some fact about APN over GF(2)

This page reports numerical facts related on APN mappings, started on Automn 2016.

### Definitions

Let m be a positive integer. A mapping from GF(2)^m into GF(2)^m is said to be APN when,

for all non zero u, all v :  N(u,v)=#{ x | f(x+u) + f(x) = v} = 0 or 2.

We denote by N_u the Boolean function corresponding to the counting functions v->(N(u,v)>0) and by f_b the Boolean function corresponding to the component x->b.f(x). An automorphism of f is an element of GL(2, 2m) preserving the graph of f.

### linearity of counting

The following APN is an example of APN permutation whose the counting functions are not linear.
f=  0  1  2  6  4 27 12 11  8 30 23 18 24 10 22 25 16  3 29 21 15  9  5 28 17 26 20 14 13  7 19 31
cycle:   1   1   1   5   1   4   4   1   4   4   4   1   1
spectrum:
10:  1 [-12]  3 [-8]  6 [-4]  6 [0]  9 [4]  7 [8]
11:  5 [-8]  5 [-4]  6 [0] 10 [4]  5 [8]  1 [12]
10:  3 [-8]  9 [-4]  6 [0]  6 [4]  7 [8]  1 [12]
1: 31 [0]  1 [32]
poly : a^18 X^1 + a^19 X^2 + a^27 X^3 + a^13 X^4 + a^11 X^5 + a^11 X^6 + a^20 X^7 + a^25 X^8
+ a^29 X^9 + a^17 X^11 + a^13 X^12 + a^22 X^13 + a^3 X^14 + a^8 X^15 + a^6 X^16 + a^27 X^17
+ a^28 X^18 + a^5 X^19 + a^12 X^20 + a^22 X^21 + a^20 X^22 + a^1 X^23 + a^25 X^24 + a^11 X^25
+ a^25 X^26 + a^6 X^27 + X^28 + a^10 X^29 + a^6 X^30
degree : 0 [0]  0 [1]  0 [2]  31 [3]  0 [4]
It has an automorphism of order 5 (x,y)->(xA,yA) where A:=compagnon( 1+T+T^2+T^3+T^4).

### long cycle

An APN permutaion may have various cycle structures. Here is an example of APN permutation of GF(2,6) with a long cycle.
f=  0 15 60 44 14 18 49 10 56 41 55  1 58  3 40 20 30 45 37 13 34 54  4 62 22 57 12 23 33  8 47 29  7 17 53 24 21 51 52 63  9 27 38 25 16 19  6 36 39  2 26 46 48 50 35 43  5 31 32 42 61 59 11 28
cycle:   1  63
spectrum:
7:  6 [-16] 48 [0] 10 [16]
7:  3 [-16] 18 [-8] 12 [0] 30 [8]  1 [16]
21:  2 [-16] 20 [-8] 12 [0] 28 [8]  2 [16]
21:  1 [-16] 22 [-8] 12 [0] 26 [8]  3 [16]
7: 24 [-8] 12 [0] 24 [8]  4 [16]
1: 63 [0]  1 [64]
poly : a^39 X^1 + a^43 X^2 + a^57 X^3 + a^48 X^4 + a^12 X^5 + a^26 X^6 + a^5 X^7 + a^13 X^8
+ a^28 X^9 + a^59 X^10 + a^19 X^11 + a^17 X^12 + a^35 X^13 + a^36 X^14 + a^37 X^15 + a^3 X^16
+ a^18 X^17 + a^61 X^18 + a^47 X^19 + a^33 X^20 + a^29 X^21 + X^22 + a^16 X^23 + a^23 X^24
+ a^43 X^25 + a^10 X^26 + a^12 X^28 + a^2 X^29 + a^23 X^30 + a^48 X^32 + a^23 X^33 + a^13 X^34
+ a^16 X^35 + a^23 X^36 + a^33 X^37 + a^51 X^38 + a^61 X^39 + a^33 X^40 + a^51 X^41 + a^4 X^42
+ a^3 X^43 + a^6 X^44 + a^26 X^45 + a^27 X^46 + a^39 X^48 + a^35 X^49 + a^19 X^50 + a^45 X^51
+ a^25 X^52 + a^47 X^53 + a^34 X^54 + a^31 X^56 + a^31 X^57 + a^52 X^58 + a^51 X^60
degree : 0 [0]  7 [1]  0 [2]  0 [3]  56 [4]  0 [5]
It has an automorphism of order 7 (x,y)->(xA,yA^2) where A:=compagnon( 1+T+T^2+T^3+T^4+T^5+T^6).

## Involution

f=  0 11 30 37 47 52 57 18 59  9 12  1 10 32 60 33 42 21  7 31 41 17 24 23 22 56 26 28 27 58  2 19 13 15 51 35 48  3 39 38 55 20 16 46 44 54 43  4 36 50 49 34  5 53 45 40 25  6 29  8 14 63 62 61

poly : a^44 X^1 + a^43 X^2 + a^60 X^3 + a^50 X^4 + a^59 X^5 + a^37 X^7 + X^8 + a^10 X^9 + a^48 X^10 + a^4 X^11 + a^36 X^12 + a^24 X^13 + a^12 X^14 + a^3 X^15 + a^28 X^16 + a^55 X^17
+ a^40 X^18 + a^27 X^19 + a^39 X^20 + a^54 X^21 + a^29 X^22 + a^8 X^23 + a^35 X^24 + a^50 X^25+ a^23 X^26 + a^35 X^27 + a^56 X^28 + a^6 X^29 + a^56 X^32 + a^23 X^33 + a^41 X^34 + a^48 X^35
+ a^49 X^36 + a^50 X^37 + a^38 X^38 + a^56 X^39 + a^54 X^40 + a^35 X^41 + a^6 X^42 + X^43 + a^26 X^44 + a^16 X^45 + a^13 X^48 + a^38 X^49 + a^7 X^50 + a^1 X^51 + a^58 X^52 + a^7 X^53 + a^6 X^56 + a^12 X^57
where a is a root of the primitive polynomial T^6 + T^3 + T^1 + T^0.
It has an automorphism of order 7 (x,y)->(xA,yA) where A:=compagnon( 1+T+T^2+T^3+T^4+T^5+T^6).

## Permutation

There is only one ccz class of APN permutation of degree 4 whose component space contains a 3-dimensionnal space of cubics.
Philippe Langevin, Last modification, june 2018.