Homework 3

due Saturday, October 14, 2006, midnight
 


Re: Lecture 4 - Groups, Rings, and Fields
 

Required reading

  1. W. Stallings, Cryptography and Network Security, 4th Edition


  2.  
  3. D. Hankerson, A. Menezes, S. Vanstone, Guide to Elliptic Curve Cryptography


  4.  
  5. A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography


  6.  

Written assignment
 

Problem 6 (6 points)

For the cyclic groups

1.  (Z21*, multiplication modulo 21)

2. (Z20*, multiplication modulo 20)

3. F2^4* = multiplicative group of the field  F2^4=Z2[x]/(x4+x3+1)

find

a. order of the group

b. all generators of this group

c. all subgroups generated by the elements of this group.

 

Problem 7 (2 points)

Determine whether the polynomial

m(x) = x8+x4+x3+x+1

used in AES is primitive.

 

Problem 8 (4 points)

Generate tables of logarithms and antilogarithms that can be used to speed up computations in the Galois field

GF(25) = Z2[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