FGLM and coprime polynomial pairs

To obtain coprime polynomial pairs, it is often used an approach that consists of taking two random polynomials of the required degrees and testing whether they are relatively prime. If they are not, they are discarded and another pair is tried. The test can obviously be carried out very efficiently. Although in many cases the probability of success for the test is close to 1, this test approach is not deterministic. In [4], a deterministic method to generate coprime polynomial pairs has been proposed. Starting from such a method, we have implemented an algorithm that generates, in a finite field of characteristic 2, at most 2λ distinct polynomials ã such that gcd(ã, b̃) = 1 and deg ã = deg b̃ + d, where d ∈ N and λ is a parameter that depends on our setting. This is accomplished by an application of the FGLM algorithm.