Paper 2021/1281

Leveled Fully Homomorphic Encryption Schemes with Hensel Codes

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


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)
Public-key cryptography
Publication info
Preprint. Minor revision.
Rational numbersFully homomorphic encryptionHensel codesPublic-key encryptionExtended Euclidean algorithm
Contact author(s)
dsilva @ algemetric com
2021-09-24: received
Short URL
Creative Commons Attribution


      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{}},
      url = {}
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