key, encrypted message

if authentication needed use a digital signature with a digital certificate:A→B: M || EKRa[H(M)] || EKRas[T||IDA||KUa] with message, signature, certificate



Hash AlgorithmsXiang-Yang Li


Hash Function Map a message to a smaller value Requirements

Be applied to a block of data of any size Produced a fixed length output H(x) is easy to compute (by hardware, software) One-way: given code h, it is computationally infeasible

to find x: H(x)=h Weak collision resistance: given x, computationally

infeasible to find y so H(x)=H(y) Strong collision resistance: Computationally infeasible

to find x, y so H(x)=H(y)


Hash Algorithms see similarities in the evolution of hash

functions & block ciphers increasing power of brute-force attacks leading to evolution in algorithms from DES to AES in block ciphers from MD4 & MD5 to SHA-1 & RIPEMD-160 in hash

algorithms likewise tend to use common iterative

structure as do block ciphers


Basic Uses of Hash FunctionSix basics usages

Ek(M||H(M)) Confidentiality and authentication

M|| Ek(H(M)) Authentication

M|| EKRa(H(M)) Authentication and digital signature

Ek(M|| EKRa(H(M))) Authentication, digital signature and confidentiality

M||H(M||S) Authentication (S shared by both sides)

Ek(M||H(M||S)) Confidentiality and authentication


Birthday Attacks If 64-bits hash code is used

On average, how many messages need to try to find one match the intercepted hash code?

Birthday paradox A will sign a message appended with m-bits hash code Attacker generates some variations of fraud message,

also variations of good message Find pair of message each from the two sets messages

Such that they have the same hash code Give good message to A to get signature Replace good message with fraud message


Analysis Using birthday attack, given 64-bits

hash code How many message variations needed so the success

probability is large, say 90%?


Examples Simple hash functions

XOR of the input message H(M)=X1 X2 … Xm-1 Xm

But not secure Ym=H(M) Y1 Y2 … Ym-1 has same hash value

as (X1X2 … Xm-1 Xm), where Yi is any value


Cont. Based on DES, block chaining technique

Rabin, 1978 Divide message M into fix-sized blocks Mi

Assume total n data blocks H0=initial value Hi=Emi[Hi-1] Hn is the hash value

Birthday attack still applies If still 64-bits code used


More Attacks Birthday attack applied if chosen plaintext Meet in the middle attack if known

plaintext Known signed hash code G Construct n-2 desired message block Qi

Compute Hi=EQi[Hi-1] Generate 2m/2 random blocks X

For each X, Compute Hn-1=EX[Hn-2] Generate 2m/2 random blocks Y

For each Y, Compute H’n-1=DY[G] Find X, Y such that Hn-1= H’n-1

Then Q1, Q2,…Qn-2, X,Y is a fraud message


Security The size of hash code determines

security 128bits is not secure Currently, most use 160 bits hash code

Attack MAC Object find valid (x, Ck(x)) pair Attack the key space: roughly 2k, k =key size Attack the MAC value


More Hash Algorithms Algorithms

Message Digest:MD5 (was mostly widely used) Secure Hash Algorithm: SHA-1 (from MD4) RIPEMD-160 HMAC


MD5 designed by Ronald Rivest (the R in

RSA) latest in a series of MD2, MD4 produces a 128-bit hash value until recently was the most widely used

hash algorithm in recent times have both brute-force & cryptanalytic

concerns specified as Internet standard RFC1321


MD5 Overview1. pad message so its length is 448 mod 512 2. append a 64-bit length value to message 3. initialise 4-word (128-bit) MD buffer

(A,B,C,D) 4. process message in 16-word (512-bit)

blocks: using 4 rounds of 16 bit operations on message block &

buffer add output to buffer input to form new buffer value

5. output hash value is the final buffer value


MD5 Overview


MD5 Compression Function each round has 16 steps of the form:

a = b+((a+g(b,c,d)+X[k]+T[i])<<<s) a,b,c,d refer to the 4 words of the

buffer, but used in varying permutations note this updates 1 word only of the buffer after 16 steps each word is updated 4 times

where g(b,c,d) is a different nonlinear function in each round (F,G,H,I)

T[i] is a constant value derived from sin


MD5 Compression Function


MD4 precursor to MD5 also produces a 128-bit hash of

message has 3 rounds of 16 steps vs 4 in MD5 design goals:

collision resistant (hard to find collisions) direct security (no dependence on "hard" problems) fast, simple, compact favours little-endian systems (eg PCs)


Strength of MD5 MD5 hash is dependent on all message

bits Rivest claims security is good as can be known attacks are:

Berson 92 attacked any 1 round using differential cryptanalysis (but can’t extend)

Boer & Bosselaers 93 found a pseudo collision (again unable to extend)

Dobbertin 96 created collisions on MD compression function (but initial constants prevent exploit)

conclusion is that MD5 looks vulnerable soon


Bad news Chinese authors (Wang, Feng, Lai, and Yu)

reported a family of collisions in MD5 (fixing the previous bug in their analysis), and also

reported that their method can efficiently (2^40 hash steps) find a collision in SHA-0.

August Crypto 2004, MD5 is fatally wounded; its use will be phased

out. SHA-1 is still alive but the vultures are circling. A gradual transition away from SHA-1 will now start. The first stage will be a debate about alternatives, leading to a consensus among practicing cryptographers about what the substitute will be.


Why collisions are bad An example of what you might do with this.

You could request an SSL certificate (for your real identity) from a certificate authority. After the response comes back, you can then use that response (which is based on the MD5 of your identity+key) to "authenticate" a carefully chosen different certificate, one which claims that you are LargeBankOrSoftwareCorp., but which has the same MD5 as your real identity. You can then present this to other people in order to convince them that you are someone whom you are not.

Another example, core internet routers use md5 to exchange passwords. I

simply sniff the md5sum, and if I can find a string that generates the same sum, easily, I can send my own routing update that takes down the internet. More examples, since a LOT of applications use md5, but you get the idea.


Further detail Obviously the above attack isn't quite so simple,

but this research makes it *possible*. Before, it was believed to be sufficiently difficult to find a collision, that nobody worried about it. Now they are saying its feasible to do it in hours.

The question hanging around right now is that these researchers managed to find collisions easily, but not for an artbitrary string. The questions is how long before someone modifies this method to find any colllision. That is how much time the world has to move away.

More at http://www.freedom-to-tinker.com/archives/000664.html


Secure Hash Algorithm (SHA-1) SHA was designed by NIST & NSA in

1993, revised 1995 as SHA-1 US standard for use with DSA signature

scheme standard is FIPS 180-1 1995, also Internet RFC3174 nb. the algorithm is SHA, the standard is SHS

produces 160-bit hash values now the generally preferred hash

algorithm based on design of MD4 with key

differences


SHA Overview1. pad message so its length is 448 mod 512 2. append a 64-bit length value to message3. initialise 5-word (160-bit) buffer

(A,B,C,D,E) to (67452301,efcdab89,98badcfe,10325476,c3d2e1f0)

4. process message in 16-word (512-bit) chunks: expand 16 words into 80 words by mixing & shifting use 4 rounds of 20 bit operations on message block &

buffer add output to input to form new buffer value



SHA-1 Compression Function each round has 20 steps which replaces

the 5 buffer words thus:(A,B,C,D,E) <-(E+f(t,B,C,D)+(A<<5)+Wt+Kt),A,

(B<<30),C,D)

a,b,c,d refer to the 4 words of the buffer t is the step number f(t,B,C,D) is nonlinear function for

round Wt is derived from the message block Kt is a constant value derived from sin


SHA-1 Compression Function


SHA-1 verses MD5 brute force attack is harder (160 vs 128

bits for MD5) not vulnerable to any known attacks

(compared to MD4/5) a little slower than MD5 (80 vs 64 steps) both designed as simple and compact optimised for big endian CPU's (vs MD5

which is optimised for little endian CPU’s)


Revised Secure Hash Standard NIST have issued a revision FIPS 180-2 adds 3 additional hash algorithms SHA-256, SHA-384, SHA-512 designed for compatibility with

increased security provided by the AES cipher

structure & detail is similar to SHA-1 hence analysis should be similar


RIPEMD-160 RIPEMD-160 was developed in Europe as part

of RIPE project in 96 by researchers involved in attacks on MD4/5 initial proposal strengthen following analysis

to become RIPEMD-160 somewhat similar to MD5/SHA uses 2 parallel lines of 5 rounds of 16 steps creates a 160-bit hash value slower, but probably more secure, than SHA


RIPEMD-160 Overview1. pad message so its length is 448 mod 512 2. append a 64-bit length value to message3. initialise 5-word (160-bit) buffer (A,B,C,D,E)

to (67452301,efcdab89,98badcfe,10325476,c3d2e1f0)

4. process message in 16-word (512-bit) chunks:

use 10 rounds of 16 bit operations on message block & buffer – in 2 parallel lines of 5

add output to input to form new buffer value



RIPEMD-160 Round


RIPEMD-160 Compression Function


RIPEMD-160 Design Criteria use 2 parallel lines of 5 rounds for

increased complexity for simplicity the 2 lines are very similar step operation very close to MD5 permutation varies parts of message

used circular shifts designed for best results


RIPEMD-160 verses MD5 & SHA-1 brute force attack harder (160 like SHA-1

vs 128 bits for MD5) not vulnerable to known attacks, like

SHA-1 though stronger (compared to MD4/5)

slower than MD5 (more steps) all designed as simple and compact SHA-1 optimised for big endian CPU's vs

RIPEMD-160 & MD5 optimised for little endian CPU’s


Keyed Hash Functions as MACs have desire to create a MAC using a

hash function rather than a block cipher because hash functions are generally faster not limited by export controls unlike block ciphers

hash includes a key along with the message

original proposal:KeyedHash = Hash(Key|Message) some weaknesses were found with this

eventually led to development of HMAC


HMAC specified as Internet standard RFC2104 uses hash function on the message:

HMACK = Hash[(K+ XOR opad) ||

Hash[(K+ XOR ipad)||M)]] where K+ is the key padded out to size and opad, ipad are specified padding

constants overhead is just 3 more hash calculations than

the message needs alone any of MD5, SHA-1, RIPEMD-160 can be used


HMAC Overview


HMAC Security know that the security of HMAC relates

to that of the underlying hash algorithm attacking HMAC requires either:

brute force attack on key used birthday attack (but since keyed would need to observe

a very large number of messages) choose hash function used based on

speed verses security constraints


Summary have considered:

some current hash algorithms: MD5, SHA-1, RIPEMD-160

HMAC authentication using hash function



Digital SignatureXiang-Yang Li


Digital Signatures have looked at message authentication

but does not address issues of lack of trust digital signatures provide the ability to:

verify author, date & time of signature authenticate message contents be verified by third parties to resolve disputes

hence include authentication function with additional capabilities


Digital Signature Properties must depend on the message signed must use information unique to sender

to prevent both forgery and denial must be relatively easy to produce must be relatively easy to recognize & verify be computationally infeasible to forge

with new message for existing digital signature with fraudulent digital signature for given message

be practical save digital signature in storage


Classification of Digital Signature Undeniable Fail-Stop Blind One-time Multi-party (group signature) (n,k)-multi-party Oblivious Multi-undeniable


Algorithm and legal concerns several prior requirements

quality algorithms. Some public key algorithms are known to be insecure, practicable attacks against them having been identified.

quality implementations. An implementation of a good algorithm with mistake(s) will not work. (about 1 defect per 1,000 lines).

the private key must remain actually secret; if it becomes known to some other party, that party can produce perfect digital signatures of anything whatsoever.

distribution of public keys must be done in such a way that the public key claimed to belong to Bob actually belongs to Bob, and vice versa. This is commonly done using a public key infrastructure and the public key user association is attested by the operator of the PKI (called a certificate authority). For 'open' PKIs in which anyone can request such an attestation, the possibility of mistake is non trivial.

users (and their software) must carry out the signature protocol properly. Legal concerns


Direct Digital Signatures involve only sender & receiver assumed receiver has sender’s public-key digital signature made by sender signing

entire message or hash with private-key can encrypt using receivers public-key important that sign first then encrypt

message & signature security depends on sender’s private-key


Arbitrated Digital Signatures involves use of arbiter A

validates any signed message then dated and sent to recipient

requires suitable level of trust in arbiter can be implemented with either private

or public-key algorithms arbiter may or may not see message


RSA signature N=p q,where p and q are large primes Alice’s private key (e,n), Alice’s public key (d,n)

Signature of message m by Alice S=H(m)e mod n

Verification of signature by Bob Check if h(m) = Sd mod n


Cont. Typically d is chosen small (3 or 216+1) Problem:

Easy to create the signature of h(m1)h(m2)


ElGamal Signature Global public components

Prime number p with 512-1024 bits Primitive element g in Zp

Users private key Random integer x less than p

Users public key Integer y=gx mod p


Elgamal Signature

For each message M, generates random k Computes r=gk mod p Computes s=k-1(H(M)-xr) mod (p-1) Signature is (r,s)

Verifying Computes v1=gH(M) mod p Computes v2=yrrs mod p Test if v1= v2


Proof of Correctness Computes v2=yrrs mod q

So v2=yrrs mod q =gxr gks mod p = gxr+k k-1(H(M)-xr) mod (p-1) mod p =gH(M) mod p=v1

Notice that here it uses Fermat theorem to show That g (H(M)-xr) mod (p-1) mod p = g (H(M)-xr) mod p


Cont. The main disadvantage of ElGamal is

the need for randomness, and its slower speed (especially for signing). Another potential disadvantage of the ElGamal system

is that message expansion by a factor of two takes place during encryption. However, such message expansion is negligible if the cryptosystem is used only for exchange of secret keys.


Digital Signature Standard FIPS PUB 186 by NIST, 1991 Final announcement 1994 It uses

Secure Hashing Algorithm (SHA) for hashing Digital Signature Algorithm (DSA) for signature The hash code is set as input of DSA The signature consists of two numbers

DSA Based on the difficulty of discrete logarithm Based on Elgamal and Schnorr system


DSA Global public components

Prime number p with 512-1024 bits Prime divisor q of (p-1) with 160 bits Integer g=h(p-1)/q mod p

Users private key Random integer x less than q

Users public key Integer y=gx mod p


DSA Signature

For each message M, generates random k Computes r=(gk mod p) mod q Computes s=k-1(H(M)+xr) mod q Signature is (r,s)

Verifying Computes w=s-1 mod q, u1=H(M)w mod q Computes u2=rw mod q,v=(gu1yu2 mod p) mod q Test if v=r


Proof of Correctness Notice that v=(gu1yu2 mod p) mod q

=(gH(M)w mod q yrw mod q mod p) mod q =(gH(M)w mod q gxrw mod q mod p) mod q =(gH(M)w +xrw mod q mod p) mod q =(g(H(M)+xr)w mod q mod p) mod q =(g(H(M)+xr)k(H(M)+xr)-1 mod q mod p) mod q =(gk mod p) mod q =r


In practice (Sun Java Library) g = F7E1A085D69B3DDE CBBCAB5C36B857B9 7994AFBBFA3AEA82 F9574C0B3D078267 5159578EBAD4594F E67107108180B449 167123E84C281613 B7CF09328CC8A6E1 3C167A8B547C8D28 E0A3AE1E2BB3A675 916EA37F0BFA2135 62F1FB627A01243B CCA4F1BEA8519089 A883DFE15AE59F06 928B665E807B5525 64014C3BFECF492A

p = FD7F53811D751229 52DF4A9C2EECE4E7 F611B7523CEF4400 C31E3F80B6512669 455D402251FB593D 8D58FABFC5F5BA30 F6CB9B556CD7813B 801D346FF26660B7 6B9950A5A49F9FE8 047B1022C24FBBA9 D7FEB7C61BF83B57 E7C6A8A6150F04FB 83F6D3C51EC30235 54135A169132F675 F3AE2B61D72AEFF2 2203199DD14801C7

q = 9760508F15230BCC B292B982A2EB840B F0581CF5 Here g and p have 1024 bits, while q has 160 bits. They fulfill the requirement

that gq = 1 mod p,


Note Can we use the random number k twice?

What will happen if k used twice? We have r=(gk mod p) mod q s1=k-1(H(M1)+xr) mod q and s2=k-1(H(M2)+xr) mod q

We have s1 - s2 =k-1(H(M1)-H(M2)) mod q Another attack (for OpenPGP)

Replace p and g http://www.tigertools.net/board/?topic=topic4&msg=14 http://www.orlingrabbe.com/DSAflaw_OpenPGP.htm


Cont. We cannot use small k


Non-deterministic Non-determined signatures

For each message, many valid signatures exist DSA, Elgamal

Deterministic signatures For each message, one valid signature exists RSA


Comparisons Speed

DSS has faster signing than verifying RSA could have faster verifying than signing Message be signed once, but verified many times

This prefers the faster verification But the signer may have limited computing power

Example: smart card This prefers the faster siging


Blind Signature (digital cash) first introduced by Chaum, allow a person to get a

message signed by another party without revealing any information about the message to the other party.

Suppose Alice has a message m that she wishes to have signed by Bob, and she does not want Bob to learn anything about m. Let (n,e) be Bob's public key and (n,d) be his private key. Alice generates a random value r such that gcd(r, n) = 1 and sends x = (re m)

mod n to Bob. The value x is ``blinded'' by the random value r; hence Bob can derive no useful information from it.

Bob returns the signed value t = xd mod n to Alice. Since xd (re m)d r md mod n, Alice can obtain the true signature s of m by computing s = r-1 t mod n.


Security Concerns GnuPG permits creating ElGamal keys

are usable for both encryption and signing. It is even possible to have one key (the primary one)

used for both operations. This is not considered good cryptographic practice,

but is permitted by the OpenPGP standard. signature is much larger than a RSA or DSA

signature verification and creation takes far longer and the use

of ElGamal for signing has always been problematic due to a couple of cryptographic weaknesses when not done properly.


Applications of Blind Signature In an online context the blind signature works

as follows. Voters encrypt their ballot with a secret key and then blinds it. Then the voter signs the encrypted vote and sends it to the

validator. The validator checks to see if the signature is valid (the signature

acts as a I.D. tag and will have to be registered with the voter before the voting process has started) and if it is the validator signs it and returns it to the voter.

The voter removes the blinding encryption layer, which then leaves behind an encrypted ballot with the validator's signature.


Cont. This is then sent to the tallier who checks to make

sure the validator's signature is present on the votes.

He then waits until all votes haven been collected and then publishes all the encrypted votes so that the voters can verify their votes have been received.

The voters then send their keys to the tallier to decrypt their ballots.

Once the vote has been counted the tallier publishes the encrypted votes and the decryption keys so that voters can then verify the results.

Next we illustrate the transfer of ballots between the various parties.


Cont.


Cont, This protocol has been implemented used in

reality and has been found that the entire voting process can be completed in a matter of minutes despite the complex nature of the voting procedure.

Most of the tasks can be automated with the only user interaction needed being the actual vote casting.

Encryption, blinding and all the verification needed can be performed by software in the background.

Of course we'd have to trust this software to handle the voting procedures correctly and accurately and to assume it has not been compromised in some way.



CertificateXiang-Yang Li


Certificate A public-key certificate is a digitally

signed statement from one entity, saying that the public key (and some other information) of another entity has some specific value.


More terms Digitally Signed

If some data is digitally signed it has been stored with the "identity" of an entity, and a signature that proves that entity knows about the data. The data is rendered unforgeable by signing with the entitys' private key.

Identity A known way of addressing an entity. In some systems

the identity is the public key, in others it can be anything from a Unix UID to an Email address to an X.509 Distinguished Name.

Entity An entity is a person, organization, program,

computer, business, bank, or something else you are trusting to some degree.


More about CA Why need it

In a large-scale networked environment it is impossible to guarantee that prior relationships between communicating entities have been established or that a trusted repository exists with all used public keys. Certificates were invented as a solution to this public key distribution problem. Now a Certification Authority (CA) can act as a Trusted Third Party. CAs are entities (e.g., businesses) that are trusted to sign (issue) certificates for other entities. It is assumed that CAs will only create valid and reliable certificates as they are bound by legal agreements. There are many public Certification Authorities, such as VeriSign, Thawte, Entrust, and so on. You can also run your own Certification Authority using products such as the Netscape/Microsoft Certificate Servers or the Entrust CA product for your organization.

http://www.verisign.com/

http://www.thawte.com/

http://www.entrust.com/


Who uses Certificate? Probably the most widely visible application of

X.509 certificates today is in web browsers (such as Netscape Navigator and Microsoft Internet Explorer) that support the SSL protocol. SSL (Secure Socket Layer) is a security protocol that provides

privacy and authentication for your network traffic. These browsers can only use this protocol with web servers that support SSL.

Other technologies that rely on X.509 certificates include: Various code-signing schemes, such as signed Java Archives, and

Microsoft Authenticode. Various secure E-Mail standards, such as PEM and S/MIME. E-Commerce protocols, such as SET.


How to create certificate? There are two basic techniques used to get

certificates: you can create one yourself (using the right tools, such as keytool)

Not everyone will accept self-signed certificates, you can ask a Certification Authority to issue you one (either directly or

using a tool such as keytool to generate the request). The main inputs to the certificate creation are:

Matched public and private keys, generated using some special tools (such as keytool), or a browser.

information about the entity being certified (e.g., you). This normally includes information such as your name and organizational address. If you ask a CA to issue a certificate for you, you will normally need to provide proof to show correctness of the information.

http://csrc.nist.gov/wireless/S10_802.11i%20Overview-jw1.pdf


business Many companies sale the service of

creating the certificate (such as SSL certificate) Comodo Verisign Thawte Entrust Geotrust


X.509 Authentication Service Public key certificate associated with user

The certificates are created by Trusted Authority Then placed in the directory by TA or user Itself is not responsible for creating certificate It includes

Version, serial number, signature algorithm identifier, Issuer name, issuer identifier, validity period, the user, user identifier, user’s public key, extensions, signature by TA

The signature by TA guarantees the authority Certificates can be used to certify other TAs Y<<X>>: certificate of user X issued by TA Y


What is inside X.509 certificate? Version

Thus far, three versions are defined. Serial Number

distinguish it from other certificates it issues. This information is used in numerous ways, for example when a certificate is revoked its serial number is placed in a Certificate Revocation List (CRL).

Signature Algorithm Identifier This identifies the algorithm used by the CA to sign the

certificate. Issuer Name

The X.500 name of the entity that signed the certificate. This is normally a CA. Using this certificate implies trusting the entity that signed this certificate. root or top-level CA certificates, the issuer signs its own certificate.


cont Validity Period

This period is described by a start date and time and an end date and time, and can be as short as a few seconds or almost as long as a century. It depends on a number of factors, such as the strength of the private key used to sign the certificate or the amount one is willing to pay for a certificate. This is the expected period that entities can rely on the public value, if the associated private key has not been compromised.

Subject Name The name of the entity whose public key the certificate

identifies. This name uses the X.500 standard, so it is intended to be unique across the Internet.

Subject Public Key Information together with an algorithm identifier


Certificate Revocation Need the private key together with the

certificate to revoke it The revocation is recorded at the

directory Each time a certificate is arrived, check

the directory to see if it is revoked


X.509 Authentication Service part of CCITT X.500 directory service

standards distributed servers maintaining some info database

defines framework for authentication services directory may store public-key certificates with public key of user signed by certification authority

also defines authentication protocols uses public-key crypto & digital signatures

algorithms not standardised, but RSA recommended


X.509 Certificates issued by a Certification Authority (CA), containing:

version (1, 2, or 3) serial number (unique within CA) identifying certificate signature algorithm identifier issuer X.500 name (CA) period of validity (from - to dates) subject X.500 name (name of owner) subject public-key info (algorithm, parameters, key) issuer unique identifier (v2+) subject unique identifier (v2+) extension fields (v3) signature (of hash of all fields in certificate)

notation CA<<A>> denotes certificate for A signed by CA


X.509 Certificates


Obtaining a Certificate any user with access to CA can get any

certificate from it only the CA can modify a certificate because cannot be forged, certificates

can be placed in a public directory


CA Hierarchy if both users share a common CA then

they are assumed to know its public key otherwise CA's must form a hierarchy use certificates linking members of

hierarchy to validate other CA's each CA has certificates for clients (forward) and

parent (backward) each client trusts parents certificates enable verification of any certificate

from one CA by users of all other CAs in hierarchy


CA Hierarchy Use


Certificate Revocation certificates have a period of validity may need to revoke before expiry, eg:

1. user's private key is compromised2. user is no longer certified by this CA3. CA's certificate is compromised

CA’s maintain list of revoked certificates the Certificate Revocation List (CRL)

users should check certs with CA’s CRL


Authentication Procedures X.509 includes three alternative

authentication procedures: One-Way Authentication Two-Way Authentication Three-Way Authentication all use public-key signatures


One-Way Authentication 1 message ( A->B) used to establish

the identity of A and that message is from A message was intended for B integrity & originality of message

message must include timestamp, nonce, B's identity and is signed by A


Two-Way Authentication 2 messages (A->B, B->A) which also

establishes in addition: the identity of B and that reply is from B that reply is intended for A integrity & originality of reply

reply includes original nonce from A, also timestamp and nonce from B


Three-Way Authentication 3 messages (A->B, B->A, A->B) which

enables above authentication without synchronized clocks

has reply from A back to B containing signed copy of nonce from B

means that timestamps need not be checked or relied upon


X.509 Version 3 has been recognised that additional

information is needed in a certificate email/URL, policy details, usage constraints

rather than explicitly naming new fields defined a general extension method

extensions consist of: extension identifier criticality indicator extension value


Certificate Extensions key and policy information

convey info about subject & issuer keys, plus indicators of certificate policy

certificate subject and issuer attributes support alternative names, in alternative formats for

certificate subject and/or issuer certificate path constraints

allow constraints on use of certificates by other CA’s



IdentificationXiang-Yang Li


Identification Identification: user authentication

convince system of your identity before it can act on your behalf sometimes also require that the computer verify its identity with the

user Based on three methods

what you know what you have what you are

Verification Validation of information supplied against a table of possible values

based on users claimed identity


What you Know Passwords or Pass-phrases

prompt user for a login name and password verify identity by checking that password is correct on some (older) systems, password was stored clear more often use a one-way function, whose output cannot

easily be used to find the input value either takes a fixed sized input (eg 8 chars) or based on a hash function to accept a variable sized

input to create the value important that passwords are selected with care to

reduce risk of exhaustive search


Weakness Traditional password scheme is

vulnerable to eavesdropping over an insecure network


Solutions? One-time password

these are passwords used once only future values cannot be predicted from older values

Password generation either generate a printed list, and keep matching list on

system to be accessed or use an algorithm based on a one-way function f (eg

MD5) to generate previous values in series (eg SKey) start with a secret password s, and number N , p0 = fN(s) ith password in series is pi = fN-i(s)

must reset password after N uses


What you Have Magnetic Card, Magnetic Key

possess item with required code value encoded Smart Card or Calculator

may interact with system may require information from user could be used to actively calculate: a time dependent password a one-shot password a challenge-response verification public-key based verification


What you Are Verify identity based on your physical

characteristics, known as biometrics Characteristics used include:

Signature (usually dynamic) Fingerprint, hand geometry face or body profile Speech, retina pattern

Tradeoff between false rejection (type I error) false acceptance (type II error)



Secret SharingXiang-Yang Li


Threshold Scheme A (t,w)-threshold scheme

Sharing key K among a set of w users Any t users can recover the key Any t-1 users can not do so

Schemes Shamir’s scheme Geometric techniques Matroid theory


Shamir’s Scheme Initialization phase

Dealer chooses a large prime number p Dealer chooses w distinct xi from Zp

Gives value xi to person pi

Share distribution of key k from Zp Dealer choose t-1 random number ai

Dealer computes yi=f(xi) Here f(x)=k+ajxj mod p

Dealer gives share yi to person pi


Geometry View

http://www.rsasecurity.com/rsalabs/faq/images/shamir.gif


Simple (t,t) Sharing Procedure

D secretly chooses t-1 random elements yi from Zn

D computes Value yt=K- yj mod n

D distributes yi to person pi for all i It is secure and easy

Number n can be any number Easy to recover the key Only t persons together can do so, assume yi random


Blakley's Scheme Secret is a point in an t-dimensional

space Dealer gives each user a hyper-plane

passing the secret point Any t users can recover the common

point


Geometry View

http://www.rsasecurity.com/rsalabs/faq/images/blakley.gif


Avoid Cheating Two major distinct weaknesses

Bogus values are undetectable. Participants need not reveal their true share.

Even if a bogus value was detected, it would not necessarily give any information about the true value

One participant did not reveal its true value after get the true values from other one


Ben-Or/Rabin Solution Using Checking Vectors For any two participants A and B

Dealer gives A (SA, YAB) Dealer gives B (BAB, CAB) Here CAB = BAB YAB+ SA mod p SA is the secret share of A A and B keep their values secret B can use (BAB, CAB) to verify the value (SA, YAB) of A


Avoid Cheating Participant B can send A bogus value

after receive A’s value Solution: bit transfer

Dealer gives A (SAi, YABi) Dealer gives B (BABi, CABi) Here CABi = BABi YABi+ SAi mod p SAi is the ith bit of the secret share of A


Cont. Protocol

Participant A gives its value (SAi, YABi) to B B verifies: CABi = BABi YABi+ SAi mod p B then sends its value (SBi, YBAi) to A A verifies: CBAi = BBAi YBAi+ SBi mod p The protocol terminates whenever

One side detects cheating, or All values transferred


Chinese Remainder Theorem Given a number m<n, and n=n1n2…nk,

Numbers ni and nj are coprimes Let ai=m mod ni

Number n is public Dealer delivers ai and ni to the ith participant Then all k users can recover the number m

Why it is not a good secret sharing scheme? Is it computationally for any k-1 users to recover the key

if n is large?


Recover method Each user pre-computes

Ni=n/ni

Inverse of Ni: yi=Ni mod ni

Compute the product si=aiNiyi mod n Recover the secret m

Each user submits si

Computes s1+s2+….+sk mod n


Access Structure Threshold scheme allows any t users to

recover key! Access structure allows some subsets to

recover the key! Example: {{p1,p2,p4},{p1,p3,p4},{p2,p3}} among

p1,p2,p3,p4,p5 able to recover the key Assume the accessing subset is minimized

No subset of any accessing subset is able to recover


Monotone Circuit Assign sharing for each accessing

subset

k

kkk

p1 p2 p3 p4

a1a2

b1b2 k-b1-b2

k-a1-a2

c1 k-c1


Cont. Distribution

(a1,b1) to p1

(a2,c1) to p2

(k-c1,b2) to p3

(k-a1-a2,k-b1-b2) to p4

The sharer needs know The circuit used by dealer Which shares corresponding to which wires

The shared value is secret


Visual Secret Sharing There is a secret picture to be shared

among n participants. The picture is divided into n transparencies (shares)

such that if any m transparencies are placed together, the picture

becomes visible but if fewer than m transparencies are placed together,

nothing can be seen.


Visual Secret Sharing Such a scheme is constructed by

viewing the secret picture as a set of black and white pixels and handling each pixel separately. The schemes are perfectly secure and easily

implemented without any cryptographic computation. A further improvement allows each

transparency (share) to be an innocent picture For example, a picture of a landscape or a picture of a

building thus concealing the fact of secret sharing


Interactive Proof Interactive proof is a protocol between

two parties in which one party, called the prover, tries to prove a certain fact to the other party, called the verifier

Often takes the form of a challenge-response protocol


cont protocol in which one or more provers

try to convince another party, called the verifier, that the prover(s) possess certain true knowledge, such as the membership of a string x in a given language, often with the goal of revealing no further details about this knowledge. The prover(s) and verifier are formally defined as probabilistic Turing machines with special "interaction tapes" for exchanging messages.

http://www.nist.gov/dads/HTML/string.html

http://www.nist.gov/dads/HTML/language.html

http://www.nist.gov/dads/HTML/probablturng.html


Desired Properties Desired properties of interactive proofs

Completeness: The verifier always accepts the proof if the prover knows the fact and both the prover and the verifier follow the protocol.

Soundness: Verifier always rejects the proof if prover doesnot know the fact, and verifier follows protocol.

Zero knowledge: The verifier learns nothing about the fact being proved (except that it is correct) from the prover that he could not already learn without the prover. In a zero-knowledge proof, the verifier cannot even later prove the fact to anyone else.


Typical Protocol A typical round in a zero-knowledge proof

consists of a "commitment" message from the prover, followed by a challenge from the verifier, and then a response to the challenge from the prover. The protocol may be repeated for many rounds. Based on the prover's responses in all the rounds, the verifier decides whether to accept or reject the proof.


An example Ali Baba’s Cave


Cont. Alice wants to prove to Bob that

she knows the secret words to open the portal at CD but does not wish to reveal the secret to Bob. In this scenario, Alice’s commitment is to go to C or D.


Proof Protocol A typical round in the proof proceeds as

follows: Bob goes to A, waits there while Alice goes to C or D. Bob then asks Alice to appear from either the right side

or the left side of the tunnel. If Alice does not know the secret words

there is only a 50 percent chance that she will come out from the right tunnel.

Bob will repeat this round as many times as he desires until he is certain that Alice knows the secret words.

No matter how many times that the proof repeats, Bob does not learn the secret words.


Graph Isomorphism Problem Instance

Two graphs G1=(V1,E1) and G2=(V2,E2) Question

Is there a bijection f from V1 to V2, so (u,v)E1 implies that (f(u),f(v))E2

If such bijection exists, then graphs G1 and G2 are said to be isomorphic

If such bijection does not exist, then graphs G1 and G2 are said to be non-isomorphic


Graph Non-isomorphism Input: graphs G1 and G2 over {1,2,…n} Prover want to prove

G1 and G2 are not isomophic Assumption

Prover has unbounded computational power Verifier has limited computational power


Proof Protocol Protocol (repeated for n rounds)

Verifier Randomly chooses i=1 or 2 Selects a random permutation f and compute H to

be the image of Gi under f, sends H to prover Prover

Determines the value j such that Gj is isomorphic to H

Sends j to verifier Verifier checks if j=i If equal for n rounds, then accepts the proof


Correctness and Soundness Correctness

If G1 and G2 are not isomorphic, then for any round, there is only one graph of G1, G2 that could produce H under a permutation f

So if the verifier knows non-isomorphism, then each round a correct j will be computed

Soundness If the verifier does not know (G1 and G2 are isomorphic),

then each round two answers possible, and it has half chance to get the correct i chosen by the prover.


Graph Isomorphism Input: graphs G1 and G2 over {1,2,…n} Prover want to prove

G1 and G2 are isomophic Assumption

Prover has unbounded computational power Verifier has limited computational power


Proof Protocol Protocol (repeated for n rounds)

Prover Selects a random permutation f and compute H to be

the image of G1 under f, sends H to prover Verifier

Randomly chooses i=1 or 2, sends it to prover Prover

Computes the permutation g such that H is the image of Gj under g, and sends g to verifier

Verifier checks if H is the image of Gj under g

If yes for n rounds, then accepts the proof


Correctness and Soundness Correctness

If G1 and G2 are isomorphic, and the verifier knows how to find the permutation between G1 and G2, then each round a correct g will be computed

Soundness If the verifier does not know (G1 and G2 are non-

isomorphic or the permutation between G1 and G2), then each round prover can deceive the verifier is to guess the value i chosen by the verifier


Perfect Zero-Knowledge The graph isomorphism proof is ZKP

All information seen by the verifier is the same as generated by a random simulator

Define transcript of the proof as t=(G1,G2,(H1,i,g1),(H2,i,g2),….(Hn,i,gn))

Anyone can generate the transcript without knowing which permutation carries G1 to G2

Hence the verifier gains nothing by knowing the transcript (I.e., the proof history)


ZKP for Verifier Perfect Zero-knowledge for verifier

Suppose we have a poly-time interactive proof system and a poly-time simulator S. Let T be all yes-instance transcripts and let F be all transcripts generated by S. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F)

We say the interactive proof system are perfect zero-knowledge for the verifier


Isomorphism Proof: ZKP-verifier Graph isomorphism is a perfect zero-

knowledge for verifier A triple (H,i,g). There are 2n! valid triples. All triples (H,i,g) occurs equiprobable in some

transcript Here, assume that both the verifier and the

prover are honest Both of them randomly chooses parameters that

supposed to be chosen randomly


Cheating Verifier What happened if verifier does not

follow the protocol (does not choose i randomly) Transcript produced by ZKP is not same as that

produced by the random simulator anymore The verifier may gain some information due to this

imbalance But, there is another expected poly-time simulator to

generate the same transcript Hence, the verifier still gains nothing


Perfect Zero-Knowledge Definition

Suppose we have a poly-time interactive proof system, a poly-time algorithm V to generate random numbers by verifier, and a poly-time simulator S. Let T be all yes-instance transcripts (depending on V) and let F be all transcripts generated by S and V. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F)

We say the interactive proof system are perfect zero-knowledge


Forging Simulator Initial transcript t=(G1,G2), repeat n

rounds Let old-state=state(V), repeat follows

Chooses ij from {1,2} randomly Chooses gj to be a random permutation over {1,...n} Compute Hj to be the image of Gi under g Call V with input Hj, obtaining a challenge ij’ If ij=ij’, then concatenate (Hj, ij, gj) onto the end of t Else reset V by state(V)=old-state

Until ij=ij’


Perfect Zero-knowledge The graph isomorphism is perfect ZKP

The expected running time of simulator is 2n For the kth round of the interactive proof system

Let pk be the probability that verifier chooses i=1 Then (H,1,g) occurs in actual transcript with pk/n!,

(H,2,g) occurs in actual transcript with (1-pk)/n! For simulator, when it terminates the simulation

for the kth round, same probability distribution for (H,1,g) and (H,2,g)

Therefore, all transcripts by simulator or actual has the same probability distribution


Quadratic Residue Fiat-Shamir Identification Question

Given integer n=pq, here p, q are primes. Prover wants to prove

Integer x is a quadratic residue mod n In other words, knows u so x=u2 mod n

Quadratic residue is hard to solve if do not knowing the factoring of n


Proof Protocol Repeat the following for log2n times

Prover Chooses random v less than n and computes y=v2

mod n. Sends y to verifier Verifier

Chooses a random I from {0,1}, sends it to prover Prover

Computes z=u2v mod n, sends z to verifier Verifier

Checks if z2=xiy mod n Accepts the proof if equation holds all log2n rounds


Cont Correctness

Show that verifier will accept the prover if indeed knows

Soundness Show that verifier will detect the prover if it does not

know with a good probability Zero-knowledge

Show that verifier gets nothing from the protocol


Guillou Quisquater Protocol The GQ protocol is an extension of the Fiat

Shamir protocol that limits the number t of rounds required.

One Time Set-up: 1. A trusted authority T selects two random

primes p and q and forms a modulus n = p · q.

2. T defines a public exponent v > 4 with gcd(v, (p-1)(q -1) = 1 so that T can compute s = v-1 mod (p-1) (q-1).

3. T publishes parameters n and v.


Cont. Selection of per-user parameters: 1. Each entity A has a unique identification

Id(A). Everyone can calculate a value J(A) = f(Id(A)) mod n (the redundant identity).

2. T gives to each entity A the secret data secret(A) = J(A)-s, which it can calculate.


Cont. Protocol: A proves her identity to B using t rounds, each of

which consists of: 1. A selects a random secret r and sends her identity Id(A)

and x = rv mod n to B. 2. B selects a random challenge e in {1, 2, ... , v}. 3. A computes and sends the following response to B: y = r ·

secret(A)e mod n. 4. B receives y, constructs J(A) = f(Id(A)) mod n, computes z

= J(A)eyv, and accepts this round if z = x mod n. In this protocol, v determines the security level. In Fiat

Shamir, v = 2 and there are many rounds. A fraudulent claimant can defeat the protocol by correctly guessing the challenge e (with a 1 in v chance.) GQ seems secure, because we need to extract v-roots modulo n.


Discrete Logarithm Question:

Prover wants to prove to verifier that he knows x such that y=gx mod p .

Here g, y, and p are public information Prover does not want to publicize the value of x.


Proof Protocol Repeat the following for log2n times

Prover Chooses random j < p-1 and computes r=gj mod

p. Sends r to verifier Verifier

Chooses a random i from {0,1}, sends it to prover Prover

Computes h=i x +j mod p-1, sends h to verifier Verifier

Checks if gh=yir mod n Accepts the proof if equation holds all log2n rounds


Cont Correctness

Show that verifier will accept the prover if indeed knows

Soundness Show that verifier will detect the prover if it does not

know with a good probability Zero-knowledge

Show that verifier gets nothing from the protocol


Bit Commitments Bit commitment

Sometimes, it is desirable to give someone a piece of information, but not commit to it until a later date. It may be desirable for the piece of information to be held secret for a certain period of time.

Example: stock up and down


Properties Bit commitment scheme

The sender encrypts the b in some way The encrypted form of b is called blob Scheme f: (X,b)Y

Properties Concealing: verifier cannot detect b from f(x,b) Binding: sender can open the blob by revealing x Hence, the sender must use random x to mask b


Methods One can choose any encryption method E

Function f((x0,k),b)=Ek((x0,b)) Need supply decryption k to reveal b Assume the decryption method D is known

Choose any integer n=pq, p and q are large primes Function f(x,b)=mbx2 mod n

Goldwasser-Micali Scheme Here n=pq, m is not quadratic residule, m,n public mx1

2 mod n x22 mod n

So sender can not change mind after commitment


Coin Flip Even protocols

Alice has a coin flip result i or j Bob wants to guess the result Alice has a message M that is commitment If bob guesses correct, Bob should have M received Alice starts with 2 pairs of public keys (Ei,Di) and

(Ej,Dj) Bob starts with a symmetric encryption S and a key k


Protocol Procedure

Alice sends Ei, Ej to Bob Bob guess h and sends y=Eh(k) to Alice Alice computes p=Dj(y) and sends the encryption z of

M by p using S to Bob Bob decrypts the encryption z using S and key k If the guess is correct, then Bob gets the commitment


Oblivious Transfer What is oblivious transfer

Alice wants to send Bob a secret in such a way that Bob will know whether he gets it, but Alice won't. Another version is where Alice has several secrets and transfers one of them to Bob in such a way that Bob knows what he got, but Alice doesn't. This kind of transfer is said to be oblivious (to Alice).


Transfer Factoring By means of RSA, oblivious transfer of

any secret amounts to oblivious transfer of the factorization of n=pq Bob chooses x and sends x2 mod n to Alice Alice (who knows p,q) computes the square roots x,-

x,y,-y of x2 mod n and sends one of them to Bob. Note that Alice does not know x.

If Bob gets one of y or -y, he can factor n. This means that with probability 1/2, Bob gets the secret. Alice doesn't know whether Bob got one of y or -y because she doesn't know x.


Factoring If one knows x and y such that

1) x2=y2 mod n 2) 0<x,y<n, xy and x+y0 mod n Number n is the production of two primes

Then n can be factored First gcd(x+y,n) is a factor of n And gcd(x-y,n) is a factor of n


Quadratic Solution Given n=p, and a is a quadratic residue

Then there is two positive integers x less than n Such that x2=a mod n

Given n=pq, and a is a quadratic residue Then there is four positive integers x less than n Such that x2=a mod n


Oblivious Transfer of Message Alice has a message M, bob wants to

get M through oblivious transfer Alice does not know if Bob get M or not Bob knows if he gets it or not Bob gets M with probability ½ Coin flipping can be used to achieve this


Contract Signing It requires two things

Commitment: after certain point, both parties are bound by the contract, until then, neither is

Unforgeability: it must be possible for either party to prove the signature of the other party

With Pen and Paper Two party together, face to face Sign simultaneously (or one character by one)


Remote Contract Signing Simple one

Alice generate a signature, divided into SL, SR Alice randomly select two keys KL, KR Encrypt the signatures SL, SR Transfer encrypted SL,SR to Bob Obliviously transfer KL, KR to bob

Bob gets one, but Alice does not know which one Bob decrypts the encrypted SL or SR

Verify the decrypted signature, if invalid, stop Alice sends the ith bits of keys KL and KR to Bob

Here i=1 to the length of the keys


Cont. The protocol will be conducted by Bob

also What is the chance of Alice to cheat successfully?

Alice can guess which key will be transferred obliviously ---(1/2 chance)

Then send wrong signature for the other half or send the wrong key of the other half

Bob can not detect it if Alice can guess which key Bob got

How about Alice stop prematurely? One bit advance over Bob

Enhanced protocol Use many pair of keys and signatures instead of one



Pseudo-random NumberXiang-Yang Li


Random number, Pseudorandom The outputs of pseudorandom number

generators are not truly random they only approximate some of the properties of

random numbers. "Anyone who considers arithmetical methods of

producing random digits is, of course, in a state of sin.”--- John von Neumann

Truely random numbers can be generated using hardware random number generators


Inherent non-randomness Because any PRNG run on a deterministic

computer (contrast quantum computer) is deterministic, its output will inevitably have certain properties that a true random sequence would not exhibit. guaranteed periodicity—it is certain that if the generator uses only

a fixed amount of memory then, given a sufficient number of iterations, the generator will revisit the same internal state twice, after which it will repeat forever. A generator that isn't periodic can be designed, but its memory requirements would grow as it ran. In addition, a PRNG can be started from an arbitrary starting point, or seed state, and will always produce an identical sequence from that point on.


cont In practice, many PRNGs exhibit artifacts which can

cause them to fail statistically significant tests. These include, but are certainly not limited to: Shorter than expected periods for some seed

states (not full period) Poor dimensional distribution Successive values are not independent Some bits may be 'more random' than others Lack of uniformity


Pseudo-random Bit Generator Several applications

Key generation Some encryption algorithms, or one-time pad

Let l>k be integers Function f: Z2

k Z2l computable in poly-time

Then f called (k,l)-pseudo-random bit generator The input s0 Z2

k is called the seed Output f(s0) is called the pseudo-random string


Desired Properties Three important properties:

Unbiased (uniform distribution): All values of whatever sample size is collected are

equiprobable Unpredictable (independence):

It is impossible to predict what the next output will be, given all the previous outputs, but not the internal "hidden" state.

Irreproducible: Two of the same generators, given the same

starting conditions, will produce different outputs.


Desired Properties Usually when a person says

A "good" pseudo-random number generator they mean it is unbiased.

A "true" PRNG they usually mean it's irreproducible

A "cryptographically strong" PRNG they mean it's unpredictable

Very rarely they mean it's all threes


More Properties Long period

The generator should be of long period Fast computation

The generator should be reasonably fast Security

The generator should be secure What is security level of PRNG?


Security A PRNG suitable for cryptographic applications is called

a cryptographically secure PRNG (CSPRNG). Its output should not only pass all statistical tests for randomness but satisfy

some additional cryptographic requirements. Used in many aspects of cryptography require random numbers, for example:

Key generation Nonces Salts in certain signature schemes, (ECDSA, RSASSA-PSS). One-time pads


CSPRNG CSPRNG requirements fall into two groups:

their statistical properties are good (passing tests of randomness), they hold up well in case of attack, even when (part of) their secrets are

revealed. A CSPRNG should satisfy the 'next-bit test'.

Given the first l bits of a random sequence there is no polynomial-time algorithm that can predict the next bit with probability of success significantly higher than 1/2.

It has been proven that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness.

should withstand state compromise extensions. That is, in the unfortunate case that part or all of the state has been revealed (or

guessed correctly), it should be impossible to reconstruct the stream of random numbers prior to the incident. Also if there is an input of entropy, it should be infeasible to use knowledge of the state to predict future conditions of the state.


Example the CSPRNG being considered produces

output by computing some function of the next digit of pi (ie, 3.1415...),

it may well be random as pi appears to be a random sequence.

However, this does not satisfy the next-bit test, and thus is not cryptographically secure. There exists an algorithm that will predict the next bit.


Design divide designs of CSPRNGs into classes:

those based on block ciphers; those based upon hard mathematical problems, and special-purpose designs.


Designs based on cryptographic primitives Designs based on cryptographic primitives

A secure block cipher can also be converted into a CSPRNG by running it in counter mode.

This is done by choosing an arbitrary key and encrypting a zero, then encrypting a 1, then encrypting a 2, etc. The counter can also be started at an arbitrary number other than zero. Obviously, the period will be 2n for an n-bit block cipher; equally obviously, the initial values (i.e. key and 'plaintext') must not become known to an attacker lest, however good this CSPRNG construction might be otherwise, all security be lost.

A cryptographically secure hash of a counter might also act as a good CSPRNG in some cases. it is necessary that the initial value of this counter is random and secret.

If the counter is a bignum, then CSPRNG could have an infinite period.


DES Based Generator ANSI X9.17 PRNG (used by PGP,..)

Inputs: two pseudo-random inputs one is a 64-bit representation of date and time The other is 64-bit seed values

Keys: three 3DES encryptions using same keys Output:

a 64-bit pseudorandom number and A 64-bit seed value for next-round use


ANSI X9.17

EDE

EDE

EDE

DT

Si

Ri

Si+1

K1,K2


Linear Congruential Generator Protocol

Let M be an integer and a, b less than M Let k be number of bits of M Integer l is between k+1 and M-1 Let s0 be a seed less than M Define si=asi-1+b mod M Then the ith random bit is si mod 2 It is not proved to be secure


Parameter Setting Not all a, b are good and m should be

large For example, m is a large prime number For fast computation, usually m=231-1

And b is set to 0 often For this m, there are less than 100

integers a It generates all numbers less than m The generated sequences appear to be random

One such a=7516807

Used in IBM 360 family of computers


RSA Generator Protocol

Let p, q be two k/2 bits primes and define n=pq Integer b: gcd(b, (n))=1 Public: n, b; Private p,q A seed s0 with k bits Sequence si+1=si

b mod n Then the ith random bit is si mod 2 It is proved to be secure!


BBS Generator Blum-Blum-Shub Generator

Let p, q be two k/2 bits primes and define n=pq Here p=q=3 mod 4

this guarantees that each quadratic residue has one square root which is also a quadratic residue

gcd(φ(p-1), φ(q-1)) should be small this makes the cycle length large.

Let QR(n) be all quadratic residues modulo n Public: n; Private p,q A seed s0 with k bits from QR(n) Sequence si+1=si

2 mod n Then the ith random bit is si mod 2


Cont on BBS Provably “secure”

When the primes are chosen appropriately, and O(log log n) bits of each Si are output, then in the limit as n grows large, distinguishing the

output bits from random will be at least as difficult as factoring n.

However, it's theoretically possible that a fast algorithm for

factoring will someday be found, so BBS is not yet guaranteed to be secure.


Discrete Logarithm Generator Protocol

Let p be a k-bit prime, Let be primitive element modulo p A seed s0 is any non-zero integer less than p Define si+1 = si mod p Then the ith random bit is

1 if si is larger than p/2 0 if si is less than p/2


Standards A number of designs of CSPRNGs have

been standardized. They can be found in: FIPS 186-2 ANSI X9.17-1985 Appendix C ANSI X9.31-1998 Appendix A.2.4 ANSI X9.62-1998 Annex A.4


Network Security


Topics to be covered Applications

Email security www security Malicious software

Networks Wireless LAN security 802.11 IPsec Firewall Intrusions



Email SecurityXiang-Yang Li


Electronic Mail Security

Despite the refusal of VADM Poindexter and LtCol North to appear, the Board's access to other sources of information filled much of this gap. The FBI provided documents taken from the files of the National Security Advisor and relevant NSC staff members, including messages from the PROF system between VADM Poindexter and LtCol North. The PROF messages were conversations by computer, written at the time events occurred and presumed by the writers to be protected from disclosure. In this sense, they provide a first-hand, contemporaneous account of events.—The Tower Commission Report to President Reagan on the Iran-Contra Affair, 1987


Email Security email is one of the most widely used

and regarded network services currently message contents are not

secure may be inspected either in transit or by suitably privileged users on destination system


Email Security Enhancements confidentiality

protection from disclosure authentication

of sender of message message integrity

protection from modification non-repudiation of origin

protection from denial by sender


Pretty Good Privacy (PGP) widely used de facto secure email developed by Phil Zimmermann selected best available crypto algs to

use integrated into a single program available on Unix, PC, Macintosh and

Amiga systems originally free, now have commercial

versions available also


PGP Five services

Authentication, confidentiality, compression, email compatibility, segmentation

Functions Digital signature Message encryption Compression Email compatibility segmentation


PGP Operation – Authentication1. sender creates a message2. SHA-1 used to generate 160-bit hash code of

message3. hash code is encrypted with RSA using the

sender's private key, and result is attached to message

4. receiver uses RSA or DSS with sender's public key to decrypt and recover hash code

5. receiver generates new hash code for message and compares with decrypted hash code, if match, message is accepted as authentic


PGP Operation – Confidentiality1. sender generates message and random 128-

bit number to be used as session key for this message only

2. message is encrypted, using CAST-128 / IDEA/3DES with session key

3. session key is encrypted using RSA with recipient's public key, then attached to message

4. receiver uses RSA with its private key to decrypt and recover session key

5. session key is used to decrypt message


PGP Operation – Confidentiality & Authentication uses both services on same message

create signature & attach to message encrypt both message & signature attach RSA encrypted session key


PGP Operation – Compression by default PGP compresses message

after signing but before encrypting so can store uncompressed message & signature for

later verification & because compression is non deterministic

uses ZIP compression algorithm


PGP Operation – Email Compatibility when using PGP will have binary data to

send (encrypted message etc) however email was designed only for

text hence PGP must encode raw binary data

into printable ASCII characters uses radix-64 algorithm

maps 3 bytes to 4 printable chars also appends a CRC

PGP also segments messages if too big


PGP Operation – Summary


Segmentation & Reassembly Email systems impose maximum length

50 Kb, for example PGP provides automatic segmentation

Done after all other operations Thus only one session key needed


Key management Generating unpredictable session keys Identifying keys

Multiple public, private key pairs for a user Maintain keys

Its own public, private keys of a PGP entity Public keys of correspondents


Session Key Generation Algorithm used: CAST-128 Input to CAST-128

A 128-bit key Two 64 bits plaintexts to be encrypted

Output using cipher feedback mode Generates 2 64-bits ciphers form session key

Plaintexts are from 128-bits randomized number Based on key stroke of user (timing and actual keys) Then combined with previous session key


Key Identifiers Receiver has multiple public keys

How to know which private key is proper? Approach

Sending the least significant 64 bits as key ID Need send the receiver’s public key ID used for

encrypting the session key Need send the sender’s public key ID, whose

corresponding private key used for signature


Key Rings Private key rings

Timestamp, Key ID, public key, encrypted private key, user ID

Public key rings Timestamp, Key ID, public key, owner trust, user ID,

key legitimacy, signature, signature trust


Public Key Management A public key attributed to B may belong

to C C can send messages to A forge B’s sigC can read any encrypted message to B

Approach to true public key Physically get key from B Obtain B’s key from mutual trusted authority Using key legitimacy field

computed from the signature trust field and number of certificates for the key


Revoking Public Key Reason

It is compromised: private key is open Simply to avoid use of same key for a period

Approach Owner issues key revocation certificate, signed by

owner Using corresponding private key to sign the certificate Disseminate the certificate as widely and as quickly as

possible


S/MIME (Secure/Multipurpose Internet Mail Extensions) security enhancement to MIME email

original Internet RFC822 email was text only MIME provided support for varying content types and

multi-part messages with encoding of binary data to textual form S/MIME added security enhancements

have S/MIME support in various modern mail agents: MS Outlook, Netscape etc


S/MIME Functions enveloped data

encrypted content and associated keys signed data

encoded message + signed digest clear-signed data

cleartext message + encoded signed digest signed & enveloped data

nesting of signed & encrypted entities


S/MIME Cryptographic Algorithms hash functions: SHA-1 & MD5 digital signatures: DSS & RSA session key encryption: ElGamal & RSA message encryption: Triple-DES, RC2/40

and others have a procedure to decide which

algorithms to use


S/MIME Certificate Processing S/MIME uses X.509 v3 certificates managed using a hybrid of a strict

X.509 CA hierarchy & PGP’s web of trust each client has a list of trusted CA’s

certs and own public/private key pairs & certs certificates must be signed by trusted

CA’s


Certificate Authorities have several well-known CA’s Verisign one of most widely used Verisign issues several types of Digital IDs with increasing levels of checks & hence

trustClass Identity Checks Usage1 name/email check web browsing/email2+ enroll/addr check email, subs, s/w validate3+ ID documents e-banking/service access



secure email PGP S/MIME



Security on WWWXiang-Yang Li


Introduction Introduction Presentation of SSL

• The inner workings of SSL• Attacks on SSL

Presentation of S-HTTP• Comparison with SSL/TLS• Attacks on S-HTTP

Other aspects of Web security• TLS• IPSec, Kerberos, SET

Conclusion


Web Security Web now widely used by business,

government, individuals but Internet & Web are vulnerable have a variety of threats

integrity confidentiality denial of service authentication

need added security mechanisms


SSL (Secure Socket Layer) transport layer security service originally developed by Netscape version 3 designed with public input subsequently became Internet standard

known as TLS (Transport Layer Security) uses TCP to provide a reliable end-to-

end service SSL has two layers of protocols


Location of SSLApplication Layer

Internet Protocol(IP)

Transmission Control Protocol (TCP)

Secure Socket Layer(SSL)

SSL is build on top of TCP

Provides a TCP like interface

In theory can be used by all type of applications in a transparent manner


SSL Architecture


SSL Architecture SSL session

an association between client & server created by the Handshake Protocol define a set of cryptographic parameters may be shared by multiple SSL connections

SSL connection a transient, peer-to-peer, communications link associated with 1 SSL session


SSL Record Protocol confidentiality

using symmetric encryption with a shared secret key defined by Handshake Protocol

IDEA, RC2-40, DES-40, DES, 3DES, Fortezza, RC4-40, RC4-128

message is compressed before encryption message integrity

using a MAC with shared secret key similar to HMAC but with different padding


SSL Change Cipher Spec Protocol one of 3 SSL specific protocols which

use the SSL Record protocol a single message causes pending state to become current hence updating the cipher suite in use


SSL Alert Protocol conveys SSL-related alerts to peer entity severity

warning or fatal specific alert

unexpected message, bad record mac, decompression failure, handshake failure, illegal parameter

close notify, no certificate, bad certificate, unsupported certificate, certificate revoked, certificate expired, certificate unknown

compressed & encrypted like all SSL data


SSL Handshake Protocol allows server & client to:

authenticate each other to negotiate encryption & MAC algorithms to negotiate cryptographic keys to be used

comprises a series of messages in phases Establish Security Capabilities Server Authentication and Key Exchange Client Authentication and Key Exchange Finish


General purpose

Two step process:• Handshake : exchange private keys using a public key encryption

algorithm• Data transmission: exchange the required data using a private key

encryption

`

1.Handshake

2. Data transmission


SSL Handshake Protocol


handshake

`

Client ServerClient HelloServer Hello

Client Key ExchangeChange Cipher Specification

Server CertificateServer Hello Done

Handshake FinishedChange Cipher Specifications

Handshake Finished


hello Client “Hello”:

• List of supported private key encryptions +

• Client random number Server “Hello”:

• Selected encryption algorithm

• Server Random number• Session ID

Server Certificate: • Verify server’s identity

`

Client Client HelloServer Hello




Handshake Finished

Server


Key exchange Client Key Exchange:

• Client Generate second

random: Pre Master Key

Send Pre Master Key Calculate Master Key Calculate Secret Key Calculate MAC Key

• Server Calculate Master Key Calculate Secret Key Calculate MAC Key

`

Client Client HelloServer Hello




Handshake Finished

Server


Resumed based on Session Id

`

Client ServerClient HelloServer Hello

Change Cipher Specification Handshake Finished

Change Cipher SpecificationsHandshake Finished


Certificate authority Certificate Authority (CA) is a trusted

third party that helps identify the server.

How does everything work?• Server sends ID, public key to CA• CA creates and signs the server’s Certificate• Client receives the Certificate from Server• Client verifies the Certificate using the signature and

the CA’s public key


MAC MAC = Message Authentication Code The initial message is split into

fragments For each fragment a “fingerprint” is

calculated using the MAC key The fragment, fingerprint and record

header are encrypted and sent Receiver checks the “fingerprint” using

MAC key to detect inconsistent messages


Attacks on SSL Certificate Injection Attack

• The list of trusted Certificate Authorities is altered• Can be avoided by upgrading the OS or switching to a safer one.

Man in the Middle • Cipher Spec Rollback : regresses the public key encryption algorithms • Version Rollback : regression from SSL 3.0 to weaker SSL 2.0• Algorithm rollback : modify public encryption method• Truncation attack : TCP FIN|RST used to terminate connection

Timing attack• Can be avoided by randomly delaying the computations

Brute force• Can be used on servers that accept small key sizes: 40 bits for symmetric

encryptions and 512 for the asymmetric one.


TLS (Transport Layer Security) IETF standard RFC 2246 similar to SSLv3 with minor differences

in record format version number uses HMAC for MAC a pseudo-random function expands secrets has additional alert codes some changes in supported ciphers changes in certificate negotiations changes in use of padding


TLSTLS was developed by IETF to replace SSL version 3.

• Based on SSL version 3, with some changes:

• Replaced FORTEZZA key exchange option with DSS.

• Include the hash method HMAC used by IPSec for authentication in IP headers.

• More differentiation between sub-protocols.

• TLS has mechanisms for backwards compatibility with SSL.


TLSTLS has about 30 possible cipher ‘suites’, combinations of key exchange, encryption method, and hashing method.

• Key exchange includes: RSA, DSS, Kerberos

• Encryption includes: IDEA(CBC), RC2, RC4, DES, 3DES, and AES

• Hashing: SHA and MD5

(Note: Some of the suites are intentionally weak export versions.)


Secure Electronic Transactions (SET) open encryption & security specification to protect Internet credit card

transactions developed in 1996 by Mastercard, Visa

etc not a payment system rather a set of security protocols &

formats secure communications amongst parties trust from use of X.509v3 certificates privacy by restricted info to those who need it


SET Components


SET Transaction1. customer opens account2. customer receives a certificate3. merchants have their own certificates4. customer places an order5. merchant is verified6. order and payment are sent7. merchant requests payment authorization8. merchant confirms order9. merchant provides goods or service10. merchant requests payment


Dual Signature customer creates dual messages

order information (OI) for merchant payment information (PI) for bank

neither party needs details of other but must know they are linked use a dual signature for this

signed concatenated hashes of OI & PI


Purchase Request – Customer


Purchase Request – Merchant


Purchase Request – Merchant1. verifies cardholder certificates using CA sigs2. verifies dual signature using customer's

public signature key to ensure order has not been tampered with in transit & that it was signed using cardholder's private signature key

3. processes order and forwards the payment information to the payment gateway for authorization (described later)

4. sends a purchase response to cardholder


Payment Gateway Authorization1. verifies all certificates2. decrypts digital envelope of authorization block to

obtain symmetric key & then decrypts authorization block

3. verifies merchant's signature on authorization block4. decrypts digital envelope of payment block to obtain

symmetric key & then decrypts payment block5. verifies dual signature on payment block6. verifies that transaction ID received from merchant

matches that in PI received (indirectly) from customer7. requests & receives an authorization from issuer8. sends authorization response back to merchant


Payment Capture merchant sends payment gateway a

payment capture request gateway checks request then causes funds to be transferred to

merchants account notifies merchant using capture

response

Cryptography and Network Security 540Security on the WWW

C- Secure-HTTP

Presentation of S-HTTP

Designed by E. Rescorla and A. Schiffman of EIT to secure HTTP connections

Proposed in 1994 but never used commercially

Not to be confused with HTTPS: encrypts HTTP messages at the application level

ABCD


C- Secure-HTTP

Location of S-HTTPABCD

Internet Protocol(IP)

Transmission Control Protocol (TCP)

Secure-HTTPMessage encryption and

signature

Application Layer:HTTP message

HTTP-specific message encryption

Can possibly be used over a secure channel

Designed to be compatible with HTTP for handling at lower layers


C- Secure-HTTP

S-HTTP vs. SSL/TLS

HTTP-specific vs. general purpose SSL (IMAPS, POPS, LDAPS…)

Burden of encryption not on transmission/reception but rather on message production/unpacking

Similar set of available ciphers, plus added capabilities for signing (DSS, RSA)

Very general specifications, leaving a lot to implement and a potential for incompatible implementations

Only one reference implementation in NCSA Mosaic

ABCD


C- Secure-HTTP

S-HTTP vs. SSL/TLS: functionalitiesABCD

Security Service S-HTTP SSL

Privacy Public or private cryptosystemEncryption of the complete HTTP transaction

Symmetric key cryptosystemComplete communication encryption

Integrity Simple MAC or signing MAC onlyAuthentication Key management on the keys

used, or digital signatureDuring the initial public key exchange (server auth. mandatory, client auth. optional)

Non-repudiation Digital signature Not provided

S-HTTP can make use of key management Non-repudiation is not provided by SSL Signing is optional, but a major attraction to S-HTTP


C- Secure-HTTP

S-HTTP vs. SSL/TLS: proxy traversalABCD

SSL-aware proxyExternal

secure server Enterprise environment

Proxy traversal: SSL connection

OR

S-HTTP-aware proxyExternal

secure server Enterprise environment

Proxy traversal: S-HTTP messaging

SSL tunnel SSL tunnel

Encrypted data

Authentication

cleartext


C- Secure-HTTP

S-HTTP inner working

Message-based encryption Superset of HTTP: “outer” envelope Specific headers added

ABCD

S-HTTP messageS-HTTP headers

HTTP payload headers:Security-Scheme, Encryption-Identity,Certificate-Info… + regular HTTP headers

HTTP message body

Request:Secure*Secure-HTTP/1.2

Response:Secure-HTTP/1.2 200 OK


C- Secure-HTTP

S-HTTP attacks

Basically the same as on SSL, since the ciphers are the same

Default values more secure in S-HTTP than SSL at the time of proposal (e.g. DES vs. RC4)

S-HTTP generally stronger by design (more resilient to proxy compromising)

More complex and wider specifications create a potential for faulty implementations

No real-world use to field test the actual security of S-HTTP

ABCD


D- Other protocols

HTTP has an authentication scheme as part of its original protocol.

HTTP Basic Authentication

ABCD

• Supported by almost all browsers and web servers.

• Password and username are sent in clear text (base64 encoded) in the HTTP request message.

• Obviously not secure enough for sensitive information.

This scheme is being replaced by the slightly more secure HTTP Digest Authentication, which sends a MD5 hash of the password and other information.


IPsecIPSec is a security layer added to a computer’s protocol stack in the kernel (Below TCP). It is invisible to the application. It is implemented by adding additional protocol numbers in the IP protocol field.

• Good for implementing a VPN.

• Packets can be either tunneled inside IPSec packets, or Transported with only the data portion of the original packet encrypted.

• Every IPSec end machine (which could be a LAN’s router) must implement IPSec for it to work.



need for web security SSL/TLS transport layer security protocols

de facto standard, versatile and low-level enough to accommodate many types of payloads

SET secure credit card payment protocols IPSec: true network-layer security for any applications

(not just the Web) Kerberos: robust 2-way authentication framework with

emphasis on security manageability


Web Security

• SSL/TLS: de facto standard, versatile and low-level enough to accommodate many types of payloads

• S-HTTP: never took off, restricted to HTTP messages

• IPSec: true network-layer security for any applications (not just the Web)

• Kerberos: robust 2-way authentication framework with emphasis on security manageability

D- ConclusionABCD


Cryptography & Network Security

Malicious SoftwareXiangYang Li


Malicious Software

What is the concept of defense: The parrying of a blow. What is its characteristic feature: Awaiting the blow.—On War, Carl Von Clausewitz


Viruses and Other Malicious Content computer viruses have got a lot of

publicity one of a family of malicious software effects usually obvious have figured in news reports, fiction,

movies (often exaggerated) getting more attention than deserve are a concern though


Malicious Software


Trapdoors secret entry point into a program allows those who know access bypassing

usual security procedures have been commonly used by developers a threat when left in production programs

allowing exploited by attackers very hard to block in O/S requires good s/w development & update


Logic Bomb one of oldest types of malicious software code embedded in legitimate program activated when specified conditions met

eg presence/absence of some file particular date/time particular user

when triggered typically damage system modify/delete files/disks


Trojan Horse program with hidden side-effects which is usually superficially attractive

eg game, s/w upgrade etc when run performs some additional

tasks allows attacker to indirectly gain access they do not

have directly often used to propagate a virus/worm or

install a backdoor or simply to destroy data


Zombie program which secretly takes over

another networked computer then uses it to indirectly launch attacks often used to launch distributed denial

of service (DDoS) attacks exploits known flaws in network

systems


Viruses a piece of self-replicating code attached

to some other code cf biological virus

both propagates itself & carries a payload carries code to make copies of itself as well as code to perform some covert task


Virus Operation virus phases:

dormant – waiting on trigger event propagation – replicating to programs/disks triggering – by event to execute payload execution – of payload

details usually machine/OS specific exploiting features/weaknesses


Virus Structureprogram V :=

{goto main;1234567;subroutine infect-executable :={loop:

file := get-random-executable-file;if (first-line-of-file = 1234567) then goto loopelse prepend V to file; }

subroutine do-damage := {whatever damage is to be done}subroutine trigger-pulled := {return true if some condition holds}main: main-program := {infect-executable;

if trigger-pulled then do-damage;

goto next;}next:

}


Types of Viruses can classify on basis of how they attack parasitic virus memory-resident virus boot sector virus stealth polymorphic virus macro virus


Macro Virus macro code attached to some data file interpreted by program using file

eg Word/Excel macros esp. using auto command & command macros

code is now platform independent is a major source of new viral infections blurs distinction between data and

program files making task of detection much harder

classic trade-off: "ease of use" vs "security"


Email Virus spread using email with attachment

containing a macro virus cf Melissa

triggered when user opens attachment or worse even when mail viewed by

using scripting features in mail agent usually targeted at Microsoft Outlook

mail agent & Word/Excel documents


Worms replicating but not infecting program typically spreads over a network

cf Morris Internet Worm in 1988 led to creation of CERTs

using users distributed privileges or by exploiting system vulnerabilities

widely used by hackers to create zombie PC's, subsequently used for further attacks, esp DoS

major issue is lack of security of permanently connected systems, esp PC's


Worm Operation worm phases like those of viruses:

dormant propagation

search for other systems to infect establish connection to target remote system replicate self onto remote system

triggering execution


Morris Worm best known classic worm released by Robert Morris in 1988 targeted Unix systems using several propagation techniques

simple password cracking of local pw file exploit bug in finger daemon exploit debug trapdoor in sendmail daemon

if any attack succeeds then replicated self


Recent Worm Attacks new spate of attacks from mid-2001 Code Red

exploited bug in MS IIS to penetrate & spread probes random IPs for systems running IIS had trigger time for denial-of-service attack 2nd wave infected 360000 servers in 14 hours

Code Red 2 had backdoor installed to allow remote control

Nimda used multiple infection mechanisms

email, shares, web client, IIS, Code Red 2 backdoor


Virus Countermeasures viral attacks exploit lack of integrity

control on systems to defend need to add such controls typically by one or more of:

prevention - block virus infection mechanism detection - of viruses in infected system reaction - restoring system to clean state


Anti-Virus Software first-generation

scanner uses virus signature to identify virus or change in length of programs

second-generation uses heuristic rules to spot viral infection or uses program checksums to spot changes

third-generation memory-resident programs identify virus by actions

fourth-generation packages with a variety of antivirus techniques eg scanning & activity traps, access-controls


Advanced Anti-Virus Techniques generic decryption

use CPU simulator to check program signature & behavior before actually running it

digital immune system (IBM) general purpose emulation & virus detection any virus entering org is captured, analyzed,

detection/shielding created for it, removed


Behavior-Blocking Software integrated with host O/S monitors program behavior in real-time

eg file access, disk format, executable mods, system settings changes, network access

for possibly malicious actions if detected can block, terminate, or seek ok

has advantage over scanners but malicious code runs before

detection



various malicious programs trapdoor, logic bomb, trojan horse, zombie viruses worms countermeasures



Wireless LAN SecurityRoad to 802.11i

Xiangyang Li


Contents Introduction Problem: 802.11b Not Secure! Wired Equivalent Privacy – WEP Types of Attacks 802.11b Proposed Solutions 802.1X Wi-Fi Protected Access (WPA) 802.11i: The Solution Conclusion


Introduction

Popular in offices, homes and public spaces (airport, coffee shop)

Most popular: 802.11b Example: Yahoo! DSL Wireless Kit Theoretical max @ 11Mbps Operate at 2.4GHz band DSSS/FSSS modulation – similar to CDMA phones


Introduction Standards: IEEE 802.11 Series

802.11b – 11Mbps @ 2.4GHz 802.11a – 54Mbps @ 5.7GHz band 802.11g – 54Mbps @ 2.4GHz band 802.1X – security add-on 802.11i – high security


Problem: 802.11b Not Secure! “No inherent security”

Wired Wireless media change was the objective Wired Equivalent Privacy (WEP)

The only “security” built into 802.11 Uses RC4 Stream Cipher – in a bad way Vulnerable to several types of attacks

Sometimes not even turned ON


Wired Equivalent Privacy – WEP RC4 stream cipher

Designed by Ron Rivest for RSA Security Very simple

Initialization Vector (IV) Shared Key

The issue is in the way RC4 is used IV (24 bits) reuse and fixed key Early versions used 40-bit key 128-bit mode effectively uses 104 bits


Wired Equivalent Privacy – WEP

RC4 Key Stream Encryption (source: http://mason.gmu.edu/~gharm/wireless.html)


Types of AttacksAttacks

ConfidentialityIntegrityAvailability


Types of Attacks Attacks on Confidentiality

Traffic Analysis Passive Eavesdropping

Very easy to do Active Eavesdropping Unauthorized Access


Types of Attacks

Attacks on Confidentiality and/or Integrity Man-In-The-Middle

Attacks on Integrity Session Hijacking Replay

Attacks on Availability Denial of Service


802.11b Proposed Solutions Virtual Private Network Closed Network

Through the use of SSID Ethernet MAC address control lists Replace RC4 with block cipher Don’t reuse IV Automatic Key Assignment


802.1X: Interim Solution Port-based authentication

Not specific to wireless networks Authentication servers

RADIUS Client authentication

EAP


802.1X Problems 802.1X still has problems

Extensible Authentication Protocol (EAP) One-way authentication

Attacks Man-in-Middle Session Hijacking


802.1X Proposed Solutions Per-packet authenticity and integrity

Lots of overhead Authenticity and integrity of EAPOL

messages Two-way authentication


Wi-Fi Protected Access (WPA) Addresses issues with WEP

Key management TKIP Key expansion

Message Integrity Check Software upgrade only Compatible with 802.1X Compatible with 802.11i


802.11i Finalized: June, 2004 Robust Security Network Wi-Fi Alliance: WPA2 Improvements made

Authentication enhanced Key Management created Data Transfer security enhanced


802.11i - Authentication Authentication Server Two-way authentication

Prevents man-in-the-middle attacks Master Key (MK) Pairwise Master Key (PMK)


802.11i – Key Management Key Types

Pairwise Transient Key Key Confirmation Key Key Encryption Key Group Transient Key Temporal Key


802.11i – Key Management

Source: http://csrc.nist.gov/wireless/S10_802.11i%20Overview-jw1.pdf

mailto:[email protected]


802.11i – Data Transfer CCMP

Long term solution – mandatory for 802.11i compliance

Latest AES encryption Requires hardware upgrades

WRAP Provided for early vendor support

TKIP Carried over from WPA


802.11i – Additional Enhancements Pre-authentication

Roaming clients Client Validation Password-to-key mappings Random number generation


Conclusion Basic 802.11b (with WEP)

Massive security holes Easily attacked

802.1X Good interim solution Allows use of existing hardware Can still be attacked


Conclusion Wi-Fi Protected Access

Allows use of existing hardware Compatible with 802.1X Compatible with 802.11i

802.11i May require hardware upgrades Very secure Nothing is ever guaranteed



IPsec

XiangYang Li


IP Security

If a secret piece of news is divulged by a spy before the time is ripe, he must be put to death, together with the man to whom the secret was told.—The Art of War, Sun Tzu


IP Security have considered some application

specific security mechanisms eg. S/MIME, PGP, Kerberos, SSL/HTTPS

however there are security concerns that cut across protocol layers

would like security implemented by the network for all applications


IPSec general IP Security mechanisms provides

authentication confidentiality key management

applicable to use over LANs, across public & private WANs, & for the Internet


IPSec Uses


Benefits of IPSec in a firewall/router provides strong

security to all traffic crossing the perimeter

is resistant to bypass is below transport layer, hence

transparent to applications can be transparent to end users can provide security for individual users

if desired


IP Security Architecture specification is quite complex defined in numerous RFC’s

incl. RFC 2401/2402/2406/2408 many others, grouped by category

mandatory in IPv6, optional in IPv4


IPSec Services Access control Connectionless integrity Data origin authentication Rejection of replayed packets

a form of partial sequence integrity Confidentiality (encryption) Limited traffic flow confidentiality


Security Associations a one-way relationship between sender

& receiver that affords security for traffic flow

defined by 3 parameters: Security Parameters Index (SPI) IP Destination Address Security Protocol Identifier

has a number of other parameters seq no, AH & EH info, lifetime etc

have a database of Security Associations


Authentication Header (AH) provides support for data integrity &

authentication of IP packets end system/router can authenticate user/app prevents address spoofing attacks by tracking sequence

numbers based on use of a MAC

HMAC-MD5-96 or HMAC-SHA-1-96 parties must share a secret key


Authentication Header


Transport & Tunnel Modes


Encapsulating Security Payload (ESP) provides message content

confidentiality & limited traffic flow confidentiality

can optionally provide the same authentication services as AH

supports range of ciphers, modes, padding incl. DES, Triple-DES, RC5, IDEA, CAST etc CBC most common pad to meet blocksize, for traffic flow


Encapsulating Security Payload


Transport vs Tunnel Mode ESP transport mode is used to encrypt &

optionally authenticate IP data data protected but header left in clear can do traffic analysis but is efficient good for ESP host to host traffic

tunnel mode encrypts entire IP packet add new header for next hop good for VPNs, gateway to gateway security


Combining Security Associations SA’s can implement either AH or ESP to implement both need to combine

SA’s form a security bundle

have 4 cases (see next)


Combining Security Associations


Key Management handles key generation & distribution typically need 2 pairs of keys

2 per direction for AH & ESP manual key management

sysadmin manually configures every system automated key management

automated system for on demand creation of keys for SA’s in large systems

has Oakley & ISAKMP elements


Oakley a key exchange protocol based on Diffie-Hellman key exchange adds features to address weaknesses

cookies, groups (global params), nonces, DH key exchange with authentication

can use arithmetic in prime fields or elliptic curve fields


ISAKMP Internet Security Association and Key

Management Protocol provides framework for key

management defines procedures and packet formats

to establish, negotiate, modify, & delete SAs

independent of key exchange protocol, encryption alg, & authentication method


ISAKMP



IPSec security framework AH ESP key management & Oakley/ISAKMP



FirewallsXiangYang Li


Firewalls

The function of a strong position is to make the forces holding it practically unassailable—On War, Carl Von Clausewitz


Introduction seen evolution of information systems now everyone want to be on the

Internet and to interconnect networks has persistent security concerns

can’t easily secure every system in org need "harm minimisation" a Firewall usually part of this


What is a Firewall? a choke point of control and monitoring interconnects networks with differing trust imposes restrictions on network services

only authorized traffic is allowed auditing and controlling access

can implement alarms for abnormal behavior is itself immune to penetration provides perimeter defence


Firewall Limitations cannot protect from attacks bypassing it

eg sneaker net, utility modems, trusted organisations, trusted services (eg SSL/SSH)

cannot protect against internal threats eg disgruntled employee

cannot protect against transfer of all virus infected programs or files because of huge range of O/S & file types


Firewalls – Packet Filters


Firewalls – Packet Filters simplest of components foundation of any firewall system examine each IP packet (no context)

and permit or deny according to rules hence restrict access to services (ports) possible default policies

that not expressly permitted is prohibited that not expressly prohibited is permitted


Firewalls – Packet Filters


Attacks on Packet Filters IP address spoofing

fake source address to be trusted add filters on router to block

source routing attacks attacker sets a route other than default block source routed packets

tiny fragment attacks split header info over several tiny packets either discard or reassemble before check


Firewalls – Stateful Packet Filters

examine each IP packet in context keeps tracks of client-server sessions checks each packet validly belongs to one

better able to detect bogus packets out of context


Firewalls - Application Level Gateway (or Proxy)


Firewalls - Application Level Gateway (or Proxy) use an application specific gateway /

proxy has full access to protocol

user requests service from proxy proxy validates request as legal then actions request and returns result to user

need separate proxies for each service some services naturally support proxying others are more problematic custom services generally not supported


Firewalls - Circuit Level Gateway


Firewalls - Circuit Level Gateway relays two TCP connections imposes security by limiting which such

connections are allowed once created usually relays traffic

without examining contents typically used when trust internal users

by allowing general outbound connections

SOCKS commonly used for this


Bastion Host highly secure host system potentially exposed to "hostile" elements hence is secured to withstand this may support 2 or more net connections may be trusted to enforce trusted

separation between network connections runs circuit / application level gateways or provides externally accessible services


Firewall Configurations


Access Control given system has identified a user determine what resources they can access general model is that of access matrix

with subject - active entity (user, process) object - passive entity (file or resource) access right – way object can be accessed

can decompose by columns as access control lists rows as capability tickets


Access Control Matrix


Trusted Computer Systems information security is increasingly important have varying degrees of sensitivity of

information cf military info classifications: confidential, secret etc

subjects (people or programs) have varying rights of access to objects (information)

want to consider ways of increasing confidence in systems to enforce these rights

known as multilevel security subjects have maximum & current security level objects have a fixed security level classification


Bell LaPadula (BLP) Model one of the most famous security models implemented as mandatory policies on system has two key policies: no read up (simple security property)

a subject can only read/write an object if the current security level of the subject dominates (>=) the classification of the object

no write down (*-property) a subject can only append/write to an object if the current security

level of the subject is dominated by (<=) the classification of the object


Reference Monitor


Evaluated Computer Systems governments can evaluate IT systems against a range of standards:

TCSEC, IPSEC and now Common Criteria define a number of “levels” of

evaluation with increasingly stringent checking

have published lists of evaluated products though aimed at government/defense use can be useful in industry also



firewalls types of firewalls configurations access control trusted systems



Third Editionby William Stallings

Lecture slides by Lawrie Brown


Intruders

They agreed that Graham should set the test for Charles Mabledene. It was neither more nor less than that Dragon should get Stern's code. If he had the 'in' at Utting which he claimed to have this should be possible, only loyalty to Moscow Centre would prevent it. If he got the key to the code he would prove his loyalty to London Central beyond a doubt.—Talking to Strange Men, Ruth Rendell


Intruders significant issue for networked systems

is hostile or unwanted access either via network or local can identify classes of intruders:

masquerader misfeasor clandestine user

varying levels of competence


Intruders clearly a growing publicized problem

from “Wily Hacker” in 1986/87 to clearly escalating CERT stats

may seem benign, but still cost resources

may use compromised system to launch other attacks


Intrusion Techniques aim to increase privileges on system basic attack methodology

target acquisition and information gathering initial access privilege escalation covering tracks

key goal often is to acquire passwords so then exercise access rights of owner


Password Guessing one of the most common attacks attacker knows a login (from email/web page etc) then attempts to guess password for it

try default passwords shipped with systems try all short passwords then try by searching dictionaries of common words intelligent searches try passwords associated with the user (variations on

names, birthday, phone, common words/interests) before exhaustively searching all possible passwords

check by login attempt or against stolen password file success depends on password chosen by user surveys show many users choose poorly


Password Capture another attack involves password

capture watching over shoulder as password is entered using a trojan horse program to collect monitoring an insecure network login (eg. telnet, FTP,

web, email) extracting recorded info after successful login (web

history/cache, last number dialed etc) using valid login/password can

impersonate user users need to be educated to use suitable

precautions/countermeasures


Intrusion Detection inevitably will have security failures so need also to detect intrusions so can

block if detected quickly act as deterrent collect info to improve security

assume intruder will behave differently to a legitimate user but will have imperfect distinction between


Approaches to Intrusion Detection statistical anomaly detection

threshold profile based

rule-based detection anomaly penetration identification


Audit Records fundamental tool for intrusion detection native audit records

part of all common multi-user O/S already present for use may not have info wanted in desired form

detection-specific audit records created specifically to collect wanted info at cost of additional overhead on system


Statistical Anomaly Detection threshold detection

count occurrences of specific event over time if exceed reasonable value assume intrusion alone is a crude & ineffective detector

profile based characterize past behavior of users detect significant deviations from this profile usually multi-parameter


Audit Record Analysis foundation of statistical approaches analyze records to get metrics over time

counter, gauge, interval timer, resource use use various tests on these to determine

if current behavior is acceptable mean & standard deviation, multivariate, markov

process, time series, operational key advantage is no prior knowledge

used


Rule-Based Intrusion Detection observe events on system & apply rules

to decide if activity is suspicious or not rule-based anomaly detection

analyze historical audit records to identify usage patterns & auto-generate rules for them

then observe current behavior & match against rules to see if conforms

like statistical anomaly detection does not require prior knowledge of security flaws


Rule-Based Intrusion Detection rule-based penetration identification

uses expert systems technology with rules identifying known penetration, weakness

patterns, or suspicious behavior rules usually machine & O/S specific rules are generated by experts who interview & codify

knowledge of security admins quality depends on how well this is done compare audit records or states against rules


Base-Rate Fallacy practically an intrusion detection

system needs to detect a substantial percentage of intrusions with few false alarms if too few intrusions detected -> false security if too many false alarms -> ignore / waste time

this is very hard to do existing systems seem not to have a

good record


Distributed Intrusion Detection traditional focus is on single systems but typically have networked systems more effective defense has these

working together to detect intrusions issues

dealing with varying audit record formats integrity & confidentiality of networked data centralized or decentralized architecture


Distributed Intrusion Detection - Architecture


Distributed Intrusion Detection – Agent Implementation


Honeypots decoy systems to lure attackers

away from accessing critical systems to collect information of their activities to encourage attacker to stay on system so

administrator can respond are filled with fabricated information instrumented to collect detailed

information on attackers activities may be single or multiple networked

systems


Password Management front-line defense against intruders users supply both:

login – determines privileges of that user password – to identify them

passwords often stored encrypted Unix uses multiple DES (variant with salt) more recent systems use crypto hash function


Managing Passwords need policies and good user education ensure every account has a default password ensure users change the default passwords to

something they can remember protect password file from general access set technical policies to enforce good

passwords minimum length (>6) require a mix of upper & lower case letters, numbers, punctuation block know dictionary words


Managing Passwords may reactively run password guessing tools

note that good dictionaries exist for almost any language/interest group

may enforce periodic changing of passwords

have system monitor failed login attempts, & lockout account if see too many in a short period

do need to educate users and get support balance requirements with user acceptance be aware of social engineering attacks


Proactive Password Checking most promising approach to improving

password security allow users to select own password but have system verify it is acceptable

simple rule enforcement (see previous slide) compare against dictionary of bad passwords use algorithmic (markov model or bloom filter) to

detect poor choices



problem of intrusion intrusion detection (statistical & rule-based) password management

Chapter 9 slides, 3rdedition - University of North Floridawkloster/crypto/CryptographyNetSecurity...

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