Chapter 2 Mathematical Foundations Cryptography-Application and Practice Harbin Institute of...

Chapter 2 Mathematical Foundations

Cryptography-Application and PracticeHarbin Institute of TechnologySchool of Computer Science and Technology

Zhijun Lihttpcsthiteducn~lizhijun

lizhijun_oshiteducn

Zhijun Li S1034040Autumn08HIT 2

Outline

bull Number Theoryndash Prime Numbersndash Modular Arithmetic

bull Algebraic Structures in Computing

bull Security Foundations of Cryptographyndash Information Theory ndash Complexity of Computingndash Semantic Security


Why Prime Numbers

bull Why numbersndash Encryption Mathematical Transformationndash Information Discrete eventsndash So Numbersndash Example A0 B1 hellip Z25

bull Why prime numbersndash Prime numbers is the foundation of numbers


Divisibility

bull Definition b is divisible by a if and only if a divides b without a remainder (or b=ma) denoted as a|bndash Example 3|6 5|20 -5|25 3|8

bull Properties ndash a|b b|c a|c

bull Proof a|bb=ma b|cc=nb=mna a|c

ndash a|b b|a |a|=|b|ndash a|b a|c a|bxcy

bull Proof


Prime and Composite Numbers

bull Definition An integer p is a prime if p is divisible only by +p-p1 and -1ndash Example 2357111317hellip101hellip

bull Denotation The set of all primes is P

bull Definition An integer n that is not a prime is called a composite numberndash Example 4625900hellip


bull Theorem(Fundamental Theorem of Arithmetic)

nN n= p1e1 p2

e2hellippkek ( or piP pei)

where ep is the exponent of the prime factor p

bull Note the result of factorization is uniquebull Example 84=2237

Factorizing in Product of Primes


Primes Amount

bull Theorem There are infinitely many primes

bull Proof (by contradiction) Assume Pmax build a number N is

N=P1P2hellipPmax+1

N can not be represented as P1e

1 P2e

2 hellip Pmaxe

max Therefore N is a new prime


Primes Density

(x) is a function gives the number of all primes contained in lt1xgt

(x)xln(x)

bull Gauss (x)xln(x)bull Legendre (x)x(ln(x)-108366) bull Hadamard de la Vallee-Poussion

((x)ln(x)x)x = 1


Greatest Common Divisor

bull Definition the greatest common divisor(GCD) is the number c

c=gcd(ab)=maxd|d|ad|bbull Properties

ndash d|n1d|n2hellipd|nk d|gcd(n1n2hellipnk)ndash gcd(ab) = gcd(b a mod b)

bull a mod b = r = a-qb 0ltrltb bull Hint a|b b|a a = b

bull Definition if gcd(ab)=1 then a and b are relatively prime(coprime)


Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63

bull Complexityndash Need factoring the a and b ndash T(n) = O(cO(n))


Euclidean Algorithm Example

1970 = 1 x 1066 + 904 gcd(1066 904)1066 = 1 x 904 + 162 gcd(904 162)904 = 5 x 162 + 94 gcd(162 94)162 = 1 x 94 + 68 gcd(94 68)94 = 1 x 68 + 26 gcd(68 26)68 = 2 x 26 + 16 gcd(26 16)26 = 1 x 16 + 10 gcd(16 10)16 = 1 x 10 + 6 gcd(10 6)10 = 1 x 6 + 4 gcd(6 4)6 = 1 x 4 + 2 gcd(4 2)4 = 2 x 2 + 0 gcd(2 0)

GCD(19701066)


Euclidean Algorithm

bull Step 1 r0 =a and r1 =b

bull Step 2 r0 =q1r1+ r2

r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn

until rn=0 and rn-1 0

bull Step 3 rn-1 = gcd(ab)


Euclidean Algorithm Proof

bull rn = 0 rn-1| rn-2

rn-1| rn-3 hellip rn-1| a and rn-1| b

rn-1| gcd(ab)

bull gcd(ab)| r0 and gcd(ab)|r1

gcd(ab) | r0 ndash q1r1 gcd(ab)| r2

hellip gcd(ab) | rn-1

bull rn-1| gcd(ab) gcd(ab) | rn-1 gcd(ab) = rn-1


Euclidean Algorithm Complexity

bull Time Complexityndash Iteration

bull Each rigt1 iteration = O(log2a)

ndash Division

bull O(log22a)

ndash T(n)=O(log32n)


Modular Arithmetic

bull Why modular arithmetic

01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n

+ addition multiplication

Problems 1 the set of plaintext (and ciphertext) finite 2 how to define +-x operations in finite


Modular Operation

bull Definition a mod n(modulo operator) is the remainder when a is divided by n

bull a mod n is r a=qn+r (0ltrltn)


Congruence Modulo n

bull Definition Integers a and b are congruence modulo nndash If a mod n=b mod nndash Denoted as a b mod nndash Example 100 34 mod 11ndash a b mod n a = b+kn n|(a-b)


Congruence Properties

bull Propertiesndash a b (mod n) c d (mod n)

a+c b+d (mod n)

ac bd (mod n)ndash Example

10 1 mod 9 102 1 mod 9 hellip

a = am10m+hellip+a110+a0 am+hellip+a1+a0 mod 9

ndash a b (mod n) d|n a b (mod d)


Congruence Relation

bull Congruence relation is an equivalence relationndash Reflexive

bull a andash Symmetric

bull a b b a ndash Transitive

bull a b b c a c

bull Equivalence relation Partition


Example Integers Modulo 7

bull Integers can always write a = qn + r (0 r n-1) -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 congruence

Z7 residues modulo 7


Set of Residues

bull Definition if a b mod n and bltn b is called a residue of a modulo n

bull The complete set of residues modulo nndash The set r1r2helliprn if for every integer b exactly one ri i

n the set satisfies bri mod n

bull The reduced set of residues(RSR)ndash A subset of the complete set of residues that each ele

ment is relatively prime to n


Set of Residues Example

bull N=10ndash The complete set of residues is 012hellip9ndash The reduced set of residues is 1379

bull N=p (p is a prime)ndash The complete set of residues is 012hellipp-1ndash The reduced set of residues is 12hellipp-1


Modular Arithmetic

bull We can define the modular arithmetic in the set of integers Zn=0 1 2 hellip n-1

bull Under normal arithmetic (+ )ndash [(a mod n) + (b mod n)] mod n = (a+b) mod n

bull Proof Let a=q1n+r1 b=q2n+r2

bull (r1+r2) mod n = (q1n+r1+q2n+r2) mod n = (r1+r2) mod n

ndash [(a mod n) (b mod n)] mod n = (ab) mod n

bull (+ ) (-)


Modular Arithmetic Examplebull (Z8=0 1 2 hellip 7 +)

What


Modular Arithmetic Examplebull (Z8=0 1 2 hellip 7 )

NOT all elements in Z8


Modular Arithmetic Remark

bull Additive Inverse Always Existsndash (a+(-a)) = 0 mod n -a = n-andash if (a+b) (a+c) mod n then b c mod n

bull ((-a)+a+b) ((-a)+a+c) mod n

bull Multiplicative Inverse NOT Always Existsndash Example 6 in Z8

ndash When


Multiplicative Inverse in Modular

bull Definition a-1 mod n is the multiplicative inverse of a12hellipn-1 when ax1 mod n

bull Theorem If and only if gcd(an)=1 then the a-1 mod n exists

bull Lemma If gcd(an)=1 then ai aj mod n for all 0iltjltn (i j)ndash Proof assume ai aj mod n n|a(i-j) N|i-j i-j=

0


Multiplicative Inverse Theorem

bull Proof

ndash gcd(an)=1 a1hellipn-1 mod n is the permutation of 1hellipn-1

ndash So there exists only an i that ai 1 mod nndash Therefore i is a-1 mod n

ndash Suppose a-1 exists call it xndash ax 1 (mod n) and ax + yn = 1 for some integer yndash gcd(a n)=1 (gcd(an)|ax+yn gcd(an)|1)


Eulerrsquos phi Function

bull Eulerrsquos ldquophirdquo function (n)ndash Is the number of invertible elements of Zn

ndash Is the number of elements in the reduced set of residues for modulo n

(n) calculation


Chinese Remainder Theorem

bull Theorem let n1 n2 hellip nk be integers st gcd(ninj)=1(ij)

x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk

There exists a unique solution of x modulo n=n1

n2hellipnk


Proof of CRT

bull Consider a one-to-one map Zn Zn1

Zn2 hellipZnk

(x) = (x mod n1 x mod n2 hellip x mod nk)ndash CRT is find -1 = (a1 a2 hellip ak)

(a1 a2 hellip ak) = i=1kaimiyi mod n

ndash mi = nni gcd(mini)=11ik ndash yi = mi

-1 mod ni1ikndash mi 0 mod nj for ji aimiyi 0 mod nj for ji ndash yj mj

-1 mod nj mjyj1 mod nj ajmjyjaj mod nj

(a1 a2 hellip ak) mod nj = aj


Finding Eulerrsquos phi Function

bull Theorem if gcd(mn)=1 then (mn)=(m)(n)

bull Proof ndash A number x mod mn is represented uniquely by the p

air (a b) (x mod m x mod n) (CRT)

ndash A number is relatively prime to both m and n iff a and b are relatively prime to m and n respectively

ndash So (mn)=(m)(n)



(n) calculation for n = i=1k pi

ei

(p) = p-1 (pe) = pe ndash pe-1

ndash The numbers with gcd(kp pe) = p when k=1hellippe-1-1 ndash So (pe) = pe -1ndash pe-1 -1

( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem

bull Theorem If gcd(an)=1 then a(n) 1 mod n

bull Proof 1 Let R= r1 r2 hellip r(n)be a residue set of residues

modulo n

2 ar1 ar2 hellip ar(n) is a permutation of R (gcd(arin)=1)

3 i=1(n) ri = i=1

(n) ari a(n) i=1(n) ri mod n

4 gcd( i=1(n) rin)=1 Hence a(n) 1 mod n


Fermatrsquos Little Theorem

bull For Eulerrsquos Theorem

Let n is p and p is a prime

Because for all 0ltaltp gcd(ap)=1 then

ap-1 1 mod p


bull Search through 1hellipn-1 until an a-1 is found such that aa-1 1 (mod n)ndash T(n)=O(n)

bull Compute a-1 = a(n)-1 mod nndash Find (n) factoring nndash T(n)=O(na)

bull Use the Extended Euclidean Algorithmndash T(n)=O(logan)

Finding a-1 mod n


From Euclidean Algorithm

bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1

bull r2 = r0-q1r1 = s2r0+t2r1

bull r3 = r1-q2r2 = r1-q2(r0-q1r1) = s3r0+t3r1

helliphellipbull rn-1 = sn-1r0+tn-1r1

bull rn-1=gcd(r0 r1)bull sn-1r0+tn-1r1 =1 tn-1r1 1 mod r0 tn-1 r1

-1 mod r0

bull Find tn-1


si and ti

bull r0 = s0r0+t0r1 s0=1 t0=0bull r1 = s1r0+t1r1 s1=0 t1=1 helliphellipbull ri = ri-2-qi-1ri-1 = si-2r0+ti-2r1-qi-1si-1r0-qi-1ti-1r1 = sir0+tir1

si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1

bull Only ti is needed bull ri = sir0+tir1 = tir1 mod r0


Step 1 r0 =n and r1 =a

Step 2 r0 =q1r1+ r2 r2 =r0 - q1r1 =-q1r1 mod n let x2= -q1 then r2 =x2r1 mod n

r1 =q2r2+ r3 r3 =r1 ndash q2r2 =(1-x2q2 ) r1 mod n

let x3= 1-x2q2 then r3 =x3r1 mod n helliphellip

rn-3 = qn-2rn-2+ rn-1 rn-1 =rn-3 ndash qn-2rn-2 mod n let xn-1= xn-3-xn-2qn-2 then rn-1 =xn-1r1 mod n Now rn-1=1

Step 3 Result is xn-2 =a-1 mod n

Extended Euclidean Algorithm


Outline





bull Why Structuresndash Cryptographic transforms are normally in

some structures

bull Definition It is the collection of a set (S) with one (more) operations () which performed on the elements of the Sndash Different algebraic structures is ltS gt satisfying

some specific conditions

Algebraic Structures


Group

bull Definition A group G= ltS gt is an algebraic structure that satisfies the following conditions

ndash G1 abS abS (closure rule) ndash G2 abcS (ab)c = a(bc) ndash G3 (eS)aS ae = ea = a ndash G4 (aS)a-1S aa-1 = a-1a = e ndash [G5 Abelian group] ab ab = ba


Group Example

bull ltZ +gt e=0 a-1 = - a

bull ltZn +gt (ZN=01hellipn-1 lsquo+rsquo is addition modulo n) e=0 a-1 = n ndash a

bull ltZ2 gt is also a group

bull ltZp gt (Zp

=1hellipp-1 lsquorsquo is multiplicative)


Group Properties

bull Definition The order of a finite group is the number of the elements in the group

bull Definition A group G1=ltS1 gt is a subgroup of the group G=ltS gt if S1 S

bull Lagrange Theorem The order of a subgroup H of a finite group G divides the order of G


Lagrange Theorem Proof

(1) Definition of Coset Let G be a group and H is the subgroup of G for aG set aH=ah | hH is called a coset of H

(2) abG aH = bH or aHbH=(3) Each element of G is in some coset(eH) So a

ll coset is a partition of G

(4) the order of aH = the order of H

So the order of H divides the order of G


The Order of Group Element

bull Definition The order of group element a is defined as the least positive integer n that

an = a a hellip a =e

bull The order of a is denoted as ord(a)

bull Corollary Suppose G is a finite group whose order is n and aG then ord(a)|n


ord(a)|n Proof

bull Proofndash For group G and aG the set a a2 hellip aord

(a) and the operator of G forms a subgroup of G

aord(a)=e (ai)-1=a(ord(a)-i)

ndash This subgroup is denoted as ltagt


The Generator of Group

bull Definition A group whose all elements can be generated by a single element g g is called the generator of the group

bull Examplendash Considering the ltZ7

gtndash 2 generates 21=2 22=4 23=8equiv1 mod 724=16equiv2 mod

7hellipbull lt124gt subgroup

ndash 3 generates 31=3 32equiv2 33equiv634equiv435equiv536equiv1hellipbull 3 is the generator


Ringbull Definition A ring R=ltS+gt (+ is addition is

multiplication) such thatndash R1 abS a+bS abS (closure rule) ndash R2 ltS+gt is an additive Abelian groupndash R3 abcS (ab)c = a(bc) ndash R4 abcS a(b+c) = ab + ac

bull Examplendash The set of even integers with + and are ringndash pq are two primes n=pq Zn=01hellipn-1 with additio

n and multiplication mod n is a ring


Field

bull Definition Field F is ltS+gt withndash F1 ltS+gt is a commutative ring (ab=ba)ndash F2 Exist identity element 1 with respected to multiplic

ation a1 = 1 a = andash F3 Any nonzero element aS a has its unique invers

e a-1 a = a-1 a = 1

bull Example ndash Zn=lt01hellipn-1+gt is a field if n is a prime


Field +-x

Relationship

Ring +-x

+-Group

Used In Cryptography


Several Important Structures

bull Zn

bull Zn

bull Zp

bull Polynomial Rings

bull Galois Field F2n


Zn

bull Zn is the ldquoringrdquo of integers modulo n

ndash Zn=0 1 2 hellipn-1

ndash Normal addition modulo nndash Normal multiplication modulo n

bull Example

ndash Z26 for English

ndash Z256 for ASCII


Zn Properties

bull Additive and multiplicative identities in Zn

ndash Additive identity is 0 mod nndash Multiplicative identity is 1 mod n

bull Additive inverse Yesndash A number y such that x + y = x for all x in Zn

bull Multiplicative inverse Not for all xndash A number y such that xy = 1 for all x in Zn

ndash The elements in Zn have inverse x|gcd(xn)=1


Zn

bull Zn is a ldquogrouprdquo of integers modulo n

ndash Zn=x| gcd(xn)=1

ndash |Zn|=(n)

ndash Normal multiplication modulo n

bull Zn may not a group over +

ndash Example n=20 35 Zn but 3+5=8 Zn

bull So Zn may not a field


Zp

bull Zp is the ldquofieldrdquo of integers modulo n

ndash Zp=012hellipp-1

ndash Normal addition modulo pndash Normal multiplication modulo p

bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring

bull Definition If R is a commutative ring then a polynomial in x over the ring R is an expression of the form f(x)=anxn+an-1xn-1+ hellip + a1x+a0 where each aiR and n0

bull Note If R is a commutative ring the polynomial ring R[x] is the ring formed by the set of all polynomials in x having coefficients from R


Polynomial Ring Example

bull Example Let f(x)=x3+x+1 and g(x)=x2+x be elements in Z2[x]

ndash f(x)+g(x) = x3+x2+1ndash f(x)-g(x) = f(x)+g(x) = x3+x2+1ndash f(x)g(x) = x5+x4+x3+x


Set of Residues Modulo p(x)

bull Definition A set of residues modulo p(x) of R[x] is the set of R[x] mod p(x)ndash p(x) is a irreducible polynomial over R

bull A set of residues modulo p(x) of R[x] is denoted as R[x]p(x)


Congruent Polynomials

bull Definition a(x)b(x) are congruent modulo p(x) or a(x) b(x) mod p(x) iff p(x)|(a(x)-b(x))

bull Example

ndash Z5[x] 3x3+4x+4 x+4 mod x2+1bull 3x3+4x+4 ndash (x+4) = 3x3+3x = 3x(x2+1)

bull Definition If a(x)b(x) 1 mod p(x) then b(x)=a-1(x) mod p(x)


Irreducible Polynomials

bull Definition if a(x)=q(x)b(x) then a(x) is divisible by b(x) and denoted as b(x)|a(x)

bull Definition a polynomial a(x) is irreducible over a R if there is NO polynomials b(x)R[x] with deg b(x) lt deg a(x) b(x)a(x)


Example [Z2(x)]x3+x+1

bull S=01xx+1x2x2+1x2+xx2+x+1 bull p(x)=x3+x+1

bull For any f(x)Z2[x] f(x) mod p(x)S

bull Examplendash x4+1=xx3+1 mod p(x)=x(x+1)+1=x2+x+1

bull Note R[x]p(x) when p(x) is irreducible is the all polynomials in f(x)R[x] with deg f(x) lt deg p(x)


[Z2(x)]x3+x+1 Properties

bull S is group over addition in Rndash 0 is e and the inverse of aS is a itselfndash Example (x2+1) + (x2+1) = 0

bull S is group over multiplication mod p(x)ndash 1 is e and the inverse of aS is existndash Example (x2+1)x = x3+x x+1+x mod x3+x+1 = 1 ndash Note Can use Extended Euclidean Algorithm to find

bull [Z2(x)]x3+x+1 is a Field


Galois Field

bull Definition Galois field = (S + ) ndash S = F[x]p(x) ndash F is field ZN (N is prime)ndash + is addition in F is multiplication modulo p(x)ndash p(x) is irreducible

bull The Galois field over ZN[x]p(x) is denoted as GF(Nn) when n = deg p(x)


Why Galois Field

bull For a 8-bit Block

ndash Z256 =01hellip255 is not a field

ndash Z251 =01hellip250 is a fieldbull The largest prime lt 256 is 251bull 251hellip255 are wasted

bull Is that possible to find a field for Z256

ndash Yes ndash Galois Field


Galois Field for Z256

bull Build a mapping from n-bit block to polynomialsbull Example 8-bit block

bull It is Z2[x]p(x) where deg p(x) = 8

bull Note many cryptographic designs use binary Galois fields GF(2n)

10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0


Galois Field Example AES

s in AES uses GF(28) = Z2[x]p(x) where p(x) = x8 + x4 + x3 + x + 1

bull Examplendash 01010011 x6+x4+x+1ndash (01010011)-1 (x6+x4+x+1)-1 mod p(x) = x7+x6

+x3+x 11001010


bull Examplebull 0287= bull 87 = 10000111 f(x) = x7+x2+x+1bull 02 = 00000010 f(x) = xbull 0287 xf(x) = x(x7+x2+x+1) mod p(x)

= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline





Mathematical Model for Ciphers

bull P=ML = m=(m1m2hellipmL)|miM ndash M=ai i=12hellipN p(ai)0 i=1

Np(ai)=1

bull K= BR = b=(b1b2hellipbR)|biB ndash B=bi i=12hellipS p(bi)0 i=1

Sp(bi)=1

bull C=c|c=ek(m) kK mM

mcSender Encryption Decryption Receiver

Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon

Shannon was the person who saw that the binary digit was the fundamental element in all of communication That was really his discovery and from it the whole communications revolution has sprung

R G GallagerClaude Shannon

1916-2001


Probability Distribution of C

bull Pr[C = c] = kKcC(k) Pr[K = k]Pr[m = dk(c)]

ndash C(k) = ek(m)|mP

bull Pr[C = c|P = m] = kKm=dk(c) Pr[K = k]

bull Pr[P = m|C = c]

= Pr[C = c|P = m]Pr[P = m]Pr[C = c]

= (Pr(m)k|m=dk(c)Pr(k))kKcC(k) Pr(k)Pr(dk(c))


Shannon Security

bull Basic Idea Ciphertext should provide no ldquoinformationrdquo about Plaintextndash How to define the ldquoinformationrdquo

bull Definition An encryption scheme is Shannon secure if probability distribution of P

cm Pr[P=m] = Pr[P=m|C=c]bull Also say such a scheme has perfect secrecy


Perfect Secrecy Theorem

bull Suppose (P C K E D) is a cryptosystem where |K| = |P| = |C|

bull Then the cryptosystem provides perfect secrecy if and only if ndash every key is used with equal probability 1|K|ndash and x P and y C there is a unique key k such t

hat ek(x) = y


Proof of

kKx=dk(y) Pr[K = k]Pr[x = x]

bull Pr[x = x|y = y]=

kKyC(k) Pr[K = k]Pr[x = dk(y)]

kKx=dk(y) Pr[K = k]=1|K|


=1|K|kKyC(k) Pr[x = dk(y)]=1|K|xPPr[x=x]=1|K|


Proof of

bull Pr[y] gt 0bull Pr[x|y]=Pr[x] Pr[y|x]=Pr[y]

bull Fixed xP yC we have Pr[y|x]=Pr[y]gt0 thus at least exists k that satisfies ek(x)=y

bull |ek(x)|kK|=|C|=|K|

ndash There is not ek1 ek2 and ek1(x) = ek2(x)

ndash There is a unique key k such that ek(x) = y


Proof of

bull Let n=|K| and P=xi|1in

bull Given a fixed y assume k1k2hellipkn where eki (xi)

=y

bull Pr[xi|y]=Pr[y|xi]Pr[xi]|Pr[y]=Pr[K=ki]Pr[xi]|Pr[y]

bull Pr[xi|y]=Pr[xi] Pr[K=ki]=Pr[y] ndash Every key is used with equal probability 1|K|


ldquoBad Newsrdquo for Perfect Secrecy

bull Theorem Let the plaintext space P has size n any cipher with perfect secrecy has |K|n

bull Proof ndash Consider a nonezero probability distribution of Pndash Given any C=c for every m

Pr[P=m|C=c] = Pr[P=m] gt 0 thus there must exist one key that decrypts c into m

ndash As one key can decrypt c into one message at least n keys are needed


Introduction of the Complexity

bull Perfect Secrecy key-length msg-lengthndash Can NOT use one key to encrypt many messagendash Such as use 56-bit key to encrypt a document(gt56-bi

t)

bull So in modern cryptographyndash NOT perfect secrecy ndash IS secure under limited resource (complexity)ndash Mean that the key(or plaintext) recovery is difficultndash Complexity is the foundation of modern cryptography


Problems

bull Definition A problem is a general question with associated parameters whose values are not specified

bull Examplendash Name GCD problemndash Instance Two natural numbers ab Nndash Question What is the greatest common divisor of a a

nd bndash An instance of GCD problem what is gcd(2416)


bull Definition A Turing Machine is SB S SB Blrndash When S is State B is Characters l is shift left r is shi

ft right

Turing Machine

hellip 0 1 1 0 0 0 1 1 hellip

State Machine


Algorithms

bull Definition An algorithm is a step-by-step procedure (based on Turing Machine) which for an instance produces the correct answer

bull Description An algorithm is said to solve a problem if it produces the correct answers for all instances of a problem


PROBLEM

INSTANCES

ALGORITHM

Input

Output

Problems and Algorithms


Complexity of Algorithm

bull Definition time complexity of an algorithm is how many steps (based on TM) are necessary to produce the solution for a given instance of the size n

bull Time complexity function (TCF)ndash Logarithmic functions f(n)=log(n)ndash Polynomial functions

f(n)=na aNndash Exponential functions

f(n)=(an) aN and exist bN f(n)=O(bn)


Complexity of Problem

bull Definition The complexity of a problem is complexity of the best algorithm for a problem OR the least complexity of all algorithmsndash The complexity of problem is much more difficultndash It relay on the mathematical analysis

bull Definition Complexity theory is mathematical discipline that classifies problems based on the difficulty to solve them


Classes of Problem

bull Undecidable ndash provably intractable ndash There is no algorithm solves itndash Such as Turing Halting Problem

bull Decidable ndash P ndash can be solved in polynomial time using DTM

bull Example ABndash NP ndash can be solved in polynomial time using NDTM

SB 2S

ndash BPP ndash can be ldquosolvedrdquo in polynomial time using PTM SB01 S (Randomized algorithms)


NP Problem

bull NPP A problem is solved by the NDTM if NTDM produces ldquoYESrdquo whenever there is a solution

bull Remarkndash P Problem NP Problemndash NP Problem is ldquomore difficultrdquo from intuition

bull NPC The most difficult problem in NP ndash How to define the difficultyndash P problem is considered as easy problem


Polynomial Reduction

bull Definition Assume Q1Q2NP and I1+ and I2

+ are all lsquoyesrsquo instances

Q1 is polynomially reducible to Q2 if there is a function f I1I2 stndash f is computable in polynomial time by a DTM

ndash for all x I1+ iff f(x)I2

+

bull Note ndash Q2 is more complexity than Q1

ndash Denoted as Q1polyQ2


bull Definition The NPC problem is the Qc for any QNP we have Q polyQc

bull Examplendash Name Satisfiability problem(SAT)ndash Instance A set U of variables and a collection C of cla

uses over Undash Question Is there a satisfying truth assignment for C

bull How to prove a problem Q NPCndash SAT polyQ

NP Completeness (NPC)


P=NP

bull P=NP or pNPpP

bull If P=NPndash Modern cryptography is NOT secure

bull If pNPpPndash A one-way function can be build based on pndash Indistinguishable pseudorandom functions existndash They are foundations of modern cryptography


BPP

bull There are random algorithms in cryptographyndash Especially in cryptanalysisndash BPP is the model for random algorithms

bull Remark of BPPndash The problem can be solved with error probabilityndash The error probability can be reduced by repeatndash Security of cryptography is meaningful under BPPndash Will be deeply analyzed in future


Problems in Cryptography

bull FACTORING Factorize n (= p1e1 p2

e2hellippkek)

bull DLP Find x satisfying x (mod p)

bull DHP Find ab (mod p) from a (mod p) and b (mod p)

bull Subset-Sum Given a set of positive integers A=a1a2hellipan and a positive integer s determine there is a subset of A whose sum is s

bull QRP Decide a is a quadratic residue modulo n

bull SQROOT Find x satisfying x2 a (mod n)


bull Ideas of Semantic Securityndash Between Shannon secure and security based on Com

plexityndash Leak partial informationndash Approximate perfect secrecy

bull Definition of Semantic Securityndash Indistinguishability of Ciphertextsndash Difficult to distinguish ciphertext for different plaintextsndash Difficult to distinguish ciphertext from random bits

Semantic Security


IND-CPA

bull Definition IND-CPA(indistinguishability under a chosen-plaintext attack)ndash A cipher is (t) IND-CPA secure if no t-time adversary

wins the following game with probability of 05 +

Challenger Adversary

picks random k andpicks random b 01isin

picks M0 M1 of equal length

Attacker wins game if b=brsquo


Bit Secure

bull A cipher is (t ) bit secure if no t-time adversary wins the following game with probability 05 + Challenger Adversary1 Picks random k2 Picks random M

Attacker wins game if a=irsquoth bit of M

3 Picks i


Semantic Security

bull Any cipher that is (t) IND-CPA secure against eavesdroppers is also (t) bit secure

bull Proof Given a (t) attacker against bit security build a (t) attacker against IND-CPA security


Summary

bull Number Theoryndash Prime gcd Eulerrsquo Theorem CRTndash Modula arithmetic

bull Algebraic Structuresndash Group Fieldndash ZnZpGF(2n)

bull Shannon Securityndash Perfect secrecyndash Probability

bull Complexity of Computingndash P NP NPC BPP


Homework

bull HTTP URL

httpcsthiteducncryptography

bull Due Date

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Outline





Why Prime Numbers




Divisibility





bull Proof








nN n= p1e1 p2






Primes Amount



N=P1P2hellipPmax+1


1 P2e

2 hellip Pmaxe



Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Why Prime Numbers




Divisibility





bull Proof








nN n= p1e1 p2






Primes Amount



N=P1P2hellipPmax+1


1 P2e

2 hellip Pmaxe



Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Divisibility





bull Proof








nN n= p1e1 p2






Primes Amount



N=P1P2hellipPmax+1


1 P2e

2 hellip Pmaxe



Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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nN n= p1e1 p2






Primes Amount



N=P1P2hellipPmax+1


1 P2e

2 hellip Pmaxe



Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Primes Amount



N=P1P2hellipPmax+1


1 P2e

2 hellip Pmaxe



Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Primes Density


(x)xln(x)


((x)ln(x)x)x = 1









Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Finding GCD

bull Theorem a=i pi

ai b=i pi

bi

gcd(ab)= i pi

min(ai bi)

bull Example

a=882=2 32 72

b=3465=33 5 7 11

gcd(ab)= 32 7 = 63





GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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GCD(19701066)


Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Euclidean Algorithm



r1 =q2r2+ r3

helliphellip

rn-2 = qn-1rn-1+ rn







rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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rn-1| gcd(ab)









ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ndash Division

bull O(log22a)



Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Modular Arithmetic


01100hellip0100

Plaintext 2n

11010hellip0111

Ciphertext 2n




Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Modular Operation




Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Congruence Modulo n





a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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a+c b+d (mod n)






Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Congruence Relation




bull a b b c a c







Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Set of Residues











Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Modular Arithmetic






bull (+ ) (-)



What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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What









ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ndash When






0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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0



bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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bull Proof








(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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(n) calculation




x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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x a1 mod n1

x a2 mod n2

helliphellip

x ak mod nk


n2hellipnk


Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Proof of CRT


Zn2 hellipZnk

















ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ei



( i=1kpi

ei ) = i=1k(pi

ei ndash piei -1)

(n) = n(1-1p1)(1-1p2)hellip(1-1pk)


Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Eulerrsquos Theorem



modulo n


3 i=1(n) ri = i=1








ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ap-1 1 mod p





Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Finding a-1 mod n



bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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bull r0 =1r0+ 0r1 = s0r0+t0r1

bull r1 =0r0+ 1r1 = s1r0+t1r1





-1 mod r0

bull Find tn-1


si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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si and ti


si=si-2-qi-1si-1

ti=ti-2-qi-1ti-1











Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Outline






some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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some structures





Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Group




Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Group Example




bull ltZp gt (Zp



Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Group Properties


















ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ord(a)|n Proof


















Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Field



e a-1 a = a-1 a = 1



Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Field +-x

Relationship

Ring +-x

+-Group




bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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bull Zn

bull Zn

bull Zp




Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Zn




bull Example




Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Zn Properties







Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Zn



ndash |Zn|=(n)






Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Zp




bull ExampleZ2+ 0 1

01

0 11 0

0 1

01

0 00 1

w -w w-1

01

01 1

XOR AND


Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Polynomial Ring














bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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bull Example



















Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Galois Field




Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Why Galois Field











10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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10010111

1x7+0x6+0x5+1x4+0x3+1x2+1x1+1x0





+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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+x3+x 11001010



= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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= x8 + x(x2+x+1) mod p(x)

(b6b5b4b3b2b1b00)(00011011)

xf(x) 02= (b6b5b4b3b2b1b00)

(b6b5b4b3b2b1b00) (00011011)

if b7=0

if b7=1



Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Outline







Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Np(ai)=1


Sp(bi)=1



Secure Channel

Cryptanalysism

k

Key Space(K)

Message Space(P)

Ciphertext Space(C)


Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Claude Shannon



1916-2001










Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Shannon Security








hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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hat ek(x) = y


Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Proof of








Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Proof of







Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Proof of



=y












t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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t)



Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Problems






ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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ft right

Turing Machine


State Machine


Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Algorithms




PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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PROBLEM

INSTANCES

ALGORITHM

Input

Output













Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Classes of Problem




SB 2S



NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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NP Problem










+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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+










P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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P=NP

bull P=NP or pNPpP




BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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BPP






e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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e2hellippkek)










Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Semantic Security


IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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IND-CPA








Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Bit Secure



3 Picks i


Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Semantic Security




Summary






Homework

bull HTTP URL


bull Due Date

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Summary






Homework

bull HTTP URL


bull Due Date

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Slide 96

Slide 97

Slide 98


Homework

bull HTTP URL


bull Due Date

Slide 1

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Chapter 2 Mathematical Foundations Cryptography-Application and Practice Harbin Institute of...

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Transcript of Chapter 2 Mathematical Foundations Cryptography-Application and Practice Harbin Institute of...