New Construction of Power Functions with Low c-Differential Uniformity over Finite Fields
Abstract
This paper investigates the c-differential uniformity of power functions over finite fields, an important class of cryptographic functions with favorable differential properties. Specifically, for finite fields 𝔽q satisfying q−1=en with e≥ 3 and e| n, we prove that there exists c∈𝔽q∖{0,1,ε,…,εe−1}, where ε is an e-th primitive root of unity in 𝔽q*, the constructed power functions f(x)=xln+1 with 1≤ l≤ e−1 and gcd(l,e)=1 satisfy the upper bound Δ(f,c)≤ e, provided that certain cyclotomic conditions hold. We show that our conditions are mild; namely, such power functions can be constructed over infinitely many extension fields 𝔽q of 𝔽p for any given e≥ 3 and prime p with pnmid e. Furthermore, based on the Weil bound for multiplicative character sums, we prove that the obtained upper bound is tight for sufficiently large q, demonstrating the optimality of our results. We also analyze the special case c=−1 and derive simplified explicit conditions. In particular, we explicitly characterize the admissible parameters for the case e=3 and present concrete function examples for practical validation.
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BibTeX
@misc{yang2026construction,
title = {{New Construction of Power Functions with Low c-Differential Uniformity over Finite Fields}},
author = {Zhiye Yang and Yan Wang and Keqin Feng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22763},
eprint = {2609.22763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22763},
}
RIS
TY - UNPB
TI - New Construction of Power Functions with Low c-Differential Uniformity over Finite Fields
AU - Zhiye Yang
AU - Yan Wang
AU - Keqin Feng
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22763
UR - https://arxiv.org/abs/2609.22763
AB - This paper investigates the c-differential uniformity of power functions over finite fields, an important class of cryptographic functions with favorable differential properties. Specifically, for finite fields 𝔽_q satisfying q−1=en with e≥ 3 and e| n, we prove that there exists c∈𝔽_q∖{0,1,ε,…,ε^e−1}, where ε is an e-th primitive root of unity in 𝔽_q^*, the constructed power functions f(x)=x^ln+1 with 1≤ l≤ e−1 and gcd(l,e)=1 satisfy the upper bound Δ(f,c)≤ e, provided that certain cyclotomic conditions hold. We show that our conditions are mild; namely, such power functions can be constructed over infinitely many extension fields 𝔽_q of 𝔽_p for any given e≥ 3 and prime p with pnmid e. Furthermore, based on the Weil bound for multiplicative character sums, we prove that the obtained upper bound is tight for sufficiently large q, demonstrating the optimality of our results. We also analyze the special case c=−1 and derive simplified explicit conditions. In particular, we explicitly characterize the admissible parameters for the case e=3 and present concrete function examples for practical validation.
ER -