Stability of the Walsh-Hadamard spectrum of cryptographic Boolean functions with biased inputs

We propose a notion of stability of the Walsh-Hadamard spectrum of Boolean functions when inputs are independent and identically distributed Bernoulli random variables. We study the stability spectrum of bent Boolean functions and obtain a bound for it. We also derive the formula for the stability transform of Maiorana-McFarland type bent functions. We analyze the stability spectrum of symmetric Boolean functions and characterize it for symmetric bent Boolean functions and symmetric Boolean functions in an odd number of variables with maximum nonlinearity. We show that the stability spectrum is not, in general, invariant under extended affine transformations. Further, we display some non-bent symmetric Boolean functions whose stability spectra are flatter than that of symmetric bent Boolean functions.

Automated summary

In this article, we propose a concept of stability for the Walsh-Hadamard spectrum of Boolean functions. We investigate the stability spectrum of bent Boolean functions and derive a bound for it. Additionally, we obtain the formula for the stability transform of Maiorana-McFarland type bent functions. We analyze the stability spectrum of symmetric Boolean functions, characterizing it for symmetric bent Boolean functions and symmetric Boolean functions in an odd number of variables with maximum nonlinearity. Furthermore, we demonstrate that the stability spectrum is not generally invariant under extended affine transformations, and present some non-bent symmetric Boolean functions with flatter stability spectra than that of symmetric bent Boolean functions.