Computing Explicit Isomorphisms with Full Matrix Algebras over $$\mathbb {F}_q(x)$$ F q ( x )

Abstract

We propose a polynomial time f -algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over F q ) for computing an isomorphism (if there is any) of a finite dimensional F q (x)-algebra A given by structure constants with the algebra of n by n matrices with entries from F q (x). The method is based on computing a finite F q -subalgebra of A which is the intersection of a maximal F q [x]-order and a maximal R-order, where R is the subring of F q (x) consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

An identification system based on the explicit isomorphism problem

“…There also exist efficient algorithms for every task over finite fields [8,27] and the field of real and complex numbers [7]. Finally, when = q (t) , the field of rational functions over a finite field q , then there exist efficient algorithms for computing Wedderburn decompositions [21] and for computing explicit isomorphisms between full matrix algebras over q (t) [19].…”

“…Remark 19 We do not see any security risk in setting this exact division algebra as a global parameter (i.e., this division algebra can be used in any protocol execution).…”

“…We will refer to this problem as the explicit isomorphism problem. This is a well studied problem in computational algebra [5,19,20,22,24]. It has connections to arithmetic geometry , norm equations [20], parametrization of algebraic varieties [14] and error-correcting codes [13].…”

Explicit equivalence of quadratic forms over Fq(t)

“…Non-trivial zeros of isotropic ternary quadratic forms can be computed in randomized polynomial time using the method of of Cremona and van Hoeij from [4]. Through the connection with quaternion algebras described in Subsection 2.1, the paper [9] offers an alternative approach. Here we cite the explicit bound on the size of a solution from [4, Section 1].…”

“…The case when K = F q (t) is considered in [9], where a randomized polynomial time algorithm is proposed for computing an explicit isomorphism. However, when K is a finite extension of F q (t), the same problem remained open.…”