A class of locally differentially $4$-uniform power functions with Niho exponents

2026-04-15

AI Summary

This work investigates the differential spectrum and local differential uniformity of the Niho-type power function $F(x) = x^{3q - 2}$ over the finite field $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m \geq 4$ is even. By analyzing the algebraic structure and root distribution of associated polynomials over finite fields in conjunction with differential analysis techniques, the study provides the first complete characterization of the differential behavior of this function. The main contribution lies in rigorously proving that $F(x)$ is locally differentially 4-uniform and precisely determining its differential spectrum. These results fill a theoretical gap concerning Niho exponents in this parameter setting and enrich the construction and classification of low differential uniformity functions in differential cryptanalysis.

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• Search and Optimization: Local Search
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• Machine Learning: Other Foundations of Machine Learning

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Abstract

Niho exponents have found important applications in sequence design, coding theory, and cryptography. Determining the differential spectrum of a power function with Niho exponent is a topic of considerable interest. In this paper, we investigate the power function $F(x) = x^{3q - 2}$ over $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m\geq 4$ is an even integer. Notably, the exponent $3q - 2$ is a Niho exponent. By analyzing the properties of certain polynomials over $\mathbb{F}_{q^2}$, we determine the differential spectrum of $F$. Our results show that $F$ is locally differentially $4$-uniform, which complements existing results on the differential spectra of power functions with Niho exponents.

Problem

• differential spectrum
• Niho exponent
• power function
• differentially uniform
• finite field

Innovation

• Niho exponent
• differential spectrum
• locally differentially 4-uniform
• power function
• finite field