Paper 2016/250

Fixed Point Arithmetic in SHE Scheme

A. Costache, N. P. Smart, S. Vivek, and A. Waller


The purpose of this paper is to investigate fixed-point arithmetic in ring-based Somewhat Homomorphic Encryption (SHE) schemes. We provide three main contributions: firstly, we investigate the representation of fixed-point numbers. We analyse the two representations from Dowlin et al, representing a fixed-point number as a large integer (encoded as a scaled polynomial) versus a polynomial-based fractional representation. We show that these two are, in fact, isomorphic by presenting an explicit isomorphism between the two that enables us to map the parameters from one representation to another. Secondly, given a computation and a bound on the fixed-point numbers used as inputs and scalars within the computation, we achieve a way of producing lower bounds on the plaintext modulus $p$ and the degree of the ring $d$ needed to support complex homomorphic operations. Finally, as an application of these bounds, we investigate homomorphic image processing.

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Publication info
Published elsewhere. SAC 2016
Contact author(s)
anamaria costache @ bristol ac uk
nigel @ cs bris ac uk
sv venkatesh @ bristol ac uk
Adrian Waller @ uk thalesgroup com
2016-09-25: revised
2016-03-07: received
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      author = {A.  Costache and N. P.  Smart and S.  Vivek and A.  Waller},
      title = {Fixed Point Arithmetic in SHE Scheme},
      howpublished = {Cryptology ePrint Archive, Paper 2016/250},
      year = {2016},
      note = {\url{}},
      url = {}
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