I am new to NTL library for its GF2X
, GF2E
, GF2EX
, etc. Now, I want to perform multiplication on the Galois field GF(2^8)
. The problem is as following:
Rijndael (standardised as AES) uses the characteristic 2 finite field with 256 elements,
which can also be called the Galois field GF(2^8).
It employs the following reducing polynomial for multiplication:
x^8 + x^4 + x^3 + x^1 + 1.
For example, {53} • {CA} = {01} in Rijndael's field because
(x^6 + x^4 + x + 1)(x^7 + x^6 + x^3 + x)
= (x^13 + x^12 + x^9 + x^7) + (x^11 + x^10 + x^7 + x^5) + (x^8 + x^7 + x^4 + x^2) + (x^7 + x^6 + x^3 + x)
= x^13 + x^12 + x^9 + x^11 + x^10 + x^5 + x^8 + x^4 + x^2 + x^6 + x^3 + x
= x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + x
and
x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + x modulo x^8 + x^4 + x^3 + x^1 + 1
= (11111101111110 mod 100011011)
= {3F7E mod 11B} = {01}
= 1 (decimal)
My question is how to represent the reducing polynomial x^8 + x^4 + x^3 + x^1 + 1
and the polynomials x^6 + x^4 + x + 1
and x^7 + x^6 + x^3 + x
in NTL
. Then perform multiplication on these polynomials, and get the result {01}
.
This is a good example for me to use this library.