irreducibility test for multinomials over a finite field

Question

I am working in an algebraic cryptosystem, and I need, in the process, ensure that a 3-variables polynomial in a finite field is irreducible but I can't find a practical method to do that. Do you know some algorithm for build a 3-variables irreducible polynomial or to check if one is irreducible?

Thanks

Answer 1

For what it's worth, sage has a good algorithm for factoring a multivariate polynomial over a finite field. You can try:

F.<a>=GF(2^8,'a')
A.<x,y,z>=PolynomialRing(F,['x','y','z'])
A.random_element(degree=10).factor()

Strangely enough, there does not seem to be an is_irreducible() method. Fortunately enough, sage is entirely open source so that you can consult the factorization algorithm, see how it works, and adapt it to make it into an is_irreducible method that stops when it has found a nontrivial factor.