Short Solutions To Homogenous Linear congruences

TL;DR

This paper extends the probability of finding short solutions to homogeneous linear congruences from prime moduli to square-free moduli, including special cases for two-variable congruences.

Contribution

It generalizes previous results to square-free moduli and analyzes the existence of short solutions in various cases, including primitive solutions for two-variable congruences.

Findings

Positive probability of short solutions for square-free moduli

Existence of short primitive solutions in two-variable cases

Results hold for all integer moduli in two-variable scenarios

Abstract

Str\"ombergsson and Venkatesh proved that a system of homogeneous linear congruence modulo prime has a positive probability to have a short non-trivial solution. We extend this result and show that the same holds for square-free moduli. In the case of 2-variables single linear congruence, we show that there is a positive probability to have a short solution for all integer moduli as well as positive probability for having short non-trivial solutions which are primitive in a suitable sense.