Finite Field elements in terms of powers of a generator

[adeines]

For large values of q, a prime power, GF(q) has elements represented as polynomials over a generator.

sage: F.<a> = GF(2^8)
sage: a^10
a^6 + a^5 + a^4 + a^2

If you further want to compute in a polynomial ring over F, then the polynomials aren't very pretty as they are polynomials with polynomial coefficients.

sage: R.<x> = F[]
sage: a^10*x+1
(a^6 + a^5 + a^4 + a^2)*x + 1

It would be nice to be able to be able to print and work with the elements as powers of the generator.

Component: finite rings

Keywords: GF, finite field

Reviewer: Jeroen Demeyer

Issue created by migration from https://trac.sagemath.org/ticket/10971

[sagetrac-lmartel]

comment:1

But I expect it to be slow...
How would you compute this efficiently?

sage: K.<a> = GF(17**17)
sage: a + 1
a^24137569    # this would be the result in your representation

[jdemeyer]

Reviewer: Jeroen Demeyer

[jdemeyer]

comment:8

This requires discrete log computations, which is too inefficient in general.

Close as "wontfix".