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 file :
[
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}
question : which ab-pair correspond to the ab-PAN?
Last modification Summer 2023.