Paper 2021/1033

Limits of Polynomial Packings for ${\mathbb{Z}}_{p^k}$ and ${\mathbb{F}}_{p^k}$

Jung Hee Cheon and Keewoo Lee

Abstract

We formally define polynomial packing methods and initiate a unified study of related concepts in various contexts of cryptography. This includes homomorphic encryption (HE) packing and reverse multiplication-friendly embedding (RMFE) in information-theoretically secure multi-party computation (MPC). We prove several upper bounds and impossibility results on packing methods for ${\mathbb{Z}}_{p^k}$ or ${\mathbb{F}}_{p^k}$-messages into ${\mathbb{Z}}_{p^t}[x]/f(x)$ in terms of (i) packing density, (ii) level-consistency, and (iii) surjectivity. These results have implications on recent development of HE-based MPC over ${\mathbb{Z}}_{2^k}$ secure against actively corrupted majority and provide new proofs for upper bounds on RMFE.