Fully Homomorphic Encryption (FHE) gives the ability to evaluate any function over encrypted data. However, despite numerous improvements during the last decade, the computational overhead caused by homomorphic computations is still very important. As a consequence, optimizing the way of performing the computations homomorphically remains fundamental. Several popular FHE schemes such as BGV and BFV encode their data, and thus perform their computations, in finite fields. In this work, we study and exploit algebraic relations occurring in prime characteristic allowing to speed-up the homomorphic evaluation of several functions over prime fields.

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More specifically we give several examples of unary functions: “modulo”, “is power of $b$” and “Hamming weight” and “Mod2′” whose homomorphic evaluation complexity over $mathbb{F}_p$ can be reduced from the generic bound $sqrt{2p} + mathcal{O}(log(p))$ homomorphic multiplications, to $sqrt{p} + mathcal{O}(log(p))$, $mathcal{O}(log (p))$, $mathcal{O}(sqrt{p/log (p)})$ and $mathcal{O}(sqrt{p/(log (p))})$ respectively. Additionally we provide a proof of a recent claim regarding the structure of the polynomial interpolation of the “less-than” bivariate function which confirms that this function can be evaluated in $2p-6$ homomorphic multiplications instead of $3p-5$ over $mathbb{F}_p$ for $pgeq 5$.

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