Circuit optimisation problems in the context of homomorphic encryption

Fully homomorphic encryption (FHE) is an encryption scheme that enables the direct execution of arbitrary computations on encrypted data. The first generation of FHE schemes began with Gentry's groundbreaking work in 2019. It relies on a technique called bootstrapping, which reduces noise in FHE ciphertexts. This construction theoretically enables the execution of any arithmetic circuit, but remains slow in practice. A second generation of FHE schemes appeared in 2015 and is referred to as fast bootstrapping schemes. One limitation of the later schemes is that they can only bootstrap one message at a time, but the bootstrapping procedure is relatively fast.

This presentation aims to highlight some works on the optimisation of Boolean and arithmetic circuits in the context of FHE. The optimisation of the so-called multiplicative depth of circuits will be discussed. The multiplicative depth is an important metric for the first generation of FHE schemes, as ciphertext size and, consequently, execution performance depend heavily on it. In a second part, we will discuss a circuit mapping problem encountered in the practical application of fast-bootstrapping schemes.