Download presentation

Presentation is loading. Please wait.

Published byCandice Hodgkin Modified over 2 years ago

1
Finite Fields Rong-Jaye Chen

2
p2. Finite fields 1. Irreducible polynomial f(x) K[x], f(x) has no proper divisors in K[x] Eg. f(x)=1+x+x 2 is irreducible f(x)=1+x+x 2 +x 3 =(1+x)(1+x 2 ) is not irreducible f(x)=1+x+x 4 is irreducible

3
p3. Finite fields 2. Primitive polynomial f(x) is irreducible of degree n > 1 f(x) is not a divisor of 1+x m for any m < 2 n -1 Eg. f(x)=1+x+x 2 is not a factor of 1+x m for m < 3 so f(x) is a primitive polynomial f(x)= 1+x+x 2 +x 3 +x 4 is irreducible but 1+x 5 =(1+x)(1+x+x 2 +x 3 +x 4 ) and m=5 < 2 4 -1=15 so f(x) is not a primitive polynomial

4
p4. Finite fields 3. Definition of K n [x] The set of all polynomials in K[x] having degree less than n Each word in K n corresponds to a polynomial in K n [x] Multiplication in K n modulo h(x), with irreducible h(x) of degree n If we use multiplication modulo a reducible h(x), say, 1+x 4 to define multiplication of words in K 4, however: (0101)(0101) (x+x 3 )(x+x 3 ) = x 2 +x 6 = x 2 +x 2 (mod 1+x 4 ) = 0 0000 (K 4 -{0000} is not closed under multiplication.)

5
p5. Finite fields Furthermore each nonzero element in K n can have an inverse if we use irreducible h(x). But if we use reducible h(x) then there exists nonzero element, which has no inverse. Why? Let f(x) is nonzero element and h(x) is irreducible then gcd(f(x),h(x))=1 and so exists a(x)f(x)+b(x)h(x)=1 => a(x)f(x)=1 mod h(x) and so a(x) is the inverse of f(x)

6
p6. Finite fields 4. Definition of Field (K n,+,x) (K n,+) is an abelian group with identity denoted 0 The operation x is associative a x ( b x c) = ( a x b ) x c There is a multiplicative identity denoted 1, with 1 0 1 x a = a x 1 = a, a K n The operation x is distributive over + a x ( b + c ) = ( a x b ) + ( a x c ) It is communicative a x b = b x a, a,b K n All non-zero elements have multiplicative inverses Galois Fields: GF(2 r ) For every prime power order p m, there is a unique finite field of order p m Denoted by GF(p m )

7
p7. Finite fields Example Let us consider the construction of GF(2 3 ) using the primitive polynomial h(x)=1+x+x 3 to define multiplication. We do this by computing x i mod h(x): word x i mod h(x) 1001 010x 001x 2 110x 3 1+x 011x 4 x+x 2 111x 5 1+x+x 2 101x 6 1+x 2

8
p8. Finite fields 5. Use a primitive polynomial to construct GF(2 n ) Let K n represent the word corresponding to x mod h(x) i x i mod h(x) m 1 for m<2 n -1 since h(x) dose not divide 1+x m for m<2 n -1 Since j = i for j i iff i = j-i i j-i = 1 K n \{0}={ i | i = 0,1,…,2 n -2}

9
p9. Finite fields 6. GF(2 r ) is primitive (or a generator) is primitive if m 1 for 1 m<2 r -1 In other words, every non-zero word in GF(2 r ) can be expressed as a power of Eg. Construct GF(2 4 ) using the primitive polynomial h(x)=1+x+x 4. Write every vector as a power of x mod h(x)(see Table 5.1) Note the 15 =1. (0110)(1101)= 5. 7 = 12 =1111

10
p10. Minimal polynomials 1. Root of a polynomial : an element of F=GF(2 r ), p(x) F[x] is a root of a polynomial p(x) iff p( )=0 2. Order of The smallest positive integer m such that m =1 in GF(2 r ) is a primitive element if it has order 2 r -1

11
p11. Minimal polynomials 3. Minimal polynomial of The polynomial in K[x] of smallest degree having as root Denoted by m (x) m (x) is irreducible over K If f(x) is any polynomial over K such that f( )=0,then m (x) is a factor of f(x) m (x) is unique m (x) is a factor of

12
p12. Minimal polynomials Example Let p(x)=1+x 3 +x 4, and let be the primitive element in GF(2 4 ) constructed using h(x)=1+x+x 4 (see Table 5.1): p( )=1+ 3 + 4 =1000+0001+1100=0101= 9 is not a root of p(x). However p( 7 )=1+( 7 ) 3 +( 7 ) 4 =1+ 21 + 28 =1+ 6 + 13 =1000+0011+1011=0000=0 7 is a root of p(x).

13
p13. Minimal polynomials 4. Finding the minimal polynomial of Reduce to find a linear combination of the vectors{1, , 2,…, r }, which sums to 0 Any set of r+1 vectors in K r is dependent, such a solution exists Represent m (x) by m i (x) where = i eg. Find the m (x), = 3, GF(2 4 ) constructed using h(x)=1+x+x 4

14
p14. Minimal polynomials If f( )=0, then f( 2 )=(f( )) 2 =0 If is a root of f(x), so are , 2, 4,…, The degree of m (x) is |{ , 2, 4,…, }|

15
p15. Minimal polynomials Example Find the m (x), = 3, GF(2 4 ) constructed using h(x)=1+x+x 4 Let m (x)= m 3 (x)=a 0 +a 1 x+a 2 x 2 +a 3 x 3 +a 4 x 4 then we must find the value for a 0,a 1,…,a 4 {0,1} m ( )=0=a 0 1+a 1 +a 2 2 +a 3 3 +a 4 4 =a 0 0 +a 1 3 +a 2 6 +a 3 9 +a 4 12 0000=a 0 (1000)+a 1 (0001)+a 2 (0011)+a 3 (0101)+ a 4 (1111) a 0 =a 1 =a 2 =a 3 =a 4 =1 and m (x)=1+x+x 2 +x 3 +x 4

16
p16. Minimal polynomials Example Let m 5 (x) be the minimal polynomials of = 5, 5 GF(2 4 ) Since { , 2, 4, 8 }={ 5, 10 }, the roots of m 5 (x) are 5 and 10 which means that degree (m 5 (x))=2. Thus m 5 (x)=a 0 +a 1 x+a 2 x 2 : 0=a 0 +a 1 5 +a 2 10 =a 0 (1000)+a 1 (0110) +a 2 (1110) Thus a 0 =a 1 =a 2 =1 and m 5 (x)=1+x+x 2

17
p17. Minimal polynomials Table 5.2: Minimal polynomials in GF(2 4 ) constructed using 1+x+x 4 Element of GF(2 4 )Minimal polynomial 0 1 , 2, 4, 8 3, 6, 9, 12 5, 10 7, 11, 13, 14 x 1+x 1+x+x 4 1+x+x 2 +x 3 +x 4 1+x+x 2 1+x 3 +x 4

Similar presentations

Presentation is loading. Please wait....

OK

Elliptic Curve Cryptography

Elliptic Curve Cryptography

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on art of war 2 Ppt on art of war author Ppt on power generation using footsteps Ppt on limits and continuity ppt Ppt on role of individual in conservation of natural resources Ppt on phonetic transcription of english words Ppt on directors under companies act 1956 Ppt on hindi class 10 Ppt on latest technology in electrical Ppt on single phase and three phase dual converter flat