Short Overview of Cryptography (Lecture II) John C. Mitchell Stanford University.
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Transcript of Short Overview of Cryptography (Lecture II) John C. Mitchell Stanford University.
Some philosophy (my opinions)
Do something useful with your life Computers can do many things Have fun! Do something that matters
Learn something about the problems you solve If you are going to do graphics, study visual art If you work on computational biology, try to learn a
little organic chemistry If we are going to analyze security protocols, we
should learn a few things about cryptography
Some security objectives
Secrecy Info not revealed
Authentication Know identity of
individual or site Data integrity
Msg not altered Message
Authentication Know source of msg
Receipt Know msg received
Access control Revocation Anonymity Non-repudiation
Some Basic Concepts
Encryption scheme: encrypt(plaintext,key) decrypt(ciphertext,key )
Secret vs. public keyPublic key: publishing key does not reveal key
Secret key: more efficient; can have key = key Hash function
map long text to short hash key; ideally, no collision
Signature schemepublic key and private key provide “authentication”
-1
-1
-1
-1
Cryptosystem
A cryptosystem consists of five parts A set P of plaintexts A set C of ciphertexts A set K of keys A pair of functions encrypt: KPC decrypt: KCP such that for every key kK and plaintext pP decrypt(k, encrypt(k, p)) = p
Good def’n for now, but doesn’t include key generation or prob encryption.
Primitive Example: Shift Cipher
Shift letters using mod 26 arithmetic Set P of plaintexts {a, b, c, … , x, y, z} Set C of ciphertexts {a, b, c, … , x, y, z} Set K of keys {1, 2, 3, … , 25} Encryption and decryption functions encrypt(key, letter) = letter + key (mod 26)
decrypt(key, letter) = letter - key (mod 26)
Example encrypt(3, marktoberdorf) = pdunwrehugrui
Evaluation of Shift Cipher
Advantages Easy to encrypt, decrypt Ciphertext does look garbled
Disadvantages Not very good for long sequences of English words
Few keys -- only 26 possibilities Regular pattern
• encrypt(key,e) is same for all occurrences of letter e• can use letter-frequency tables, etc
Letter frequency in English
Five frequency groups [Beker and Piper]
E has probability 0.12 TAOINSHR have probability 0.06 - 0.09 DL have probability ~ 0.04CUMWFGYPB have probability 0.015 - 0.028 VKJXQZ have probability < 0.01
Possible to break many letter-to-letter substitution ciphers.
One-time Pad
Secret-key encryption scheme (symmetric) Encrypt plaintext by xor with sequence of bits Decrypt ciphertext by xor with same bit sequence
Scheme for pad of length n Set P of plaintexts: all n-bit sequences Set C of ciphertexts: all n-bit sequences Set K of keys: all n-bit sequences Encryption and decryption functions encrypt(key, text) = key text (bit-by-bit)
decrypt(key, text) = key text (bit-by-bit)
Example one-time pad
10
01010
PlaintextKey Ciphertext KeyPlaintext
11
01001
01
00011
11
01001
10
01010
= =
01
00011
Ciphertext
Evaluation of one-time pad
Advantages Easy to compute encrypt, decrypt from key, text As hard to break as possible
This is an information-theoretically secure cipher Given ciphertext, all possible plaintexts are equally likely,
assuming that key is chosen randomly
Disadvantage Key is as long as the plaintext
How does sender get key to receiver securely?
Idea can be combined with pseudo-random generators ...
What is a “secure” cryptosystem?
Idea If an enemy intercepts your ciphertext, cannot
recover plaintext Issues in making this precise
What else might your enemy know? The kind of encryption function you are using Some plaintext-ciphertext pairs from last year Some information about how you choose keys
What do we mean by “cannot recover plaintext” ? Ciphertext contains no information about plaintext No efficient computation could make a reasonable guess
Information-theoretic Security
Remember conditional probability... Random variables X, Y, … Conditional probability P(X=x|Y=y)
Probability that X takes value x, given that Y=y
Apply to plaintext, ciphertext Cryptosystem is info-theoretically secure if
P(Plaintext=p | Ciphertext=c) = P(Plaintext=p)
Ciphertext gives no advantage in guessing the plaintext.
Data Encryption Standard
Developed at IBM, widely used Regular structure
Permute input bits Repeat application of a certain function Apply inverse permutation to produce output
Appears to work well in practice Efficient to encrypt, decrypt Not provably secure
One round of DES
Function f(Ri-1 ,Ki) Expand Ri-1 and XOR w/ Ki
Divide into 8 6-bit blocks Apply “S-box” table-lookup
functions to each block Permute resulting bits
Ki is permutation of key K Invertible if K known
See Biham and Shamir for analysis
L i-1 R i-1
R iL i
f
K i
Properties of DES
Not a simple mathematical function Difficult to analyze All operations are linear except “S-boxes”
Security depends on “magic” S-box functions These were designed secretly by NSA
• No S-box is a linear function• Changing one input bit changes two output bits
Efficient to compute Combination of bit operations and table lookup
Differential cryptanalysis of DES Can break 8-round DES, but not 16-round DES (yet)
Complexity-based Cryptography
Some computational problems provably hard Undecidability of halting problem Presburger arithmetic is non-elementary Commutative semi-groups require exponential space
Some problems are believed intractable NP-complete optimization problems
Traveling salesman as hard as any problem in NP No known polynomial time algorithm, in spite of effort
Factoring is not believed to be poly-time Not NP-complete, but many years of effort
Still, useful to relate crypto to standard problems
Review: Complexity Classes
Answer in polynomial space may need exhaustive search
If yes, can guess and check in polynomial time
Answer in polynomial time, with high probability
Answer in polynomial time compute answer directly
P
BPP
NP
PSpace
easy
hard
One-way functions
A function f is one-way if it is Easy to compute f(x), given x Hard to compute x, given f(x), for most x
Examples (we believe) f(x) = divide bits, x = yz, and multiply f(x)=y*z f(x) = 3x mod p, where p is prime f(x) = x3 mod pq, where p,q are primes with |p|
=|q|
Easy and hard (more precisely)
For any finite f, can build a table and invert f Measure “hardness” using classes of functions
Want this to be hard as a function of choice of f
A class {fa :Df Rf | aA} is one-way if Efficient algorithm for fa (x), given a, x
No efficient alg computes x, given a, fa (x)
where we assume Df , Rf finite and measure running time as a function of |a|
One-way trapdoor
A function f is one-way trapdoor if Easy to compute f(x), given x Hard to compute x, given f(x), for most x There is extra “trapdoor” information making it
easy to compute x from f(x)
Example (we believe) f(x) = x3 mod pq, where p,q are primes with |p|
=|q| Compute cube root using (p-1)*(q-1)
Group theory for RSA
Group G = G, , e, ( )-1 Set of elements with
associative “multiplication” identity e with ex = xe = x inverse ( )-1 with xx-1 = x-1 x = e
Cyclic group Group G = G, , e, ( )-1 with
G = { g0, g1 , g2 , ... , gk = g0} element g is called a generator of G number of distinct elements if called the order of group
Number theory for RSA
Group Zn* of integers relatively prime to n
multiplication mod n is associative operation 1 is identity x-1 computed by Euclidean algorithm for gcd order of group is (n) = | { k<n | gcd(k,n) =1 } |
What if x not relatively prime to n? Can have zero divisors, no multiplicative inverse
If y divides x and n, then yi=x, yj=n and therefore xj = yij 0 mod n
Only numbers relatively prime to n form group
RSA Encryption
Let p, q be two distinct primes and let n=p*q Encryption, decryption based on group Zn
* For n=p*q product of primes, (n) = (p-1)*(q-1)
Proof: (p-1)*(q-1) = p*q - p - q + 1
Key pair: a, b with ab 1 mod (n) Encrypt(x) = xa mod n Decrypt(y) = yb mod n Since ab 1 mod (n), have xab x mod n
Proof: if gcd(x,n) = 1, then by general group theory, otherwise use “Chinese remainder theorem”.
How well does this work?
Can generate modulus, keys fairly efficiently Efficient rand algorithms for generating primes p,q
May fail, but with low probability
Given primes p,q easy to compute n=p*q and (n) Choose a randomly with gcd(a, (n))=1 Compute b = a-1 mod (n) by Euclidean algorithm
Public key n, a does not reveal b This is not proven, but believed
But if n can be factored, all is lost ...
Message integrity
Theoretically, a weak point encrypt(k*m) = (k*m)e = ke * me
= encrypt(k)*encrypt(m) This leads to “chosen ciphertext” form of attack
If someone will decrypt new messages, then can trick them into decrypting m by asking for decrypt(ke *m)
Implementations reflect this problem “The PKCS#1 … RSA encryption is intended
primarily to provide confidentiality. … It is not intended to provide integrity.” RSA Lab. Bulletin
Recall security objectives
Secrecy Info not revealed
Authentication Know identity of
individual or site Data integrity
Msg not altered Message
Authentication Know source of msg
Receipt Know msg received
Access control Revocation Anonymity Non-repudiation
Digital Signatures
Public-key encryption Alice publishes encryption key Anyone can send encrypted message Only Alice can decrypt messages with this key
Digital signature scheme Alice publishes key for verifying signatures Anyone can check a message signed by Alice Only Alice can send signed messages
RSA Signature Scheme
Publish decryption instead of encryption key Alice publishes decryption key Anyone can decrypt a message encrypted by
Alice Only Alice can send encrypt messages
In more detail, Alice generates primes p, q and key pair a, b Sign(x) = xa mod n Verify(y) = yb mod n Since ab 1 mod (n), have xab x mod n
Cryptographic hash functions
Function h with two main properties Map arbitrary strings to strings of fixed length Given h(x), impractical to find y with h(y)=h(x)
Variety of uses More efficient digital signatures
Sign hash of message instead of entire message
Data integrity Compute and store hash of some data Check later by recomputing hash and comparing
Keyed hash fctns provide message authentication ???
Iterated hash functions
Repeat use of block cipher (like DES, …) Pad input to some multiple of block length Iterate a length-reducing function f
f : 22k -> 2k reduces bits by 2 Repeat h0= some seed
hi+1 = f(hi, xi)
Some final function g completes calculation
Pad to x=x1x2 …xk
f
g
xi
f(xi-1)
x
General Basis for Cryptography
Cyclic group with one-way properties multiplication, inverse easy to compute discrete log a, an n not in O(log2 |G|)
Note: randomized algorithm in O(sqrt |G|)
Examples Integers modulo prime p Elliptic curve groups
Important: complexity depends on group presentation
Public-Key Cryptography [ElGamal]
Public encryption key: g, ga Private decryption key: a Encryption function
Choose random b [2, |G|-1] Send encrypt(msg) = gb , gab msg
Decryption Compute g-ab = ((gb)a) -1
Decrypt g-ab gab msg
This is classical algorithm; better security with hash(gab) msg