due
Monday February 25, 2008
solutions should be submitted in the
handwritten or printed form in class
Re: Lecture 3 - Mathematical Background: Groups, Rings, and Fields
W. Stallings, Cryptography and Network Security, 4th Edition
Chapter 4, Finite Fields
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.
Re: Lecture 2 - Rijndael
W. Stallings, Cryptography and Network Security,
4th Edition, Chapter
5, Advanced Encryption Standard.
Federal Information Processing Standards Publication (FIPS) 197, Specification for the Advanced Encryption Standard (AES), Nov. 26, 2001, available at http://csrc.nist.gov/CryptoToolkit/aes/, and in particular sections
3, Notation and Conventions
4, Mathematical Preliminaries.
For the cyclic group 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.
Generate tables of logarithms and antilogarithms that can be used to speed up computations in the Galois field
GF(2^{4}) = Z_{2}[x]/f(x), where f(x)=x^{4}+x^{3}+1.
Use your tables to compute
a. 'A' · '7'
b. '5' · 'F'
c. 'B'^{ -1}
d. 'C' · 'D'^{-1 }
Perform the following multiplications in GF(2^{8})=Z_{2}[x]/m(x), where m(x)=x^{8}+x^{4}+x^{3}+x+1. using three methods:
a. using conversion to the polynomial representation
b. using the xtime function
c. using the look-up tables of logarithms and antilogarithms
A.
'08' ·
'B2'
B. '17' ·
'D3'
C. 'EF' ·
'FE'
Record and report the time it took you to perform each multiplication using each of the listed above methods.
Problem
4 (5 points)
Compute an output of the MixColumns transformation for the following sequence of input bytes "C0 FF EE 01". Use multiplication based on the xtime function in your calculations.
Apply the InvMixColumns transformation to the obtained result to verify your calculations. Use the look-up tables of logarithms and antilogarithms for multiplication in GF(2^{8}).
Change the first byte of the input
from ’C0’ to ’C1’, perform the MixColumns transformation again for the new
input, and determine how many bits have changed in each output byte.
Problem 5 (3 points)
Using your knowledge about the internal structure of the ByteSub and
InvByteSub transformations, verify the correctness of the following entries of
the AES S-box and AES Inverse S-box:
a. S-box['02'] = '77'
b. InverseS-box['02'] = '6A'
Problem 6 (3 points)
Compute the first two round keys of AES corresponding to the 128-bit key of all zeros.