Finding primitive elements in finite fields of small characteristic

Abstract

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field with p n elements. In time polynomial in p and n, the algorithm either outputs an element that is provably a generator or declares that it has failed in finding one. Under a heuristic assumption, we argue that the algorithm does always succeed in finding a generator. The algorithm relies on a relation generation technique in a recent breakthrough by Antoine Joux's for discrete logarithm computation in small characteristic finite fields in L(1/4, o(1)) time. For the special case when the order of p in (Z/nZ) × is small (bounded by (log p (n)) O(1) ), we present a modified algorithm which is reliant on weaker heuristic assumptions.

Traps to the BGJT-algorithm for discrete logarithms

“…Similarly, Z q 2 +1 /L 1 is not cyclic as its quotient Z q 2 +1 /L * 2 is not cyclic if l ≥ 1. Hence the conjecture in [9] also needs modification. It seems reasonable to hope thatL 1 is a good approximation to L * 2 in the sense that the quotient L * 2 /L 1 is a direct sum of small order cyclic groups.…”

A New Index Calculus Algorithm with Complexity $$L(1/4+o(1))$$ in Small Characteristic

“…Another application of the same ideas has been described in [7], where a new deterministic algorithm (based on similar heuristics to ours) is proposed to find a provable multiplicative generator of a finite field.…”

Technical history of discrete logarithms in small characteristic finite fields

“…However, this factorization is hard to compute when q k is large. Interestingly, the techniques to solve the discrete logarithm problem that we survey in the present paper have also been used in [17] to compute a multiplicative generator of F q k without knowledge of the factorization of q k − 1.…”