Distribution on elements of cosets of small subgroups and applications

TL;DR

This paper provides estimates on the distribution of small integers and Farey fractions within cosets of small subgroups of the multiplicative group modulo a prime, with applications to polynomial distribution and fixed points of discrete logarithm.

Contribution

It introduces new bounds on the distribution of elements in cosets of small subgroups and applies these to problems in polynomial distribution and discrete logarithm fixed points.

Findings

Estimates for the number of small integers in cosets of small subgroups.

Results on the distribution of Farey fractions in these cosets.

Applications to fixed points of the discrete logarithm.

Abstract

We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order $t$ of the group of units of the residue ring modulo a prime $p$, in the case when $t$ is small compared to $p$. We give two applications of these results: to the simultaneous distribution of two high degree monomials $x^{k_1}$ and $x^{k_2}$ modulo $p$ and to a question of J.Holden and P.Moree on fixed points of the discrete logarithm.