1 / 30

I. Finite Field Algebra

I. Finite Field Algebra. Binary Operation. G is a set of elements. “*” 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 “*”. Groups.

sarmando
Download Presentation

I. Finite Field Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. I. Finite Field Algebra

  2. Binary Operation G is a set of elements “*” A binary operation on G is a rulethat assigns to each pair of elements a and ba uniquely defined element c G is closed under “*”

  3. Groups A set G on which a binary operation “*” is defined is called a Group if: • The binary operation is associative • G contains an identity element e • (a *e= e *a= a) • For any element a in G, there exists an inverse element a’ in G • (a *a’= a’ *a= e) Commutative GroupGif for any a and b in G: a*b = b*a

  4. Theorems The identity element in a group G is unique If we have two identity elements e and e’ in G, Then, e’ =e’ * e =ee, e’ are identical Proof 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

  5. Example: Modulo-2 Addition The set G={0,1} is a group of order 2 under modulo-2 addition • Modulo-2 addition is associative • The identity element is 0 • The inverse of 0 is 0 in G • The inverse of 1 is 1 in G Modulo-2 Addition

  6. j =r i + Example: Modulo-m Addition The set G={0,1,2,…,m-1} is a group of orderm under modulo-m addition Modulo-m Addition • Modulo-m addition is associative • The identity element is 0 • The inverse of i is m-i in G i+j=qm+r, 0≤r<m-1

  7. 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-p Multiplication . j =r i i.j=qp+r, 0≤r<p-1 Modulo-5 Multiplication • Modulo-5 multiplication is associative • The identity element is 1 • 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 .

  8. SubGroups Define a set G as a group under a binary operation *, A subset H is called a subgroup if • H is closed under the binary operation * • For any element ain 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

  9. Cosets H is a subgroup of a group G under binary operation * If the group G is commutative, a *H =H *a is simply labeled as: a Coset of H

  10. Example • G={0,1,2,…,15} under modulo-16 addition • H={0,4,8,12} is a subgroup of G why? • The coset H ={3,7,11,15}= H 3 7 + + Four Distinct and Disjoint Cosets of H H ={0,4,8,12} 0 + H ={1,5,9,13} 1 + + H ={2,6,10,14} 2 H ={3,7,11,15} 3 +

  11. Theorem (Read Only) Let Hbe a subgroup of a group G with binary operation *. No two elements in a Coset of Hare identical

  12. Theorem (Read Only) No two elements in two different Cosets of a subgroup Hof a group G are identical

  13. Properties of Cosets • Every element in G appears in one and only one of distinct Cosets of H • All the distinct Cosets of H are disjoint • The union of all distinct Cosets of H forms the group G

  14. 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: • F is a commutative group under addition. The identity element with respect to addition is called the zero element (denoted by 0) • 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) • Multiplication is distributive over addition: • a.(b+c) = a.b + a.c, a, b, c in F

  15. 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

  16. Binary Field GF(2) F={0,1} is a Finite field of order 2 under modulo-2 addition and modulo-2 multiplication Modulo-2 Addition Modulo-2 Multiplication Galois Field of the order 2

  17. Subtraction and Division (GF(7)) Modulo-7 Addition Modulo-7 Multiplication Ex: 3-6=3+(-6)=3+1=4 Ex: 3/2=3.2-1 =3.4=5

  18. Characteristic of a Finite Field GF(q) (Read)

  19. Theorem (Read Only) Proof

  20. The order of a Field Element (Read)

  21. Theorem (Read Only) Let a be a nonzero element of a finite field GF(q). Then aq-1=1 Proof

  22. 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

  23. A Primitive Element of GF(q) • A nonzero element a is said to be primitive if the order of a is q-1 • Example: GF(7) Order of element 4 is 3 which is a factor of 6 Element 4 is not a primitive element of GF(7) Order of element 3 is 6 Element 3 is a primitive element of GF(7)

  24. Binary Field Arithmetic

  25. 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

  26. Division of Two Polynomials over GF(2) (Quotient q(X)) (Remainder r(X))

  27. 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

  28. Theorem Any irreducible polynomial over GF(2) divides Xn+1 where n=2m-1 and m is the degree of the polynomial

  29. 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)

  30. Useful Property of Polynomials over GF(2)

More Related