Paper 2020/183

A note on secure multiparty computation via higher residue symbols

Ignacio Cascudo and Reto Schnyder

Abstract

We generalize a protocol by Yu for comparing two integers with relatively small difference in a secure multiparty computation setting. Yu's protocol is based on the Legendre symbol. A prime number $p$ is found for which the Legendre symbol $(\cdot \mid p)$ agrees with the sign function for integers in a certain range $\{-N, \ldots, N\} \subset \mathbb{Z}$. This can then be computed efficiently. We generalize this idea to higher residue symbols in cyclotomic rings $\mathbb{Z}[\zeta_r]$ for $r$ a small odd prime. We present a way to determine a prime number $p$ such that the $r$-th residue symbol $(\cdot \mid p)_r$ agrees with a desired function $f\colon A \to \{\zeta_r^0, \ldots, \zeta_r^{r - 1}\}$ on a given small subset $A \subset \mathbb{Z}[\zeta_r]$, when this is possible. We also explain how to efficiently compute the $r$-th residue symbol in a secret shared setting.