On the Non-linearity of Power Functions

Let L be an extension of degree m of the eld F2. Let be the canonical additive character of L. The Fourier coe cient of the polynomial f(X) 2 L[X ] at the point a 2 L is de ned by b f(a) = P x2L (f(x) + a:x): The spectral magnitude of f , R(f) := supa2F 2 j b f(a)j, is one of the most important cryptographic parameters associated to f [10]. The determination of the spectral magnitude of power functions x 7! x is an old open problem from di erent frameworks [5, 9] : two zeros cyclic codes (Kasami, Charpin), correlation of sequences (Gold, Niho, Welch) and boolean functions (Canteaut, Dobbertin). The main questions are : what is the spectral magnitude, say L(s;m) of the power function x 7! x, what is the minimal value L(m) of the L(s;m) and what are the good exponents s such that L(s;m) = L(m) ? Most of the papers on the subject use McEliece's theorem on the divisibility of weight of cyclic codes whose the proof [2, 3, 6] is very similar to those of the Ax 's theorem on the divisibility of the number of zeros of algebraic equations [8]. Here we apply Stickelberger's congruences on Gauss sums [7] to explore the non-linearity of power functions.