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.
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