The six infinite families of power APN functions are among the oldest known instances of APN functions, and it has been conjectured in 2000 that they exhaust all possible power APN functions. Another long-standing open problem is that of the Walsh spectrum of the Dobbertin power family, which is still unknown. Those of Kasami, Niho and Welch functions are known, but not the precise values of their Walsh transform, with rare exceptions. One promising approach that could lead to the resolution of these problems is to consider alternative representations of the functions in questions. We derive alternative representations for the infinite APN monomial families. We show how the Niho, Welch, and Dobbertin functions can be represented as the composition x i • x 1/j of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can be used to represent the functions in this way. We investigate compositions x i • L • x 1/j for a linear polynomial L, show how the Kasami functions in odd dimension can be expressed in this way with i = j being a Gold exponent and compute all APN functions of this form for n ≤ 9 and for L with binary coefficients, thereby showing that our theoretical constructions exhaust all possible cases. We present observations and data on power functions with exponent k−1 i=1 2 2ni − 1 which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for n ≤ 35, and conjecture its exact form. As an application of our results, we determine the exact values of the Walsh transform of the Kasami function at all points of a special form. Computations performed for n ≤ 21 show that these points cover about 2/3 of the field.

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In 2008 Budaghyan, Carlet and Leander generalized a known instance of an APN function over the finite field F 2 12 and constructed two new infinite families of APN binomials over the finite field F2n , one for n divisible by 3, and one for n divisible by 4. By relaxing conditions, the family of APN binomials for n divisible by 3 was generalized to a family of differentially 2 t -uniform functions in 2012 by Bracken, Tan and Tan; in this sense, the binomials behave in the same way as the Gold functions. In this paper, we show that when relaxing conditions on the APN binomials for n divisible by 4, they also behave in the same way as the Gold function x 2 s +1 (with s and n not necessarily coprime). As a counterexample, we also show that a family of APN quadrinomials obtained as a generalization of a known APN instance over F 2 10 cannot be generalized to functions with 2 t -to-1 derivatives by relaxing conditions in a similar way.

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