Cryptology ePrint Archive: Report 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.

Category / Keywords: public-key cryptography / Packing method, Homomorphic encryption, Multi-party computation, Reverse multiplication-friendly embedding

Date: received 7 Aug 2021, last revised 7 Oct 2021

Contact author: activecondor at snu ac kr

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Version: 20211007:145253 (All versions of this report)

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