On the nonlinearity of boolean functions

Boolean functions on the space F2 are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In both cases, the properties of systems depend on the nonlinearity of a boolean function. The nonlinearity is linked to the covering radius of Reed-Muller codes (cf. for instance P. Langevin [9]). It is also an important cryptographic parameter (cf. the article by F. Chabaud and S. Vaudenay [3] and the thesis of Caroline Fontaine [7] or the recent article by C. Carlet [1]). It is useful to have at one’s disposal boolean functions with highest nonlinearity. These functions have been studied in the case where m is even, and have been called “bent” functions (cf. Dillon [4]). For these, the degree of nonlinearity is well known, we know how to construct several series of them, but we do not know yet their number, nor their classification (cf. works by Carlet, in particular the article of C. Carlet and P. Guillot, [2]). In the case where m is odd, the situation is quite different. We do not know the value of the maximal nonlinearity but for some value of m, and we have only a conjecture for the other values. The problem of the research of the maximum of the degree of nonlinearity comes down to minimize the Fourier transform of boolean functions. It is a problem analogous to Fourier series on the real torus, where one wants to minimize the transform of these functions on Z which take values ±1 for a finite set (and 0 elsewhere), or one wants to minimize the values of polynomials with coefficients ±1 (random polynomials) on the set of complex numbers of module 1. In this article, we have been inspired by the works of Salem and Zygmund [15] and by Kahane [8] on random polynomials, and we have transposed them on boolean functions. In this way, we find an evaluation of the mean of the maximum of the absolute values the Fourier transforms of boolean functions, which is not very far from the theoretical minimum value 2m/2. This gives an evaluation of the mean of the degrees of nonlinearity of these functions. We find in particular the fact that the boolean functions have in majority a high nonlinearity, a result found recently by C. Carlet [1]. Moreover, by transposing a work of D. Newman and J. Byrnes [11] on the norms in L4 of polynomials, we studied also a weaker conjecture about the moments of order 4 of the Fourier transform of boolean functions.