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.

翻译：我们提出了一种高效的证明方案，适用于从左到右的模幂运算实例，该运算广泛应用于许多素性检验计算中。具体而言，我们证明对于任意 $(a,n,r,m)$，计算 $a^n\equiv r\pmod m$ 的正确性可以以相对于模幂运算计算成本可忽略的开销进行证明和验证。我们的工作推广了当 $n$ 为 $2$ 的幂时使用的 Gerbicz-Pietrzak 证明方案，并已在 PrimeGrid 成功实现，将分布式素数搜索的效率提升了一倍。