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I. Finite Field Algebra Binary Operation G is a set of elements a , b G c a* b if c G © Tallal Elshabrawy “*” A binary operation on G is a rule that assigns to each pair of elements a and b a uniquely defined element c G is closed under “*” 2 Groups A set G on which a binary operation “*” is defined is called a Group if: i. The binary operation is associative ii. G contains an identity element e (a *e = e *a = a) iii. For any element a in G, there exists an inverse element a’ in G (a *a’ = a’ *a = e) Commutative Group G if for any a and b in G: a*b = b*a © Tallal Elshabrawy 3 Theorems The identity element in a group G is unique Proof If we have two identity elements e and e’ in G, Then, e’ =e’ * e =e e, e’ are identical The inverse of any element in a group G is unique Proof If we have two inverse elements a’ and a’’ for a in G, Then, a’ =a’ *e =a’ *(a*a’’) a’, a’’ are identical © Tallal Elshabrawy 4 Example: Modulo-2 Addition The set G={0,1} is a group of order 2 under modulo-2 addition i. Modulo-2 addition is associative ii. The identity element is 0 iii. The inverse of 0 is 0 in G The inverse of 1 is 1 in G Modulo-2 Addition 00 0 01 1 1 0 1 11 0 © Tallal Elshabrawy 5 Example: Modulo-m Addition The set G={0,1,2,…,m-1} is a group of order m under modulo-m addition i. Modulo-m addition is associative ii. The identity element is 0 iii. The inverse of i is m-i in G © Tallal Elshabrawy Modulo-m Addition i + j =r i+j=qm+r, 0≤r<m-1 6 Example: Modulo-p Multiplication G={1,2,…,p-1}, p is a prime number, is a group of order p under modulo-p multiplication Modulo-5 multiplication is associative ii. The identity element is 1 iii. The inverse of 1 is 1 in G The inverse of 2 is 3 in G The inverse of 3 is 2 in G The inverse of 4 is 4 in G i. © Tallal Elshabrawy Modulo-p Multiplication . j =r i.j=qp+r, 0≤r<p-1 i Modulo-5 Multiplication . 1 . 1 1 2 2 2 2 4 3 3 1 4 4 3 3 3 1 4 2 4 4 3 2 1 7 SubGroups Define a set G as a group under a binary operation *, A subset H is called a subgroup if i. H is closed under the binary operation * ii. For any element a in H, the inverse of a is also in H Example: Let G be the set of rational numbers constitute a group under real addition. Therefore, The set of integers H is a proper (i.e., H ≠G) subgroup under real addition © Tallal Elshabrawy 8 Cosets H is a subgroup of a group G under binary operation * a G a* H H* a a * h : h H h * a : h H Left Coset of H Right Coset of H If the group G is commutative, a *H =H *a is simply labeled as: a Coset of H © Tallal Elshabrawy 9 Example G={0,1,2,…,15} under modulo-16 addition H={0,4,8,12} is a subgroup of G why? The coset 3 + H ={3,7,11,15}= 7 + H Four Distinct and Disjoint Cosets of H 0 1 2 3 + + + + © Tallal Elshabrawy H H H H ={0,4,8,12} ={1,5,9,13} ={2,6,10,14} ={3,7,11,15} 10 Theorem (Read Only) Let H be a subgroup of a group G with binary operation *. No two elements in a Coset of H are identical a G a *H a *h : h H Suppose a *h , a *h' are identical where h h' Given a -1 denotes the inverse of a , then a 1 * a *h a 1 * a *h' e *h e *h' h h' (Contradiction) © Tallal Elshabrawy 11 Theorem (Read Only) No two elements in two different Cosets of a subgroup H of a group G are identical a *H b *H , a , b G Suppose a *h a *H , b *h' b *H If a *h =b *h' a *h *h 1 b *h' *h 1 a b *h'', h'' h' *h 1 a *H b *h'' *H a *H b *h'' *h : h H a *H b *h''' : h''' H a *H b *H Contradiction © Tallal Elshabrawy 12 Properties of Cosets i. Every element in G appears in one and only one of distinct Cosets of H ii. All the distinct Cosets of H are disjoint iii. The union of all distinct Cosets of H forms the group G © Tallal Elshabrawy 13 Fields Let F be a set of elements on which two binary operations called addition “+” and multiplication “.” are defined. The set F and the two binary operations represent a field if: i. F is a commutative group under addition. The identity element with respect to addition is called the zero element (denoted by 0) ii. The set of nonzero elements in F is a commutative group under multiplication. The identity element with respect to multiplication is called the unit element (denoted the 1 element) iii. Multiplication is distributive over addition: a.(b+c) = a.b + a.c, a, b, c in F © Tallal Elshabrawy 14 Basic Properties of Fields a.0=0.a=0 If a,b≠0, a.b≠0 a.b=0 and a≠0 imply that b=0 -(a.b)=(-a).b=a.(-b) If a≠0, a.b=a.c imply that b=c © Tallal Elshabrawy 15 Binary Field GF(2) Modulo-2 Addition Modulo-2 Multiplication + 0 1 . 0 1 0 0 1 0 0 0 1 1 0 1 0 1 F={0,1} is a Finite field of order 2 under modulo-2 addition and modulo-2 multiplication Galois Field of the order 2 © Tallal Elshabrawy 16 Subtraction and Division (GF(7)) Modulo-7 Addition Modulo-7 Multiplication + 0 1 2 3 4 5 6 . 0 0 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 3 4 5 6 3 3 4 5 6 4 4 5 6 5 5 6 6 6 0 2 3 4 5 6 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 0 1 2 0 2 4 6 1 3 5 0 1 2 3 0 3 6 2 5 1 4 0 1 2 3 4 0 4 1 5 2 6 3 0 1 2 3 4 5 5 3 1 6 4 2 1 2 3 4 5 6 0 6 5 4 3 2 1 Ex: 3-6=3+(-6)=3+1=4 © Tallal Elshabrawy 0 0 1 Ex: 3/2=3.2-1 =3.4=5 17 Characteristic of a Finite Field GF(q) (Read) 1 2 k 1 1 1 1 1 , 1 1 1 , ... 1 1 1 ... 1 k i i i times Closed Field There Exists 2 positive integers m,n, m<n where n m 1 1 n m 1 1 1 0 i i i 1 Characteristic of the field Smallest integer that satisfies 10 i 1 © Tallal Elshabrawy 18 Theorem (Read Only) Characteristic of a finite field is prime Proof Suppose is not a prime number such that km k m 1 1 . 1 0 i 1 i 1 i 1 k m 1 0 , or 1 0 i 1 i 1 This contradicts the assumption that is km the smallest integer satisfying 1 0 i 1 © Tallal Elshabrawy 19 The order of a Field Element (Read) a 1 a , a 2 a.a , a 3 a.a.a , ... non zero elements in GF(q) Closed Field There Exists 2 positive integers k,m, m>k where a k a m a k .a k a m k 1 Order of a field Element a Smallest integer n that satisfies n a 1 © Tallal Elshabrawy 20 Theorem (Read Only) Let a be a nonzero element of a finite field GF(q). Then aq-1=1 Proof Let b1 , b2 , ..., bq- 1 be (q -1) non-zero distinct elements of GF(q ) If a is an element in GF(q ) a.b1 , a.b2 , ..., a.bq- 1 represent q-1 distinct nonzero elements a.b1 , a.b2 , ..., a.bq- 1 b1 .b2 .....bq- 1 a q- 1b1 .b2 .....bq- 1 b1 .b2 .....bq- 1 a q- 1 1 © Tallal Elshabrawy 21 Theorem (Read Only) Let a be a nonzero element in a finite field GF(q). Let n be the order of a. Then n divides q-1 Proof Suppose n does not divide q -1 q - 1 kn r , 0<r<n a kn r 1 kn r a .a 1 This is impossible because 0<r <n and n is the smallest integer such that a n 1 © Tallal Elshabrawy 22 A Primitive Element of GF(q) (Read) A nonzero element a is said to be primitive if the order of a is q-1 Example: GF(7) 31=3 41=4 32=2 42=2 33=6 43=1 34=4 35=5 36=1 Order of element 3 is 6 Element 3 is a primitive element of GF(7) © Tallal Elshabrawy Order of element 4 is 3 which is a factor of 6 Element 4 is not a primitive element of GF(7) 23 Binary Field Arithmetic Polynomial of degree n over GF(2) f X f 0 f 1 X f 2 X 2 ... f n X n , f 0 , f 1 , f 2 , ..., f n are in GF(2)={0,1} fn 1 Polynomials of Degree 1 over GF(2) Polynomials of Degree 2 over GF(2) Polynomials of Degree n over GF(2) X X2 1+X 1+X2 2n Polynomials over GF(2) with degree n X+X2 1+X+X2 © Tallal Elshabrawy 24 Addition of Two Polynomials over GF(2) Example: g(X) = 1+X+X3+X5 f(X) = 1+X2+X3+X4+X7 g(X)+f(X) = X+X2+X4+X5+X7 © Tallal Elshabrawy 25 Division of Two Polynomials over GF(2) X3 X 3 X 1 X 6 X6 X 2 (Quotient q(X)) X 5 X 4 X 4 X 3 X5 X 3 X5 X 3 X 2 X2 X 1 X 1 X 1 (Remainder r(X)) © Tallal Elshabrawy 26 Irreducible Polynomials A polynomial p(X) over GF(2) of degree m is said to be irreducible over GF(2) if p(X) is not divisible by any polynomial over GF(2) of degree less than m but greater than 0 © Tallal Elshabrawy 27 Theorem Any irreducible polynomial over GF(2) divides Xn+1 where n=2m-1 and m is the degree of the polynomial © Tallal Elshabrawy 28 Primitive Polynomials An irreducible polynomial p(X) of degree m is said to be primitive if the smallest positive integer n for which p(X) divides Xn+1 is n=2m-1 Example p(X)=X4+X+1 divides X15+1 but does not divide any Xn+1 for 1≤n<15 (Primitive) p(X)= X4+X3+X2+X+1 divides X5+1 (Irreducible but Not Primitive) © Tallal Elshabrawy 29 Useful Property of Polynomials over GF(2) f 2 X f 0 f 1 X f 2 X 2 ... f n X n 2 f X f 0 f 1 X f 2 X 2 ... f n X n 2 f 2 0 f f X f X 2 ... f X n 0 1 2 n 2 f X f f X f X 2 ... f X n 2 n 0 1 2 n 2 f X f X ... f X 1 2 n 2 2 f X f 0 f 1 X f 2 X 2 ... f n X n 2 f 2 f 2 f 2 © Tallal Elshabrawy X f 0 2 f 1X 2 f2X 2 X f 0 f 1 X 2 f 2 X 2 X f X 2 2 2 2 ... f n X n ... f n X n 2 2 where f .f f i i i 30