### Leveled Fully Homomorphic Encryption Schemes with Hensel Codes

David W. H. A. da Silva, Luke Harmon, Gaetan Delavignette, and Carlos Araujo

##### Abstract

We propose the use of Hensel codes (a mathematical tool lifted from the theory of $p$-adic numbers) as an alternative way to construct fully homomorphic encryption (FHE) schemes that rely on the hardness of some instance of the approximate common divisor (AGCD) problem. We provide a self-contained introduction to Hensel codes which covers all the properties of interest for this work. Two constructions are presented: a private-key leveled FHE scheme and a public-key leveled FHE scheme. The public-key scheme is obtained via minor modifications to the private-key scheme in which we explore asymmetric properties of Hensel codes. The efficiency and security (under an AGCD variant) of the public-key scheme are discussed in detail. Our constructions take messages from large specialized subsets of the rational numbers that admit fractional numerical inputs and associated computations for virtually any real-world application. Further, our results can be seen as a natural unification of error-free computation (computation free of rounding errors over rational numbers) and homomorphic encryption. Experimental results indicate the scheme is practical for a large variety of applications.

Available format(s)
Category
Public-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
Rational numbersFully homomorphic encryptionHensel codesPublic-key encryptionExtended Euclidean algorithm
Contact author(s)
dsilva @ algemetric com
History
Short URL
https://ia.cr/2021/1281

CC BY

BibTeX

@misc{cryptoeprint:2021/1281,
author = {David W.  H.  A.  da Silva and Luke Harmon and Gaetan Delavignette and Carlos Araujo},
title = {Leveled Fully Homomorphic Encryption Schemes with Hensel Codes},
howpublished = {Cryptology ePrint Archive, Paper 2021/1281},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1281}},
url = {https://eprint.iacr.org/2021/1281}
}

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