A further study of quadratic APN permutations in dimension nine

Abstract

Recently, Beierle and Leander found two new sporadic quadratic APN permutations in dimension 9. Up to EA-equivalence, we present a single trivariate representation of those two permutations as Cu:(F2m)3→(F2m)3,(x,y,z)↦(x3+uy2z,y3+uxz2,z3+ux2y), where m=3 and u∈F23∖{0,1} such that the two permutations correspond to different choices of u. We then analyze the differential uniformity and the nonlinearity of Cu in a more general case. For m≥3 being a multiple of 3 and u∈F2m not being a 7-th power, we show that the differential uniformity of Cu is bounded above by 8, and that the linearity of Cu is bounded above by 81+⌊m2⌋. Based on numerical experiments, we conjecture that Cu is not APN if m is greater than 3. We also analyze the CCZ-equivalence classes of the quadratic APN permutations in dimension 9 known so far and derive a lower bound on the number of their EA-equivalence classes. We further show that the two sporadic APN permutations share an interesting similarity with Gold APN permutations in odd dimension divisible by 3, namely that a permutation EA-inequivalent to those sporadic APN permutations and their inverses can be obtained by just applying EA transformations and inversion to the original permutations.