Abstract
In this paper we present a new class of protocols for the secure computation of the sine and cosine functions. The precision for the underlying secure fixed-point arithmetic is parametrized by the number of fractional bits f and can be set to any desired value. We perform a rigorous error analysis to provide an exact bound for the absolute error of \(2^{-f}\) in the worst case. Existing methods rely on polynomial approximations of the sine and cosine, whereas our approach relies on the random self-reducibility of the problem, using efficiently generated solved instances for uniformly random angles. As a consequence, most of the \(O(f^2)\) secure multiplications can be done in preprocessing, leaving only O(f) work for the online part. The overall round complexity can be limited to O(1) using standard techniques. We have integrated our solution in MPyC.
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Notes
- 1.
Probabilistic (or, stochastic) rounding is applied in various areas of research, including machine learning, ODEs/PDEs, quantum mechanics/computing, and digital signal processing, usually in combination with a severe limitation on numerical precision (see, for instance, [7, 14, 16, 19]). The latter condition makes probabilistic rounding desirable in these cases, because it ensures zero-mean rounding errors and avoids the problem of stagnation, where small values are lost to rounding when they are added to an increasingly large accumulator [8]. However, the use of a randomness source may be expensive, where the number of random bits (entropy) varies with the probability distribution required for the rounding errors.
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Korzilius, S., Schoenmakers, B. (2023). New Approach for Sine and Cosine in Secure Fixed-Point Arithmetic. In: Dolev, S., Gudes, E., Paillier, P. (eds) Cyber Security, Cryptology, and Machine Learning. CSCML 2023. Lecture Notes in Computer Science, vol 13914. Springer, Cham. https://doi.org/10.1007/978-3-031-34671-2_22
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