Researchers Detail Differential Spectrum for a New Family of APN Power Functions

Global: Researchers Detail Differential Spectrum for a New Family of APN Power Functions

In a recent preprint posted to arXiv (arXiv:2307.15657v2), a team of cryptography researchers presented a refined analysis of a known infinite family of almost perfect nonlinear (APN) power functions. The study, released in July 2023, re-expresses the exponents defining these functions to facilitate a comprehensive determination of their differential spectra and to identify the precise elements contained in any fiber of the discrete derivative. The work aims to deepen theoretical understanding of APN functions, which are valued for their resistance to differential cryptanalysis.

Background on APN Functions and Cryptographic Relevance

APN functions are mappings f : F → F over a finite field F such that, for every nonzero a ∈ F*, the equation f(x + a) − f(x) = b has at most two solutions for any b ∈ F. This property ensures the largest cardinality in the differential spectrum equals 2, making APN functions attractive as cryptographic primitives in block ciphers and S‑boxes.

Reformulating the Exponent Family

The authors focus on power functions of the form f(x) = x^d, where d is a positive integer. By introducing a more convenient representation of the exponent d, they align the family with a set of algebraic transformations that simplify subsequent analysis. This re‑expression does not alter the underlying function but makes the structure of its discrete derivatives more tractable.

Methodology: Composing Derivatives with Permutations

To examine the fibers of the discrete derivative Δ_a f(x) = f(x + a) − f(x), the researchers compose Δ_a f with specific permutations and a double covering of the domain. This composition yields an auxiliary function whose fibers can be enumerated directly, allowing the authors to bypass the combinatorial complexity that typically hampers differential spectrum calculations for power functions.

Results: Exact Differential Spectrum and Fiber Elements

The analysis produces a complete differential spectrum for each power function within the targeted family, confirming that the maximum fiber size remains 2, consistent with the APN definition. Moreover, the paper identifies the exact set of field elements that appear in any given fiber of Δ_a f, offering a level of detail that surpasses prior results in the literature.

Implications and Future Research

By delivering a granular view of the differential behavior of these APN power functions, the study provides valuable insights for designers of cryptographic algorithms seeking optimal resistance to differential attacks. The authors suggest that the techniques introduced could be extended to other families of power functions or to broader classes of nonlinear mappings, potentially uncovering additional APN candidates.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.