A library for affine calculations of k-algebras?

Question

I am looking for a library or computer algebra system that will help calculate operations on polynomials in a ring

F_2[x_1, ..., x_n] / <f^2 - f> 

where F_2 is a 2-element finite field, and <f^2 - f> is the ideal generated by the elements f^2 - f for all f in F_2[...] . (I think / hope / sure that this is a ring of Boolean algebra that uses xor as + and and as * [ wikipedia ]).

For instance,

 x_1 = poly_xn 1 x_2 = poly_xn 2 x_1 * x_2 * x_1 -- returns "x_1 * x_2" x_1 + x_1 + x_2 -- returns "x_2" 

I wrote the code for this in Haskell, but unfortunately the performance is not very good.

Note: the name "affine k-algebra" comes from the commutative Eisenbud algebra with the idea of ​​a book of algebraic geometry p. 35; if there is a better name, please edit the question, thanks!

Answer 1

I have worked hard in this area for many years, and consider myself using Sage as my preferred system [http://www.sagemath.org/]. It is extremely efficient and has a natural language for circuits and other algebraic structures. Others that I have used and use are OpenAxiom and Magma. I tend to avoid MathCad and Mathematica, as they tend to have a lot if the overhead is in their parsers and provide a lot of bloat related to their interfaces that are not relevant to the resolution of your calculations.

An example of affine schema support can be seen at http://www.sagemath.org/doc/reference/sage/schemes/generic/affine_space.html?highlight=affine#sage.schemes.generic.affine_space