Abstract
We present a methodology to achieve low latency homomorphic operations on approximations to complex numbers, by encoding a complex number as an evaluation of a polynomial at a root of unity. We then use this encoding to evaluate a Discrete Fourier Transform (DFT) on data which has been encrypted using a Somewhat Homomorphic Encryption (SHE) scheme, with up to three orders of magnitude improvement in latency over previous methods. We are also able to deal with much larger input sizes than previous methods. Due to the fact that the entire DFT algorithm is an algebraic operation over the underlying ring of the SHE scheme (for a suitably chosen ring), our method for the DFT utilizes exact arithmetic over the complex numbers, as opposed to approximations.
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Acknowledgements
This work has been supported in part by ERC Advanced Grant ERC-2015-AdG-IMPaCT and by the European Union’s H2020 Programme under grant agreement number ICT-644209 (HEAT). We thank the referee for helpful comments on an earlier version of this paper, and pointing out a few optimizations which we had missed.
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Costache, A., Smart, N.P., Vivek, S. (2017). Faster Homomorphic Evaluation of Discrete Fourier Transforms. In: Kiayias, A. (eds) Financial Cryptography and Data Security. FC 2017. Lecture Notes in Computer Science(), vol 10322. Springer, Cham. https://doi.org/10.1007/978-3-319-70972-7_29
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