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Almost perfect nonlinear power functions with exponents expressed as fractions

Daniel J. Katz, Kathleen R. O'Connor, Kyle Pacheco, Yakov Sapozhnikov

TL;DR

The paper addresses the problem of characterizing almost perfect nonlinear (APN) power functions on finite fields with exponents expressed as fractions. It develops a novel framework that reexpresses these exponents, then applies a sequence of fiber-permutation transformations and a finite-field double covering to transform the discrete derivative into a form whose fibers can be determined explicitly. The main contribution is a complete fiber-by-fiber determination for a family of APN power functions over characteristic $3$, yielding the reduced differential spectrum $\frac{3^n-3}{2}[0]+3[1]+\frac{3^n-3}{2}[2]$ and establishing equivalence of these exponents with a known Zha–Wang family; this shows APN behavior for $n>1$ and PN when $n=1$. The methods provide efficient, polynomial-time computation of fiber contents and deepen understanding of APN power maps in odd characteristic, with implications for cryptographic design of nonlinear primitives.

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 $Δ_a f \colon F \to F$ 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 $F^*$. An almost perfect nonlinear (APN) function is one for which the largest cardinality in its differential spectrum is $2$. Almost perfect nonlinear functions are of interest as cryptographic primitives. If $d$ is a positive integer, then the power function over $F$ with exponent $d$ is the function $f \colon F \to F$ with $f(x)=x^d$ for every $x \in F$. 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 not only to obtain the differential spectrum of each power function $f$ with an exponent in our family, but also to determine the elements that lie in an arbitrary fiber of the discrete derivative of $f$. This differential analysis, which is far more detailed than previous results, is achieved by composing the discrete derivative of $f$ with some permutations and a double covering of its domain to obtain a function whose fibers can more readily be analyzed.

Almost perfect nonlinear power functions with exponents expressed as fractions

TL;DR

The paper addresses the problem of characterizing almost perfect nonlinear (APN) power functions on finite fields with exponents expressed as fractions. It develops a novel framework that reexpresses these exponents, then applies a sequence of fiber-permutation transformations and a finite-field double covering to transform the discrete derivative into a form whose fibers can be determined explicitly. The main contribution is a complete fiber-by-fiber determination for a family of APN power functions over characteristic , yielding the reduced differential spectrum and establishing equivalence of these exponents with a known Zha–Wang family; this shows APN behavior for and PN when . The methods provide efficient, polynomial-time computation of fiber contents and deepen understanding of APN power maps in odd characteristic, with implications for cryptographic design of nonlinear primitives.

Abstract

Let be a finite field, let be a function from to , and let be a nonzero element of . The discrete derivative of in direction is with . The differential spectrum of is the multiset of cardinalities of all the fibers of all the derivatives as runs through . An almost perfect nonlinear (APN) function is one for which the largest cardinality in its differential spectrum is . Almost perfect nonlinear functions are of interest as cryptographic primitives. If is a positive integer, then the power function over with exponent is the function with for every . 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 not only to obtain the differential spectrum of each power function with an exponent in our family, but also to determine the elements that lie in an arbitrary fiber of the discrete derivative of . This differential analysis, which is far more detailed than previous results, is achieved by composing the discrete derivative of with some permutations and a double covering of its domain to obtain a function whose fibers can more readily be analyzed.
Paper Structure (19 sections, 34 theorems, 92 equations)

This paper contains 19 sections, 34 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Valuations
  4. Greatest common divisors
  5. Quadratic residues and nonresidues
  6. Power functions and roots
  7. Algebraic lemmas
  8. Permuting fibers
  9. General principles
  10. Fibers of the discrete derivative
  11. A double covering of a finite field
  12. Half field, conjugation, and the unit circle
  13. The fiber doubling map
  14. Analysis of the fibers of the function of Lemma \ref{['Dorothy']} via the fiber doubling map
  15. When $d_1=p^j+1$ and $d_2=p^k+1$ over a field of odd characteristic $p$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $F$ be a finite field of order $q=3^n$ with $n$ odd and let $k$ be an even nonnegative integer with $\gcd(k,n)=1$. Let $f\colon F \to F$ be the power function with exponent $(3^n+1)/(3^k+1)$. If $\eta$ is the extended quadratic character of $F$, then we have the following determination of the fi Therefore, for any $c \in F^*$, we have $\left|{(\Delta f)^{-1}(\{c\})}\right|=1+\eta(1-c^2)$, and

Theorems & Definitions (71)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 61 more