Distribution of Elements of Cosets of Small Subgroups and Applications

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 k1 and x k2 modulo p and to a question of J. Holden and P. Moree on fixed points of the discrete logarithm.

On the Hidden Shifted Power Problem

“…So our goal is to find a "small" set Y ⊆ F * q such that its shifts cannot be inside of any coset of G e (we note that the value of r is unknown). Questions about the distribution of cosets of multiplicative groups have been considered in a number of works and have numerous applications, see [33] and also [6,8,12,9,11,39,41,42] for several more recent results and applications to cryptographic and computational number theory problems.…”

Periodic structure of the exponential pseudorandom number generator

“…The quantity N p,g (k) for k = 1, 2, 3 has recently been studied in [5,6,12,18,21,22,23,27,31]. Fixed points with various restrictions on u have been considered as well.…”

Artin's Primitive Root Conjecture – A Survey

“…A related conjecture by Holden and Moree [236] states that one should have F (p) = (1 + o (1))p. Bourgain et al [44] showed that F (p) = p + O(p 4/5+ǫ ) for a set of primes p of relative density 1 and in a later paper, [45],…”