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On functions of low differential uniformity in characteristic 2: A close look (I)

Nurdagül Anbar, Tekgül Kalaycı, Alev Topuzoğlu

TL;DR

This work introduces APN-defect, a quantitative measure of how far a function G: F_${2^n}$ → F_${2^n}$ is from being APN, by aggregating global and row-wise derivative information encoded in difference-squares. It connects APN-defect to established notions such as the (p_a) property, x_0-partial APN-ness, and the VF_G partial quadruple system, and shows how difference squares reveal vanishing flats and spectral data that influence differential behavior. The authors derive bounds and exact values for APN-defect across function classes (including power and DO polymorphisms) and compute precise APN-defect values for two-point inverses F_{0,α}, relating them to the Carlitz-rank of inverse-based modifications. They also characterize the associated partial quadruple systems and demonstrate that APN-defect can distinguish CCZ-inequivalent functions, highlighting its potential for guiding the construction of APN permutations and informing differential-cryptanalysis considerations. The paper lays groundwork for a second part that extends the analysis to higher Carlitz ranks and infinite extension families, aiming for a broader understanding of differential properties in characteristic 2.

Abstract

We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function $G:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}$ to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) $G$ using the so-called difference squares. We describe the relations between the APN-defect and other recent concepts of similar nature. Upper and lower bounds for the values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. Its exact values in some cases are also calculated. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on $n$ is evaluated and the implications are discussed. In the forthcoming second part of this work we further examine modifications of the inverse function. We also study modifications of classes of functions of low uniformity over infinitely many extensions of $\mathbb{F}_{2^n}$. We present quantitative results on their differential behaviour, especially in connection with their APN-defects.

On functions of low differential uniformity in characteristic 2: A close look (I)

TL;DR

This work introduces APN-defect, a quantitative measure of how far a function G: F_ → F_ is from being APN, by aggregating global and row-wise derivative information encoded in difference-squares. It connects APN-defect to established notions such as the (p_a) property, x_0-partial APN-ness, and the VF_G partial quadruple system, and shows how difference squares reveal vanishing flats and spectral data that influence differential behavior. The authors derive bounds and exact values for APN-defect across function classes (including power and DO polymorphisms) and compute precise APN-defect values for two-point inverses F_{0,α}, relating them to the Carlitz-rank of inverse-based modifications. They also characterize the associated partial quadruple systems and demonstrate that APN-defect can distinguish CCZ-inequivalent functions, highlighting its potential for guiding the construction of APN permutations and informing differential-cryptanalysis considerations. The paper lays groundwork for a second part that extends the analysis to higher Carlitz ranks and infinite extension families, aiming for a broader understanding of differential properties in characteristic 2.

Abstract

We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) using the so-called difference squares. We describe the relations between the APN-defect and other recent concepts of similar nature. Upper and lower bounds for the values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. Its exact values in some cases are also calculated. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on is evaluated and the implications are discussed. In the forthcoming second part of this work we further examine modifications of the inverse function. We also study modifications of classes of functions of low uniformity over infinitely many extensions of . We present quantitative results on their differential behaviour, especially in connection with their APN-defects.
Paper Structure (13 sections, 17 theorems, 72 equations, 2 tables)

This paper contains 13 sections, 17 theorems, 72 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Difference squares
  4. Difference squares and partial quadruple systems
  5. A new measure: APN-defect
  6. APN-defect of some special functions
  7. APN-defect and modifications of the inverse function
  8. Modifications of the inverse permutation: Carlitz rank
  9. APN-defect of $F_{0,\alpha}$
  10. The partial quadruple system associated to $F_{0, \alpha}$
  11. Overview
  12. Conclusion
  13. Acknowledgement

Key Result

lemma 1

Let $F(x)=x^{2^n-2}$ and $n$ be even. Put $\omega={\zeta}^{\frac{2^n-1}{3}}$ for a primitive element $\zeta$ of $\mathbb{F}_{2^n}$.The partial quadruple system associated to $F$ is $(\mathbb{F}_{2^n}, VF_F)$, where Hence, the number of vanishing flats with respect to $F$ is given as $|VF_F|=\frac{2^n-1}{3}.$

Theorems & Definitions (45)

  • definition 1
  • definition 2
  • definition 3
  • remark 1
  • remark 2
  • lemma 1
  • lemma 2
  • proof
  • remark 3
  • remark 4
  • ...and 35 more