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 two permutations as Cu : (F2m ) 3 → (F2m ) , (x, y, z) 7→ (x + uyz, y + uxz, z + uxy), 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. In particular, 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 8 1+⌊ 2 . 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 function EA-inequivalent to those sporadic APN permutations and their inverses can be obtained by just applying EA transformations and inversion to the original permutations.