PIE: $p$-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption

Luke Harmon; Gaetan Delavignette; Arnab Roy; David Silva

Paper 2023/700

PIE: $p$-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption

Luke Harmon

Gaetan Delavignette

Arnab Roy, University of Klagenfurt

David Silva

Abstract

A large part of current research in homomorphic encryption (HE) aims towards making HE practical for real-world applications. In any practical HE, an important issue is to convert the application data (type) to the data type suitable for the HE. The main purpose of this work is to investigate an efficient HE-compatible encoding method that is generic, and can be easily adapted to apply to the HE schemes over integers or polynomials. $p$-adic number theory provides a way to transform rationals to integers, which makes it a natural candidate for encoding rationals. Although one may use naive number-theoretic techniques to perform rational-to-integer transformations without reference to $p$-adic numbers, we contend that the theory of $p$-adic numbers is the proper lens to view such transformations. In this work we identify mathematical techniques (supported by $p$-adic number theory) as appropriate tools to construct a generic rational encoder which is compatible with HE. Based on these techniques, we propose a new encoding scheme PIE, that can be easily combined with both AGCD-based and RLWE-based HE to perform high precision arithmetic. After presenting an abstract version of PIE, we show how it can be attached to two well-known HE schemes: the AGCD-based IDGHV scheme and the RLWE-based (modified) Fan-Vercauteren scheme. We also discuss the advantages of our encoding scheme in comparison with previous works.

Metadata

Available format(s): PDF
Category: Cryptographic protocols
Publication info: Published elsewhere. ACNS23 -- 21st International Conference on Applied Cryptography and Network Security
Contact author(s): lharmon @ algemetric com
gdelavignette @ algemetric com
arnab roy @ aau at
dsilva @ algemetric com
History: 2023-05-22: approved; 2023-05-16: received; See all versions
Short URL: https://ia.cr/2023/700
License: CC BY

BibTeX

@misc{cryptoeprint:2023/700,
      author = {Luke Harmon and Gaetan Delavignette and Arnab Roy and David Silva},
      title = {PIE: $p$-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption},
      howpublished = {Cryptology ePrint Archive, Paper 2023/700},
      year = {2023},
      note = {\url{https://eprint.iacr.org/2023/700}},
      url = {https://eprint.iacr.org/2023/700}
}