Construction of ab-APN mappings in dimension 6 over GF(2)


This project page reports the numerical experiments related to the construction ab-APN mappings in 6 variables.


definitions

It is well known that a vectorial function F is APN if and only if

sum_{ 0 !=f in F } L(f) = 2 q^3 (q-1)
where L(f) the fourth moment of the Walsh coefficients of a boolean function in m variables.
Using the normalisation alpha(f) := L(f) / q^3.
sum_{ 0 != f in F } alpha(f) = 2 (q-1)

objective

The goal of the numerical experiment is to construct APN mappings in dimension 6 whose non zero components are in two classes. For a such mapping, the set of alpha(f)'s where f ranges the components is just a pair {a,b}.

ab-pairs

a X + b Y = 2(q-1), X+Y= q-1, a = alpha(f), b = alpha(f).
there are 63 ab-pairs summarized in the following files :
[ ab-file ]
where a line like :
55 alpha=1.750000 (56) [..45.] { 25} beta =4.000000 ( 7) [2345.] {1142}
means that 1.75 * 56 + 4.00 * 7 = 2(q-1) and there are {25} classes of Boolean function of degree [..45.] with alpha=1.75, and {1142} classes of Boolean function of degree [2345.] with beta=4.00. In fact, we hope to find new ccz-class of APN of that type.

challenge : contruct all the APN of type (1.75,4.00).

most ofthe ab-pairs listed in that file do not give APN, sometimes for evident reason :

alpha=1.656250 (39) [...5.] { 13} beta =2.558594 (24) [....6] {5170}
After such considrations, remaining possibilities are :
[ ab-possible ] [ ab-linear ] [ ab-affine ]

question : which ab-pair correspond to the ab-APN?


Last modification Summer 2023.