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
kappa( f ) := L(f) / q^3.
We refer kappa( f ) normalized spectral momement of order 4
of the Boolean function f.
sum_{ 0 != f in F } kappa(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 affine classes. For a such mapping, the set of kappa(f)'s
where f ranges the components is just a pair {a,b}. More
generally, we say a vectorial function F has two level
when the spectral moment of order 4 takes two values
on the set of its non zero components.
ab-pairs
a X + b Y = 2(q-1), X+Y= q-1, a = kappa(f), b = kappa(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
Let us denote by K(c, F) the subspace generated by the
components f with kappa(f) = c.
We contructed all the APN F of type (1.75,4.00) such that :
- F has degree 4
- K(4,F) is a space of cubics.
all are CCZ-equivalent !
project
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 1: which ab-pair correspond to the ab-APN?
question 2: which ab-pair correspond to the ab-APN with only two affine classes ?
data
[
abstract-bfa-2025
]
[
data
]
Valérie Gillot,
Philippe Langevin,
Abdoulaye Lo;
Institut Mathématiques de Toulon,
Last modification Summer 2023, Winter 2024, April 2025.