A Further Study of Quadratic APN Permutations in Dimension Nine
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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 C_u (𝔽_2^m)^3 → (𝔽_2^m)^3, (x,y,z) ↦ (x^3+uy^2z, y^3+uxz^2,z^3+ux^2y), where m=3 and u ∈𝔽_2^3∖{0,1} such that the two permutations correspond to different choices of u. We then analyze the differential uniformity and the nonlinearity of C_u in a more general case. In particular, for m ≥ 3 being a multiple of 3 and u ∈𝔽_2^m not being a 7-th power, we show that the differential uniformity of C_u is bounded above by 8, and that the linearity of C_u is bounded above by 8^1+⌊m/2⌋. Based on numerical experiments, we conjecture that C_u 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.
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