On the Number of Solutions of Exponential Congruences

TL;DR

This paper derives upper bounds on the solutions to exponential congruences involving x^x modulo a prime p, with implications for cryptography and understanding solution distributions.

Contribution

It provides new nontrivial bounds on the number of solutions to specific exponential congruences modulo primes, advancing theoretical understanding.

Findings

Upper bounds on solutions to x^x ≡ a mod p

Estimates on solutions to x^x ≡ y^y mod p

Implications for cryptographic problem analysis

Abstract

For a prime $p$ and an integer $a \in \Z$ we obtain nontrivial upper bounds on the number of solutions to the congruence $x^x \equiv a \pmod p$, $1 \le x \le p-1$. We use these estimates to estimate the number of solutions to the congruence $x^x \equiv y^y \pmod p$, $1 \le x,y \le p-1$, which is of cryptographic relevance.