On Two Fundamental Problems on APN Power Functions

Abstract

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.

The differential spectrum of a class of power functions over finite fields

“…The differential property of F (x) = x d has also been studied for k = 2 in [3,4] while that of F (x) = x d for the case of k = 4 remains unknown. This motivated the authors in [5] and based on experimental data they proposed the following conjecture: If Conjectrue 1 is settled, then the differential property of F (x) = x d for k = 4 can be completely determined. This paper aims to settle Conjectrue 1 and then determine the differential spectrum of F (x) = x d for k = 4 by employing some particular techniques in solving the equation in Conjectrue 1.…”

“…Power functions, namely, monomial functions, as a special class of functions over finite fields, have been extensively studied in the last decades due to their simple algebraic form and lower implementation cost in hardware environment. Very recently, Budaghyan, Calderini, Carlet, Davidova and Kaleyski in [5] presented some observations and computational data on the differential spectra of power functions F (x) = x d with d = k−1 i=1 2 in − 1 over the finite field F 2 nk , where n, k are positive integers. It is worth noting that this class of power functions includes some famous functions as special cases.…”