Irreducibility and Deterministic r-th Root Finding over Finite Fields

Abstract

Constructing r-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree re (where r is a prime) over a given finite field Fq of characteristic p (equivalently, constructing the bigger field Fqre). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its discriminant is a quadratic nonresidue in the field. We give an extension of Stickelberger's Lemma; we construct r-th nonresidues from a polynomial f for which there is a d, such that, r|d and r ł#(irreducible factors of f(x) of degree d). Our theorem has the following interesting consequences: (1) we can construct Fqm in deterministic poly(deg(f),m log q)-time if $m$ is an r-power and f is known; (2) we can find r-th roots in Fpm in deterministic poly(m log p)-time if r is constant and r|gcd(m,p-1). We also discuss a conjecture significantly weaker than the Generalized Riemann hypothesis to get a deterministic poly-time algorithm for r-th root finding.