On the Properties of the Boolean Functions Associated to the Differential Spectrum of General APN Functions and Their Consequences

Gold functions and switched cube functions are not 0-extendable in dimension n > 5

“…+ l 2 m 2 + a 2 m+1 l 2 m 1 + a 2 m + a 2 m+2 = (al 2 + l 2 2 + a 4 l 2 1 + a 2 + a 8 ) 2 m−1 = 0 and applying Relation (8) to m = n − 3 (which is even) yields 4 and l 2 = a 4 + a. Plugging those two values of l 2 into the first equation of the system yields a 17 = a 14 , that is, a 3 = 1, that is, a ∈ F * 2 gcd(2,n) , that is, a = 1 since n is necessarily odd, in which case the system has l 2 = 0 and l 2 = 1 as the only solutions.…”

“…It is not clear whether this property of the known infinite classes is because they are in fact peculiar (and would therefore not be representative of general APN functions) or whether there is a significant lower bound to be found on the nonlinearity of a large subclass of APN functions. There is a lower bound stronger than strict positivity, which is valid for a large subclass of APN functions that includes all known ones, see [8], but it is not significant. We consider that determining which is true between the two terms of the above alternative is currently the most important open problem on substitution boxes (vectorial functions) along with the "big APN problem" (consisting in finding an APN permutation in an even number of variables at least 8).…”