Cryptology ePrint Archive: Report 2021/1161

Balanced Non-Adjacent Forms

Marc Joye

Abstract: Integers can be decomposed in multiple ways. The choice of a recoding technique is generally dictated by performance considerations. The usual metric for optimizing the decomposition is the Hamming weight. In this work, we consider a different metric and propose new modified forms (i.e., integer representations using signed digits) that satisfy minimality requirements under the new metric. Specifically, we introduce what we call balanced non-adjacent forms and prove that they feature a minimal Euclidean weight. We also present efficient algorithms to produce these new minimal forms. We analyze their asymptotic and exact distributions. We extend the definition to modular integers and show similar optimality results. The balanced non-adjacent forms find natural applications in fully homomorphic encryption as they optimally reduce the noise variance in LWE-type ciphertexts.

Category / Keywords: implementation / Integer recoding Lattice cryptography Fully homomorphic encryption Gadget decomposition Noise control Implementation

Original Publication (in the same form): IACR-ASIACRYPT-2021

Date: received 11 Sep 2021

Contact author: marc at zama ai

Available format(s): PDF | BibTeX Citation

Version: 20210914:175405 (All versions of this report)

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