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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.
|Title of host publication||Selected Areas in Cryptography - SAC 2016|
|Subtitle of host publication||23rd International Conference, St. John’s, NL, Canada, August 10-12, Revised Selected Papers|
|Number of pages||22|
|Publication status||E-pub ahead of print - 20 Oct 2017|
|Name||Lecture Notes in Computer Science|
Costache, A., Smart, N., Vivek, S., & Waller, A. (2017). Fixed-point arithmetic in SHE schemes. In Selected Areas in Cryptography - SAC 2016: 23rd International Conference, St. John’s, NL, Canada, August 10-12, Revised Selected Papers (pp. 401-422). (Lecture Notes in Computer Science; Vol. 10532). Springer. https://doi.org/10.1007/978-3-319-69453-5_22