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The second-order zero differential spectra of some functions over finite fields

Kirpa Garg, Sartaj Ul Hasan, Constanza Riera, Pantelimon Stanica

TL;DR

This work computes second-order zero differential spectra for a variety of low-differential-uniformity functions over finite fields, across both odd and even characteristics. By connecting the second-order spectra to FBCT, vanishing flats, and sum-freedom, the authors show that many APN, PN, and related power and permutation maps exhibit remarkably low second-order uniformity, sometimes achieving 1-, 2-, or 3-uniformity in odd characteristic and bounded multi-valued spectra in even characteristic. The results include explicit spectra for several power maps, detailed vanishing-flat counts, and the analysis of nonmonomial permutations, highlighting the independence and interaction of boomerang-type measures with classical differential uniformity. Collectively, the paper extends prior work by Boukerrou et al. and Li et al., offering new techniques for solving finite-field equations and paving ways to refine cryptographic criteria beyond differential uniformity.

Abstract

It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions over finite fields of odd characteristic, pushing further the study started in Boukerrou et al. and continued in Li et al.~[Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity. Moreover, we study the second-order zero differential spectra of certain functions with low differential uniformity over finite fields of even characteristic. We connect this new concept to the sum-freedom and vanishing flats concepts and find some counts for the number of vanishing flats via our methods. We provide detailed analyses on several equations over finite fields that may have an interest outside of the scope of our paper.

The second-order zero differential spectra of some functions over finite fields

TL;DR

This work computes second-order zero differential spectra for a variety of low-differential-uniformity functions over finite fields, across both odd and even characteristics. By connecting the second-order spectra to FBCT, vanishing flats, and sum-freedom, the authors show that many APN, PN, and related power and permutation maps exhibit remarkably low second-order uniformity, sometimes achieving 1-, 2-, or 3-uniformity in odd characteristic and bounded multi-valued spectra in even characteristic. The results include explicit spectra for several power maps, detailed vanishing-flat counts, and the analysis of nonmonomial permutations, highlighting the independence and interaction of boomerang-type measures with classical differential uniformity. Collectively, the paper extends prior work by Boukerrou et al. and Li et al., offering new techniques for solving finite-field equations and paving ways to refine cryptographic criteria beyond differential uniformity.

Abstract

It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the -boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~ on ( prime) and the one of almost perfect nonlinear functions on is~. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions over finite fields of odd characteristic, pushing further the study started in Boukerrou et al. and continued in Li et al.~[Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity. Moreover, we study the second-order zero differential spectra of certain functions with low differential uniformity over finite fields of even characteristic. We connect this new concept to the sum-freedom and vanishing flats concepts and find some counts for the number of vanishing flats via our methods. We provide detailed analyses on several equations over finite fields that may have an interest outside of the scope of our paper.
Paper Structure (7 sections, 18 theorems, 98 equations, 3 tables)

This paper contains 7 sections, 18 theorems, 98 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A connection between vanishing flats, sum-freedom, an alternative definition of APNness for $p$ odd, and the FBCT
  4. The second-order zero differential spectrum for power functions over finite fields of odd characteristic
  5. The second-order zero differential spectrum for power functions over finite fields of even characteristic
  6. The second-order zero differential spectrum for some differentially low uniform permutation polynomials in even characteristic
  7. Conclusion

Key Result

Lemma 2.5

Bouk Let $n \geq 1$ be an even integer. Let $\omega$ and $\omega + 1$ be the solutions of the equation $X^2 + X + 1$ in $\mathbb F_{2^n}$. Let $F$ be the inverse function defined over $\mathbb F_{2^n}$ by $F(0) = 0$ and $F(X) = \frac{1}{X}$ for $X \neq 0$. The FBCT of F satisfies Moreover, $X\in \{0,a,b,a+b\}$ are the four solutions in the above system.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 27 more