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Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main concept. In this article, I show that the spectral amplitude of boolean functions, which is linked to their nonlinearity, is of the order of $2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by $2^{m/2}$ and $2^m$. Moreover I examine a conjecture of Patterson and Wiedemann saying that the minimum of this spectral amplitude is as close as desired to $2^{m/2}$. I also study a weaker conjecture about the moments of order 4 of their Fourier transform. This article is inspired by works of Salem, Zygmund, Kahane and others about the related problem of real polynomials with random coefficients.

Author

Rodier, Francois

Published

2003

Language

EN