Ch04

Cryptography and Network Security Third Edition by William Stallings Lecture slides by Lawrie Brown

Chapter 4 – Finite Fields The next morning at daybreak, Star flew indoors, seemingly keen for a lesson. I said, "Tap eight." She did a brilliant exhibition, first tapping it in 4, 4, then giving me a hasty glance and doing it in 2, 2, 2, 2, before coming for her nut. It is astonishing that Star learned to count up to 8 with no difficulty, and of her own accord discovered that each number could be given with various different divisions, this leaving no doubt that she was consciously thinking each number. In fact, she did mental arithmetic, although unable, like humans, to name the numbers. But she learned to recognize their spoken names almost immediately and was able to remember the sounds of the names. Star is unique as a wild bird, who of her own free will pursued the science of numbers with keen interest and astonishing intelligence. — Living with Birds , Len Howard

Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on “numbers” where what constitutes a “number” and the type of operations varies considerably start with concepts of groups, rings, fields from abstract algebra

Group a set of elements or “numbers” with some operation whose result is also in the set (closure) obeys: associative law: (a.b).c = a.(b.c) has identity e : e.a = a.e = a has inverses a -1 : a.a -1 = e if commutative a.b = b.a then forms an abelian group

Cyclic Group define exponentiation as repeated application of operator example: a -3 = a.a.a and let identity be: e= a 0 a group is cyclic if every element is a power of some fixed element ie b = a k for some a and every b in group a is said to be a generator of the group

Ring a set of “numbers” with two operations (addition and multiplication) which are: an abelian group with addition operation multiplication: has closure is associative distributive over addition: a(b+c) = ab + ac if multiplication operation is commutative, it forms a commutative ring if multiplication operation has inverses and no zero divisors, it forms an integral domain

Field a set of numbers with two operations: abelian group for addition abelian group for multiplication (ignoring 0) ring

Modular Arithmetic define modulo operator a mod n to be remainder when a is divided by n use the term congruence for: a ≡ b mod n when divided by n, a & b have same remainder eg. 100 = 34 mod 11 b is called the residue of a mod n since with integers can always write: a = qn + b usually have 0 <= b <= n-1 -12 mod 7 ≡ -5 mod 7 ≡ 2 mod 7 ≡ 9 mod 7

Modulo 7 Example ... -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 ...

Divisors say a non-zero number b divides a if for some m have a=mb ( a,b,m all integers) that is b divides into a with no remainder denote this b|a and say that b is a divisor of a eg. all of 1,2,3,4,6,8,12,24 divide 24

Modular Arithmetic Operations is 'clock arithmetic' uses a finite number of values, and loops back from either end modular arithmetic is when do addition & multiplication and modulo reduce answer can do reduction at any point, ie a+b mod n = [a mod n + b mod n] mod n

Modular Arithmetic can do modular arithmetic with any group of integers: Z n = {0, 1, … , n-1} form a commutative ring for addition with a multiplicative identity note some peculiarities if (a+b) ≡(a+c) mod n then b≡c mod n but (ab) ≡(ac) mod n then b≡c mod n only if a is relatively prime to n

Greatest Common Divisor (GCD) a common problem in number theory GCD (a,b) of a and b is the largest number that divides evenly into both a and b eg GCD(60,24) = 12 often want no common factors (except 1) and hence numbers are relatively prime eg GCD(8,15) = 1 hence 8 & 15 are relatively prime

Euclid's GCD Algorithm an efficient way to find the GCD(a,b) uses theorem that: GCD(a,b) = GCD(b, a mod b) Euclid's Algorithm to compute GCD(a,b): A=a, B=b while B>0 R = A mod B A = B, B = R return A

Example GCD(1970,1066) 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)

Galois Fields finite fields play a key role in cryptography can show number of elements in a finite field must be a power of a prime p n known as Galois fields denoted GF(p n ) in particular often use the fields: GF(p) GF(2 n )

Galois Fields GF(p) GF(p) is the set of integers {0,1, … , p-1} with arithmetic operations modulo prime p these form a finite field since have multiplicative inverses hence arithmetic is “well-behaved” and can do addition, subtraction, multiplication, and division without leaving the field GF(p)

Finding Inverses can extend Euclid’s algorithm: EXTENDED EUCLID( m , b ) (A1, A2, A3)=(1, 0, m ); (B1, B2, B3)=(0, 1, b ) 2. if B3 = 0 return A3 = gcd( m , b ); no inverse 3. if B3 = 1 return B3 = gcd( m , b ); B2 = b –1 mod m 4. Q = A3 div B3 5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3) 6. (A1, A2, A3)=(B1, B2, B3) 7. (B1, B2, B3)=(T1, T2, T3) 8. goto 2

Polynomial Arithmetic can compute using polynomials several alternatives available ordinary polynomial arithmetic poly arithmetic with coords mod p poly arithmetic with coords mod p and polynomials mod M(x)

Ordinary Polynomial Arithmetic add or subtract corresponding coefficients multiply all terms by each other eg let f ( x ) = x 3 + x 2 + 2 and g ( x ) = x 2 – x + 1 f ( x ) + g ( x ) = x 3 + 2 x 2 – x + 3 f ( x ) – g ( x ) = x 3 + x + 1 f ( x ) x g ( x ) = x 5 + 3 x 2 – 2 x + 2

Polynomial Arithmetic with Modulo Coefficients when computing value of each coefficient do calculation modulo some value could be modulo any prime but we are most interested in mod 2 ie all coefficients are 0 or 1 eg. let f ( x ) = x 3 + x 2 and g ( x ) = x 2 + x + 1 f ( x ) + g ( x ) = x 3 + x + 1 f ( x ) x g ( x ) = x 5 + x 2

Modular Polynomial Arithmetic can write any polynomial in the form: f ( x ) = q ( x ) g ( x ) + r ( x ) can interpret r ( x ) as being a remainder r ( x ) = f ( x ) mod g ( x ) if have no remainder say g ( x ) divides f ( x ) if g ( x ) has no divisors other than itself & 1 say it is irreducible (or prime) polynomial arithmetic modulo an irreducible polynomial forms a field

Polynomial GCD can find greatest common divisor for polys c(x) = GCD( a(x), b(x) ) if c(x) is the poly of greatest degree which divides both a(x), b(x) can adapt Euclid’s Algorithm to find it: EUCLID[ a ( x ) , b ( x )] A( x ) = a ( x ); B( x ) = b ( x ) 2. if B( x ) = 0 return A( x ) = gcd[ a ( x ) , b ( x )] 3. R( x ) = A( x ) mod B( x ) 4. A( x ) ¨ B( x ) 5. B( x ) ¨ R( x ) 6. goto 2

Modular Polynomial Arithmetic can compute in field GF(2 n ) polynomials with coefficients modulo 2 whose degree is less than n hence must reduce modulo an irreducible poly of degree n (for multiplication only) form a finite field can always find an inverse can extend Euclid’s Inverse algorithm to find

Computational Considerations since coefficients are 0 or 1, can represent any such polynomial as a bit string addition becomes XOR of these bit strings multiplication is shift & XOR cf long-hand multiplication modulo reduction done by repeatedly substituting highest power with remainder of irreducible poly (also shift & XOR)

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

Ch04

More Related Content

What's hot (20)

Similar to Ch04 (20)

More from Joe Christensen (20)

Recently uploaded (20)

Ch04