An Efficient Modular Exponentiation Proof Scheme

TL;DR

This paper introduces a highly efficient proof scheme for modular exponentiation that significantly improves verification speed in primality testing, generalizing previous methods and demonstrating practical success in distributed prime searches.

Contribution

It generalizes the Gerbicz-Pietrzak proof scheme to all modular exponentiation instances, enhancing efficiency and applicability in primality testing.

Findings

Proof scheme correctness for any modular exponentiation instance

Negligible overhead compared to exponentiation cost

Doubled efficiency in distributed prime searches at PrimeGrid

Abstract

We present an efficient proof scheme for any instance of left-to-right modular exponentiation, used in many computational tests for primality. Specifically, we show that for any $(a,n,r,m)$ the correctness of a computation $a^n\equiv r\pmod m$ can be proven and verified with an overhead negligible compared to the computational cost of the exponentiation. Our work generalizes the Gerbicz-Pietrzak proof scheme used when $n$ is a power of $2$, and has been successfully implemented at PrimeGrid, doubling the efficiency of distributed searches for primes.