Abstract
In this work we propose a multi-start heuristic which aims at minimizing the multiplicative depth of boolean circuits. The multiplicative depth objective is encountered in the field of homomorphic encryption where ciphertext size depends on the number of consecutive multiplications. The heuristic is based on rewrite operators for multiplicative depth-2 paths. Even if the proposed rewrite operators are simple and easy to understand the experimental results show that they are rather powerful. The multiplicative depth of the benchmarked circuits was hugely improved. In average the obtained multiplicative depths were lower by more than 3 times than the initial ones. The proposed rewrite operators are not limited to boolean circuits and can also be used for arithmetic circuits.
This work has been supported in part by the French’s FUI project CRYPTOCOMP and by the European Union’s H2020 Programme under grant agreement number 727528 (project KONFIDO).
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Notes
- 1.
Multiplicative depth is the number of sequential homomorphic multiplications which can be done on freshly encrypted ciphertexts in order to be able to decrypt and retrieve the result of multiplications.
- 2.
As we shall further see, more precisely it depends on the relative computational cost of circuit AND gates with respect to scheme multiplicative depth.
- 3.
By abuse of language we denote so the above defined priority functions.
- 4.
We assume that multi-million gate designs are out of reach for homomorphic execution, at least for the current state of HE schemes.
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Carpov, S., Aubry, P., Sirdey, R. (2018). A Multi-start Heuristic for Multiplicative Depth Minimization of Boolean Circuits. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_23
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