A further study of quadratic APN permutations in dimension nine

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