Cryptography and Cryptography and Network SecurityNetwork Security

Chapter 4Chapter 4

Fourth EditionFourth Edition

by William Stallingsby William Stallings

ChapterChapter 44 – – FiniteFinite FieldsFields

IntroductionIntroduction

will now introduce finite fieldswill now introduce finite fields of increasing importance in cryptographyof increasing importance in cryptography

AES, Elliptic Curve, IDEA, Public KeyAES, Elliptic Curve, IDEA, Public Key concern operations on “numbers”concern operations on “numbers”

where what constitutes a “number” and the where what constitutes a “number” and the type of operations varies considerablytype of operations varies considerably

start with concepts of groups, rings, fields start with concepts of groups, rings, fields from abstract algebrafrom abstract algebra

Group {G,.}Group {G,.}

a set of elements or “numbers” with some a set of elements or “numbers” with some operation whose result is also in the set operation whose result is also in the set (closure) (closure)

obeys:obeys: . Closure: . Closure: if a and b belong to G, then if a and b belong to G, then a.ba.b is also in G. is also in G. associative law:associative law: (a.b).c = a.(b.c)(a.b).c = a.(b.c) has identity has identity ee:: e.a = a.e = ae.a = a.e = a has inverses has inverses aa-1-1:: a.aa.a-1-1 = e = e

if commutative if commutative a.b = b.aa.b = b.a then forms an then forms an abelian groupabelian group

Cyclic GroupCyclic Group

define define exponentiationexponentiation as repeated as repeated application of operatorapplication of operator example:example: aa33 = a.a.a = a.a.a

and let identity be:and let identity be: e=e=aa00

a group is cyclic if every element of G is a a group is cyclic if every element of G is a power of some fixed elementpower of some fixed element ie ie b =b = aakk for some for some aa and every and every bb in group in group

aa is said to be a generator of the group is said to be a generator of the group

Ring {R,+,x}Ring {R,+,x}

a set of “numbers” with two operations (addition a set of “numbers” with two operations (addition and multiplication) which form:and multiplication) which form:

an abelian group with addition operation and an abelian group with addition operation and multiplication:multiplication: has closurehas closure is associativeis associative distributive over addition:distributive over addition: a(b+c) = ab + aca(b+c) = ab + ac

if multiplication operation is commutative, it if multiplication operation is commutative, it forms a forms a commutative ringcommutative ring

if if multiplication operation has an identity and no multiplication operation has an identity and no zero divisors, it forms an zero divisors, it forms an integral domainintegral domain

Field {F,+,x}Field {F,+,x}

a set of numbers with two operations a set of numbers with two operations called addition and multiplication (ignoring called addition and multiplication (ignoring 0) 0) ringring

have hierarchy with more axioms/lawshave hierarchy with more axioms/laws group -> ring -> fieldgroup -> ring -> field

Modular ArithmeticModular Arithmetic

define define modulo operatormodulo operator “ “a mod n”a mod n” to be remainder to be remainder when a is divided by nwhen a is divided by n

use the term use the term congruencecongruence for: for: a = b mod na = b mod n when divided by when divided by n,n, a & b have same remainder a & b have same remainder

a mod n = b mod na mod n = b mod n eg. 100 = 34 mod 11 eg. 100 = 34 mod 11

a= a= └ a/n┘* n +(a mod n)└ a/n┘* n +(a mod n) r is called a r is called a residueresidue of a mod n of a mod n

since with integers can always write: since with integers can always write: a = qn + r, q=a = qn + r, q=└ a/n┘└ a/n┘ usually chose smallest positive remainder as residueusually chose smallest positive remainder as residue

• ie. ie. 0 <= r <= n-10 <= r <= n-1 process is known as process is known as modulo reductionmodulo reduction

• eg. -12 mod 7 eg. -12 mod 7 == -5 mod 7 -5 mod 7 == 2 mod 7 2 mod 7 == 9 mod 7 9 mod 7

DivisorsDivisors

say a non-zero number say a non-zero number bb dividesdivides aa if for if for some some mm have have a=mba=mb ( (a,b,ma,b,m all integers) all integers)

that is that is bb divides into divides into aa with no remainder with no remainder denote this denote this b|ab|a and say that and say that bb is a is a divisordivisor of of aa eg. all of 1,2,3,4,6,8,12,24 divide 24 eg. all of 1,2,3,4,6,8,12,24 divide 24

Modular Arithmetic OperationsModular Arithmetic Operations

is 'clock arithmetic'is 'clock arithmetic' uses a finite number of values, and loops uses a finite number of values, and loops

back from either endback from either end modular arithmetic is when do addition & modular arithmetic is when do addition &

multiplication and modulo reduce answermultiplication and modulo reduce answer can do reduction at any point, iecan do reduction at any point, ie

(a+b) mod n =[(a mod n)+(b mod n)] mod n(a+b) mod n =[(a mod n)+(b mod n)] mod n (a-b) mod n =[(a mod n)–(b mod n)] mod n(a-b) mod n =[(a mod n)–(b mod n)] mod n (a*b) mod n =[(a mod n)*(b mod n)] mod n(a*b) mod n =[(a mod n)*(b mod n)] mod n

Modular ArithmeticModular Arithmetic

can do modular arithmetic with any group of can do modular arithmetic with any group of integers: integers: ZZnn = {0, 1, … , n-1} = {0, 1, … , n-1}

form a commutative ring for additionform a commutative ring for addition with a multiplicative identitywith a multiplicative identity note some peculiaritiesnote some peculiarities

if if (a+b)(a+b)=(a+c) mod n =(a+c) mod n

thenthen b=c mod n b=c mod n but if but if (a.b)(a.b)=(a.c) mod n =(a.c) mod n

thenthen b=c mod n b=c mod n only ifonly if a a is relatively prime tois relatively prime to n n

Modulo 8 Addition ExampleModulo 8 Addition Examplex+y=0mod8x+y=0mod8

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

Euclidean algorithmEuclidean algorithm Greatest Common Divisor (GCD)Greatest Common Divisor (GCD) a common problem in number theorya common problem in number theory GCD (a,b) of a and b is the largest number GCD (a,b) of a and b is the largest number

that divides evenly into both a and b that divides evenly into both a and b eg GCD(60,24) = 12eg GCD(60,24) = 12

often want often want no common factorsno common factors (except 1) (except 1) and hence numbers are and hence numbers are relatively primerelatively prime eg GCD(8,15) = 1eg GCD(8,15) = 1 hence 8 & 15 are relatively prime hence 8 & 15 are relatively prime

Euclidean AlgorithmEuclidean Algorithm

an efficient way to find the GCD(a,b)an efficient way to find the GCD(a,b) uses theorem that: uses theorem that:

GCD(a,b) = GCD(b, a mod b)GCD(a,b) = GCD(b, a mod b) Euclidean Algorithm to compute GCD(a,b) is: Euclidean Algorithm to compute GCD(a,b) is:

EUCLID(a,b)EUCLID(a,b)1. A 1. A = = a; B a; B = = b b 2. if B = 0 return A = gcd(a, b) 2. if B = 0 return A = gcd(a, b) 3. R = A mod B 3. R = A mod B 4. A = B 4. A = B 5. B 5. B = = R R 6. goto 26. goto 2

Example GCD(1970,1066)Example GCD(1970,1066)

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

Galois FieldsGalois Fields

finite fields play a key role in cryptographyfinite fields play a key role in cryptography can show number of elements in a finite can show number of elements in a finite

field field mustmust be a power of a prime p be a power of a prime pnn (where P is (where P is a prime number , a prime number , nn is positive integer)is positive integer)

known as Galois fieldsknown as Galois fields denoted GF(pdenoted GF(pnn)) in particular often use the fields:in particular often use the fields:

GF(p) for n =1 GF(p) for n =1 GF(2GF(2nn) for p = 2) for p = 2

Galois Fields GF(p)Galois Fields GF(p)

GF(p) is the set of integers {0,1, … , p-1} GF(p) is the set of integers {0,1, … , p-1} with arithmetic operations modulo prime pwith arithmetic operations modulo prime p

these form a finite fieldthese form a finite field since have multiplicative inversessince have multiplicative inverses

hence arithmetic is “well-behaved” and hence arithmetic is “well-behaved” and can do addition, subtraction, multiplication, can do addition, subtraction, multiplication, and division without leaving the field GF(p)and division without leaving the field GF(p)

GF(7) Multiplication Example GF(7) Multiplication Example

0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6

2 0 2 4 6 1 3 5

3 0 3 6 2 5 1 4

4 0 4 1 5 2 6 3

5 0 5 3 1 6 4 2

6 0 6 5 4 3 2 1

Finding Inverses Finding Inverses (bb(bb-1 -1 = 1 mod m)= 1 mod m)

EXTENDED EUCLID(EXTENDED EUCLID(mm, , bb))1.1. (A1, A2, A3)=(1, 0, (A1, A2, A3)=(1, 0, mm); );

(B1, B2, B3)=(0, 1, (B1, B2, B3)=(0, 1, bb))2. if 2. if B3 = 0B3 = 0

return return A3 = gcd(A3 = gcd(mm, , bb); no inverse); no inverse3. if 3. if B3 = 1 B3 = 1

return return B3 = gcd(B3 = gcd(mm, , bb); B2 = ); B2 = bb–1–1 mod mod mm

4. 4. Q = Q = └ └ A3 div B3 A3 div B3 ┘┘5. 5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)(T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)6. 6. (A1, A2, A3)=(B1, B2, B3)(A1, A2, A3)=(B1, B2, B3)7. 7. (B1, B2, B3)=(T1, T2, T3)(B1, B2, B3)=(T1, T2, T3)8. goto 8. goto 22

Inverse of 550 in GF(1759)Inverse of 550 in GF(1759)

Q A1 A2 A3 B1 B2 B3

— 1 0 1759 0 1 550

3 0 1 550 1 –3 109

5 1 –3 109 –5 16 5

21 –5 16 5 106 –339 4

1 106 –339 4 –111 355 1

Polynomial ArithmeticPolynomial Arithmetic

can compute using polynomialscan compute using polynomialsff((xx) = a) = annxxnn + a + an-1n-1xxn-1n-1 + … + a + … + a11x + x + aa00 = ∑ a = ∑ aiixxii

• nb. not interested in any specific value of xnb. not interested in any specific value of x• which is known as the indeterminatewhich is known as the indeterminate

several alternatives availableseveral alternatives available ordinary polynomial arithmeticordinary polynomial arithmetic poly arithmetic with coefficients is perfprmed poly arithmetic with coefficients is perfprmed

mod pmod p poly arithmetic with coefficients are in GF(P) poly arithmetic with coefficients are in GF(P)

and polynomials mod m(x)and polynomials mod m(x)

Ordinary Polynomial ArithmeticOrdinary Polynomial Arithmetic

add or subtract corresponding coefficientsadd or subtract corresponding coefficients multiply all terms by each othermultiply all terms by each other egeg

let let ff((xx) = ) = xx33 + + xx22 + 2 and + 2 and gg((xx) = ) = xx22 – – x x + 1+ 1ff((xx) + ) + gg((xx) = ) = xx33 + 2 + 2xx22 – – x x + 3+ 3ff((xx) – ) – gg((xx) = ) = xx33 + + x x + 1+ 1ff((xx) x ) x gg((xx) = ) = xx55 + 3 + 3xx22 – 2 – 2x x + 2+ 2

Polynomial Arithmetic with Polynomial Arithmetic with Modulo CoefficientsModulo Coefficients

when computing value of each coefficient when computing value of each coefficient do calculation modulo some valuedo calculation modulo some value forms a polynomial ringforms a polynomial ring

could be modulo any primecould be modulo any prime but we are most interested in mod 2but we are most interested in mod 2

ie all coefficients are 0 or 1ie all coefficients are 0 or 1 eg. let eg. let ff((xx) = ) = xx33 + + xx22 and and gg((xx) = ) = xx22 + + x x + 1+ 1

ff((xx) + ) + gg((xx) = ) = xx33 + + x x + 1+ 1ff((xx) x ) x gg((xx) = ) = xx55 + + xx22

Polynomial DivisionPolynomial Division

can write any polynomial in the form:can write any polynomial in the form: ff((xx) = ) = qq((xx) ) gg((xx) + ) + rr((xx)) can interpret can interpret rr((xx) ) as being a remainderas being a remainder rr((xx) = ) = ff((xx) mod ) mod gg((xx))

if have no remainder say if have no remainder say gg((xx) divides ) divides ff((xx)) if if gg((xx) has no divisors other than itself & 1 ) has no divisors other than itself & 1

say it is say it is irreducibleirreducible (or prime) polynomial (or prime) polynomial arithmetic modulo an irreducible arithmetic modulo an irreducible

polynomial forms a fieldpolynomial forms a field

Polynomial GCDPolynomial GCD

can find greatest common divisor for polyscan find greatest common divisor for polys c(x)c(x) = GCD( = GCD(a(x), b(x)a(x), b(x)) if ) if c(x)c(x) is the poly of greatest is the poly of greatest

degree which divides both degree which divides both a(x), b(x)a(x), b(x) can adapt Euclid’s Algorithm to find it:can adapt Euclid’s Algorithm to find it:

EUCLID[EUCLID[aa((xx)), b, b((xx)])]1. 1. A(A(xx) = ) = aa((xx); B(); B(xx) = ) = bb((xx))2. if 2. if B(B(xx) = 0 ) = 0 return return A(A(xx) = gcd[) = gcd[aa((xx)), b, b((xx)])]3. 3. R(R(xx) = A() = A(xx) mod B() mod B(xx))4. 4. A(A(xx) ¨ B() ¨ B(xx))5. 5. B(B(xx) ¨ R() ¨ R(xx))6. goto 6. goto 22

Modular Polynomial Modular Polynomial ArithmeticArithmetic

can compute in field GF(2can compute in field GF(2nn) ) polynomials with coefficients modulo 2polynomials with coefficients modulo 2 whose degree is less than nwhose degree is less than n hence must reduce modulo an irreducible poly hence must reduce modulo an irreducible poly

of degree n (for multiplication only)of degree n (for multiplication only) form a finite fieldform a finite field can always find an inversecan always find an inverse

can extend Euclid’s Inverse algorithm to findcan extend Euclid’s Inverse algorithm to find

Example GF(2Example GF(233))

Computational Computational ConsiderationsConsiderations

since coefficients are 0 or 1, can represent since coefficients are 0 or 1, can represent any such polynomial as a bit stringany such polynomial as a bit string

addition becomes XOR of these bit stringsaddition becomes XOR of these bit strings multiplication is shift & XORmultiplication is shift & XOR

cf long-hand multiplicationcf long-hand multiplication modulo reduction done by repeatedly modulo reduction done by repeatedly

substituting highest power with remainder substituting highest power with remainder of irreducible poly (also shift & XOR)of irreducible poly (also shift & XOR)

Computational ExampleComputational Example

in in GF(2GF(233) have ) have (x(x22+1) is 101+1) is 10122 & (x & (x22+x+1) is 111+x+1) is 11122

so addition isso addition is (x(x22+1) + (x+1) + (x22+x+1) = x +x+1) = x 101 XOR 111 = 010101 XOR 111 = 01022

and multiplication isand multiplication is (x+1).(x(x+1).(x22+1) = x.(x+1) = x.(x22+1) + 1.(x+1) + 1.(x22+1) +1)

= x= x33+x+x+x+x22+1 = x+1 = x33+x+x22+x+1 +x+1 011.101 = (101)<<1 XOR (101)<<0 = 011.101 = (101)<<1 XOR (101)<<0 =

1010 XOR 101 = 11111010 XOR 101 = 111122 polynomial modulo reduction (get q(x) & r(x)) ispolynomial modulo reduction (get q(x) & r(x)) is

(x(x33+x+x22+x+1 ) mod (x+x+1 ) mod (x33+x+1) = 1.(x+x+1) = 1.(x33+x+1) + (x+x+1) + (x22) = x) = x22

1111 mod 1011 = 1111 XOR 1011 = 01001111 mod 1011 = 1111 XOR 1011 = 010022

Using a GeneratorUsing a Generator

equivalent definition of a finite fieldequivalent definition of a finite field a a generatorgenerator g is an element whose g is an element whose

powers generate all non-zero elementspowers generate all non-zero elements in F have 0, gin F have 0, g00, g, g11, …, g, …, gq-2q-2

can create generator from can create generator from rootroot of the of the irreducible polynomialirreducible polynomial

then implement multiplication by adding then implement multiplication by adding exponents of generatorexponents of generator

SummarySummary

have considered:have considered: concept of groups, rings, fieldsconcept of groups, rings, fields modular arithmetic with integersmodular arithmetic with integers Euclid’s algorithm for GCDEuclid’s algorithm for GCD finite fields GF(p)finite fields GF(p) polynomial arithmetic in general and in GF(2polynomial arithmetic in general and in GF(2nn) )