Arithmetic in Mathematical Fields (in particular GF(2^8) )

[war485]

Homework Statement

* note that I meant fields in the abstract mathematical sense, notphysical (i.e. electric) fields! *

Finding the determinant of a matrix in GF(2⁸)

I want to know if it is using mod 256 or mod 255 in the field of GF(2⁸)

Homework Equations

Just math

The Attempt at a Solution

I found the determinant of a matrix no problem. Everything in between is all good. What I want to know is that in the field of GF(2⁸), am I taking the number mod 255 or mod 256?

I know that there are a total of 256 elements in the field, but I'm just not sure which mod to take for the final answer, since it is a negative determinant.

 

[Mark44]

Modulo 256. The possible moduli are 0, 1, 2, ..., 255, the possible remainders when you divide by 256.

 

[war485]

thanks Mark

 

[Hurkyl]

Noooo! You're both wrong. GF(256) is a finite field, an extension field of GF(2). It is not not the same as the ring Z/256 of integers modulo 256.

 

[war485]

Hurkyl, I agree that GF(256) is a finite field, but if we're both wrong, then how would I handle whole negative numbers?

 

[Count Iblis]

You are just computing Mod 2 with the numbers. The elements are polynomials, the coefficients are from Z_2 and you're computing Modulo some primitive polynomial over Z_2 of 8th degree.