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2019-12-31 13:20:15 +0200 commented answer Symbolic arithmetic in finite fields

Thanks for the answer. But what you have shown is arithmetic operations and extraction of co-ordinates of given elements of the field. What I am asking is if an indeterminate x belongs to the field of size 2^5 and is represented in a chosen basis then x has indeterminate co-ordinates xi in GF(2) wrt that basis. What I want is to compute symbolic co-ordinates of x^2, x^3, xy etc. in the same basis. These co-ordinates will be polynomial functions of co-ordinates of x,y.

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2019-12-31 06:07:08 +0200 asked a question Symbolic arithmetic in finite fields

I want to know whether and how we can perform symbolic arithmetic for finite fields in SAGEmath. Let me explain the problem. Let GF(2^n) be a finite field extension of GF(2) with a polynomial basis with respect to a root 'a' of an irreducible polynomial. My questions are:

  1. How to declare an indeterminate element x in this field with symbolic co-ordinates xi, i=1..n in GF(2).
  2. If x,y are indeterminates how to compute x^2, xy, xy^2 etc as functions symbolically. In other words, how to extract symbolic co-ordinates of these functions.
2019-12-31 06:07:08 +0200 asked a question How to do symbolic computation over a finite field

I want to carry out finite field arithmetic symbolically in SAGEmath. Need help. Let me explain the problem. Suppose I declare the finite field GF(2^n) with an irreducible polynomial specified with root a. I want to declare indeterminates x,y in this field with co-ordinates xi,yi i=1,...,n in the polynomial basis in a. Then extract symbolic co-ordinates of functions such as x^2,xy,x^3 in the same basis. How can this be done?