Faster Homomorphic Encryption is not Enough: Improved Heuristic for Multiplicative Depth Minimization of Boolean Circuits

In somewhat homomorphic encryption schemes (e.g. B/FV, BGV) the size of ciphertexts and the execution performance of homomorphic operations depends heavily on the multiplicative depth. The multiplicative depth is the maximal number of consecutive multiplications for which the homomorphic encryption scheme was parameterized. In this work we improve a heuristic for multiplicative depth minimization of Boolean circuits found in the literature. In particular, a new circuit rewriting operator is introduced, the so called cone rewrite operator. The results we obtain using the new method are relevant in terms of accuracy and performance. The multiplicative depths for a benchmark of Boolean circuits is highly improved and the execution time of the new heuristic is significantly lower. The proposed rewrite operator and heuristic are not limited to Boolean circuits, but can also be used for arithmetic circuits.