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If used as S-boxes, APN functions provide optimal resilience against differential attacks. However, the very existence of APN permutations operating on an even number n of bits (with n ≥ 8) has been an open problem for nearly 30 years. A possible method to solve this problem consists in generating APN functions, and then exploring the CCZ-equivalence classes of these functions looking for a permutation. Following this goal, we found 5412 new quadratic APN functions on F 2 8 using an approach based on so-called Quadratic APN Matrices (QAM). This brings the number of known CCZ-inequivalent APN functions on F 2 8 to 26525. Unfortunately, none of these new functions are CCZ-equivalent to permutations. A complete list (to the best of our knowledge) of known quadratic APN functions, including our new ones, has been added to sboxU for ease of study by others. In this paper, we recall how to construct new QAMs from a known one. Based on these results and on others on smaller fields, we make two conjectures: that the total number of CCZ-inequivalent quadratic APN functions on F 2 8 exceeds 50000, and that the full list of quadratic APN functions could be obtained by modifying only a small number of entries of the QAM, though such a search remains computationally infeasible at this stage. Finally, we propose a new model which can handle the last two columns together and avoid some redundant computation. [ABSTRACT FROM AUTHOR]

Recently Budaghyan et al. (Cryptogr. Commun. 12, 85–100, 2020) introduced a procedure for investigating if CCZ-equivalence can be more general than EA-equivalence together with inverse transformation (when applicable). In this paper, we show that it is possible to use this procedure for classifying, up to EA-equivalence, all known APN functions in dimension 6. We also give some discussion for dimension 7, 8 and 9. In particular, in these cases it is possible to give an upper bound on the EA-classes contained in the CCZ-classes of the known APN functions. [ABSTRACT FROM AUTHOR]

In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n. [ABSTRACT FROM AUTHOR]

Whether there exist Almost Perfect Non-linear permutations (APN) operating on an even number of bits is the so-called Big APN Problem. It has been solved in the 6-bit case by Dillon et al. in 2009 but, since then, the general case has remained an open problem. In 2016, Perrin et al. discovered the butterfly structure which contains Dillon et al.'s permutation over F 2 6 . Later, Canteaut et al. generalised this structure and proved that no other butterflies with exponent 3 can be APN. Recently, Yongqiang et al. further generalized the structure with Gold exponent and obtained more differentially 4-uniform permutations with optimal nonlinearity. However, the existence of more APN permutations in their generalization was left as an open problem. In this paper, we adapt the proof technique of Canteaut et al. to handle all Gold exponents and prove that a generalised butterfly with Gold exponents over F 2 n can never be APN when n > 3. More precisely, we prove that such a generalised butterfly being APN implies that the branch size is strictly smaller than 5. Hence, the only APN butterflies operate on 3-bit branches, i.e. on 6 bits in total. [ABSTRACT FROM AUTHOR]

A map f : V: = GF(2) → V is APN ( almost perfect nonlinear) if its directional derivatives in nonzero directions are all 2-to-1. If m is greater than 2 and f vanishes at 0, then this derivative condition is equivalent to the condition that the binary linear code of length 2 − 1, whose parity check matrix has jth column equal to ${\omega^j \brack f(\omega^j)}$, is double-error-correcting, where ω is primitive in V. Carlet et al. (Designs Codes Cryptogr 15:125-156, ) proved that this code has dimension 2 − 1 − 2 m; but their indirect proof uses a subtle argument involving general code parameter bounds to show that a double-error correcting code of this length could not be larger. We show here that this result follows immediately from a well-known result on bent functions ...a subject dear to the heart of Jacques Wolfmann. [ABSTRACT FROM AUTHOR]

We present an infinite family of quadrinomial APN functions on GF(2) where n is divisible by 3 but not 9. The family contains inequivalent functions, obtained by setting some coefficients equal to 0. We also discuss the inequivalence proof (by computation) which shows that these functions are new. [ABSTRACT FROM AUTHOR]