**A Note on Secure Multiparty Computation via Higher Residue Symbol Techniques**

*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 in $\mathbb{F}_p$ agrees with the sign function for integers in a certain range $\{-N, \ldots, N\}$. 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 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.

**Category / Keywords: **cryptographic protocols / secure multi-party computation

**Date: **received 14 Feb 2020, last revised 10 Mar 2020

**Contact author: **reto at math aau dk

**Available format(s): **PDF | BibTeX Citation

**Version: **20200310:113722 (All versions of this report)

**Short URL: **ia.cr/2020/183

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