Cryptology ePrint Archive: Report 2020/1118

Approximate Homomorphic Encryption with Reduced Approximation Error

Andrey Kim and Antonis Papadimitriou and Yuriy Polyakov

Abstract: The Cheon-Kim-Kim-Song (CKKS) homomorphic encryption scheme is currently the most efficient method to perform approximate homomorphic computations over real and complex numbers. Although the CKKS scheme can already be used to achieve practical performance for many advanced applications, e.g., in machine learning, its broader use in practice is hindered by several major usability issues, most of which are brought about by relatively high approximation errors and the complexity of dealing with them.

We present a reduced-error CKKS variant that removes the approximation errors due to the Learning With Errors (LWE) noise in the encryption and key switching operations. We also propose and implement its RNS instantiation that has a lower error than the original CKKS scheme implementation based on multiprecision integer arithmetic. While formulating the RNS instantiation, we develop an intermediate RNS variant that has a smaller approximation error than the prior RNS variant of CKKS. The high-level idea of our main RNS-specific improvements is to remove the approximate scaling error using an automated procedure that computes different scaling factors for each level and performs all necessary adjustments. The rescaling procedure and scaling factor adjustments in our implementation are done automatically and are not exposed to the application developer.

We implement both RNS variants in PALISADE and compare their approximation error and efficiency to the prior RNS variant. Our results for uniform ternary secret key distribution, which is the most efficient setting included in the community homomorphic encryption security standard, show that the reduced-error CKKS RNS implementation typically has an approximation error that is 6 to 9 bits smaller for computations with multiplications than the prior RNS variant. For computations without a multiplication, the approximation error can be up to 20 bits lower than in the prior RNS variant. As compared to the original CKKS using multiprecision integer arithmetic, our reduced-error CKKS RNS implementation has an error that is smaller by 4 and up to 20 bits for computations with multiplications and without multiplications, respectively. For the sparse ternary secret key setting, which was used in the original CKKS paper, the approximate error reduction of reduced-error CKKS w.r.t. original CKKS typically ranges from 6 to 8 bits for computations with multiplications.

Category / Keywords: implementation / approximate homomorphic encryption, software implementation, lattice-based cryptography, CKKS

Date: received 15 Sep 2020, last revised 7 Oct 2020

Contact author: ypolyakov at dualitytech com

Available format(s): PDF | BibTeX Citation

Version: 20201007:164903 (All versions of this report)

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