due
Saturday, October 14, 2006, midnight
W. Stallings, Cryptography and Network Security, 4th Edition
Chapter 4, Finite Fields
D. Hankerson, A. Menezes, S. Vanstone, Guide to Elliptic Curve Cryptography
Chapter 2.1, Introduction to finite fields (fragment of
a chapter available at the book website)
A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography
2.5, Abstract Algebra
2.6, Finite Fields
4.5, Irreducible polynomials over Zp
4.6, Generators and elements of high order.
For the cyclic groups
1. (Z_{21}^{*}, multiplication modulo 21)
2. (Z_{20}^{*}, multiplication modulo 20)
3. F_{2^4}^{* }= multiplicative group of the field F_{2^4}=Z_{2}[x]/(x^{4}+x^{3}+1)
find
a. order of the group
b. all generators of this group
c. all subgroups generated by the elements of this group.
m(x) = x^{8}+x^{4}+x^{3}+x+1
used in AES is primitive.
Generate tables of logarithms and antilogarithms that can be used to speed up computations in the Galois field
GF(2^{5}) = Z_{2}[x]/f(x), where f(x) is the simplest possible primitive polynomial of the degree 5.
Use your tables to compute
a. '1A' · '17'
b. '05' · '0F'
c. '1B'^{ -1}
d. '1C' · '0D'^{-1}