Almost perfect nonlinear power functions with exponents expressed as fractions

2023/07/28 by Katz, Daniel J., O'Connor, Kathleen R., Pacheco, Kyle +1
#Combinatorics (math.CO) #Cryptography and Security (cs.CR) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.15657

Abstract

Let F be a finite field, let f be a function from F to F, and let a be a nonzero element of F. The discrete derivative of f in direction a is Δ_a f \colon F → F with (Δ_a f)(x)=f(x+a)-f(x). The differential spectrum of f is the multiset of cardinalities of all the fibers of all the derivatives Δ_a f as a runs through F^*. The function f is almost perfect nonlinear (APN) if the largest cardinality in the differential spectrum is 2. Almost perfect nonlinear functions are of interest as cryptographic primitives. If d is a positive integer, the power function over F with exponent d is the function f \colon F → F with f(x)=x^d for every x ∈ F. There is a small number of known infinite families of APN power functions. In this paper, we re-express the exponents for one such family in a more convenient form. This enables us to give the differential spectrum and, even more, to determine the sizes of individual fibers of derivatives.

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