Paper 2019/1375

New ideas to build noise-free homomorphic cryptosystems

Gérald Gavin and Sandrine Tainturier


We design a very simple private-key encryption scheme whose decryption function is a rational function. This scheme is not born naturally homomorphic. To get homomorphic properties, a nonlinear additive homomorphic operator is specifically developed. The security analysis is based on symmetry considerations and we prove some formal results under the factoring assumption. In particular, we prove IND-CPA security in the generic ring model. Even if our security proof is not complete, we think that it is convincing and that the technical tools considered in this paper are interesting by themselves. Moreover, the factoring assumption is just needed to ensure that solving nonlinear equations or finding non-null polynomials with many roots is difficult. Consequently, the ideas behind our construction could be re-used in rings satisfying these properties. As motivating perspectives, we then propose to develop a simple multiplicative operator. To achieve this, randomness is added in our construction giving hope to remove the factoring assumption in order to get a pure multivariate encryption scheme.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Homomorphic cryptosystemMultivariate encryption schemeGeneric ring model
Contact author(s)
gerald gavin @ univ-lyon1 fr
2019-12-02: revised
2019-12-01: received
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Creative Commons Attribution


      author = {Gérald Gavin and Sandrine Tainturier},
      title = {New ideas to build noise-free homomorphic cryptosystems},
      howpublished = {Cryptology ePrint Archive, Paper 2019/1375},
      year = {2019},
      note = {\url{}},
      url = {}
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