Numerical Experiments on Power Functions

This page reports the results of computations that I did on December 2007, using 48 processors during one week. You will find the the Walsh Fourier spectrums of all the power functions xd in the fields GF(2,m), for all integers m < 26, and all invertibles (modulo 2^m-1) exponent d, up to equivalence : inversion and cyclotomy. I did not found any counter-examples to the main conjectures : Helleseth, Leander, Dobbertin, Langevin, but I found respectively 3 and 6 counter-examples to the Michko and Langevin-Veron conjectures. In a few words.
Let nbz(d) be the number of Walsh coefficients of x^d thar are equal to zero. Helleseth conjecture claims that nbz(d) > 1, and Leander conjecture claims it is really bigger : nbz(-1) < nbz(d).
For odd m, Dobbertin conjecture claims that the number of d such that x^d is APN is equal to phi(m)+1 and Langevin conjecture claims that spec(d) can not be equal to 4.
Let M(d) be the minimal value of the absolute value of the non zero Walsh coefficient of x^d.
Michko conjecture claims that M(d) is a power of two : it is wrong, there exist exactelly 3 counter examples in dimension 24.
Langevin-Veron conjecture claims that both -M(d) and +M(d) appear in the spectrum of x^d. It is wrong there exist 3 counter-exemples in dimension 18, and 3 others in dimension 21. But no more !!!
The primes that divides Fourier coefficients and counter-exemples are updated in this file. You will find details in our for BFCA and SAGA conferences.
Note that Niho did a such computation in the seventies up to m = 17.  For each m, the spec-m.txt contains on each line the spectrum of a power function. e.g. the third line of spec-11.txt
:
`d=7 : 562 [0], 232 [-64], 55 [-96], 11 [-128], 407 [-32], 517 [32], 77 [96], 176 [64], 11 [128]`
means that the Walsh Fourier transform of x7 takes 562 times the value 0, 232 times the value -64 etc...
The data files are compressed, the biggest (m=25) has a size of 533 Mbytes. Enjoy !