Infinitely generated semigroups and polynomial complexity

This paper continues the functional approach to the [Formula: see text]-vs.-[Formula: see text] problem, begun in [J. C. Birget, Semigroups and one-way functions, Internat. J. Algebra Comput. 25 (1–2) (2015) 3–36]. Here, we focus on the monoid [Formula: see text] of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We construct a machine model for the functions in [Formula: see text], and evaluation functions. We prove that, [Formula: see text] is not finitely generated, and use this to show separation results for time-complexity.

Traps to the BGJT-algorithm for discrete logarithms

In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular when the field is not a Kummer extension. We propose a fix to the algorithm in non-Kummer cases, without altering the heuristic quasi-polynomial time complexity. Further study is required in order to fully understand the effectiveness of the new approach.