On the Powers of 2

In 2013 the function field sieve algorithm for computing discrete logarithms in finite fields of small characteristic underwent a series of dramatic improvements, culminating in the first heuristic quasi-polynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thomé. In this article we present an alternative descent method which is built entirely from the on-the-fly degree two elimination method of Göloğlu, Granger, McGuire and Zumbrägel. This also results in a heuristic quasi-polynomial time algorithm, for which the descent does not require any relation gathering or linear algebra eliminations and interestingly, does not require any smoothness assumptions about non-uniformly distributed polynomials. These properties make the new descent method readily applicable at currently viable bitlengths and better suited to theoretical analysis

Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Ces\`aro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra