double coset of AGL(4) \ S(16) / AGL(4)
We reports the numerical experiments related to the classification of
the permutation of GF(2)^4 and applications to the classification of
Boolean bent functions of Maiorana type. A joint work,
with Alexander Polujan.
definition
Let m be a positive integer. The group AG(2,m) x AG(2,m) acts over the
group of permutation permutations of the space GF(2)^m. We denote by R( m )
the number of double coset the action :
coset(f) = { L o f o R | where L, R are in AGL( m ).
class number table
The affine group AGL(m) is small,
the permutation group S(m) is large whence
the class number R(m) is veryhuge !
| m | 1 | 2 | 3 | 4 | 5 |
| R(m) | 1 | 1 | 4 | 302 | 2569966041123963092 |
Class numbers of AGL(m) \ S(2^m) / AGL(m)
Theses values appear in the article Affinity of permutations of F2n by Xiang-Dong Hou. Note that the value of R(5) is misprinted in the paper, and just for fun we give :
R(6) = 76230976900860740792605252293646252383143627390965685153124757864...
orbit representatives
Following X.-D. Hou's methode, it is not difficult to
determine a representative set of the double coset
for the dimension m=4 :
[ pi-class-4.txt ]
stabilizer
A pair (L,R) in AGL(m) stabilizes the permutation f
L o f o R = f iff f o R o f-1 = L
these pairs form a subgroup of AGL(m,2) x AGL(m,2) with the
law (L,R) (L',R') = ( L'L,RR'). For each R corresponds
at most one L. We define the stabilizer of f :
stab( f ) := { R ∈ AGL(m,2) | f o R o f-1 ∈ AGL(m,2)
Maiorana bent functions
For an arbirary permutation pi of GF(2)^m, and for an arbitrary
Boolean function g in B(m) the Boolean function of 2m variables
MMF : (x,y) mapsto < x, pi(y) > + g(y)
is a Boolean bent function.
Let us consider the action of (A, R) with A ∈ GL(m,2) and R∈ AGL(m,2)
on MMF :
MF( x, y ) o (A,R) = < A(x), π ( R(y)) > + g( R(y) ) = < x, A* o π o R(y) > + g( R(y) )
where A* is the adjoint of A .
we can add any linear function < x, v > without changing the affine class :
MF( x, y ) ≡ < x, L o π o R (y) > + g( R(y) )
where L is the composition of A* by a tanslation v.
stabilizer
[ stabiliser.txt ]
affinity
The affinity of a permutation pi is the number of 2-flats x, y z, t
such that x + y + z + t = 0 and f(x)+f(y)+f(z)+f(t) = 0. The affinity
of an APN permutation is equal to zero.
We confirm Hou's affinity table :
[ table.txt ]
pre-classification
For each permutation pi in TT format, we compute the action of stablizer(pi)
on the space B(2,2,4), orbit representives are listed in ANF format.
[ preclass.txt ]
classification
[ todo:classification.txt ]
all files
[ data ]
Philippe Langevin,
imath,
last modifiication : autumn 2023.