NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits.

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have demonstrated the central role of this problem and its variations in diverse areas such as cryptography, derandomization, proof complexity, learning theory, and circuit lower bounds. The NP-hardness of computing the minimum numbers of terms in a DNF formula consistent with a given truth table was proved by W. Masek [31] in 1979. In this work, we make the first progress in showing NP-hardness for more expressive classes of circuits, and establish an analogous result for the MCSP problem for depth-3 circuits of the form OR-AND-MOD2. Our techniques extend to an NP-hardness result for MOD m gates at the bottom layer under inputs from (Z/mZ)(n).