A new transform related to distance from a boolean function (extended abstract)

Abstract

We introduce a new transform on Boolean functions generalizing the Walsh-Hadamard transform. For Boolean functions q and f, the q-transform of f measures the proximity of f to the set of functions obtained from q by change of basis. This has implications for security against certain algebraic attacks. In this paper we derive the expected value and second moment (Parseval’s equation) of the q-transform, leading to a notion of q-bentness. We also develop a Poisson Summation Formula, which leads to a proof that the q-transform is invertible.