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 x^{d}
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 :

^{7}
takes 562 times the value 0, 232 times the value -64 etc...

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 x

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(

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 x

The data
files are compressed, the biggest (m=25) has a size of 533 Mbytes.
Enjoy !

- m = 3 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 4 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 5 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 6 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 7 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m =8 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 9 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 10 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 11 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 12 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 13 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 14 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 15 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 16 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 17 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 18 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 19 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 20 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 21 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 22 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 23 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 24 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.

- m = 25 : file of spectrum,
distribution of valuations, zeros and spectrum
sizes.