New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra
Abstract
Let q=pn>3 be an odd prime power and let g:Fq→Fq be an even perfect nonlinear (PN) function with g(0)=0. We construct almost perfect nonlinear (APN) functions Gλ,M(x)=g(x)+M(x)+λ(xq−1−1), where λ∈Fq*, −λ∉ g(Fq), and M is an Fp-linearized permutation polynomial. These one-point modifications of g+M have boomerang uniformity one, the smallest possible value for APN functions in odd characteristic. We give complete differential and boomerang spectrum formulas and obtain closed-form evaluations for selected constructions based on Coulter--Matthews monomials and Dembowski--Ostrom polynomials. For q>7, comparison of differential spectra establishes CCZ-inequivalence to all APN monomial families listed in the paper and every APN binomial in the original Ness--Helleseth family or its extension by Zeng et al. We characterize CCZ-equivalence within our construction, showing that any two linearly inequivalent even PN functions over the same field yield CCZ-inequivalent APN functions. We obtain three pairwise CCZ-inequivalent APN functions with boomerang uniformity one over Fp15ℓ for every ℓ≥1, six over Fp120r for every r≥1, and unbounded numbers of such classes in characteristic three. We further prove that CCZ-equivalence and EA-equivalence coincide for functions at Hamming distance at most m from a PN function, provided that m(m+1)<q−q/p. We also show that all our APN functions have the maximal possible algebraic degree n(p−1). Finally, we exhibit both CCZ-equivalent and CCZ-inequivalent APN pairs with boomerang uniformity one at Hamming distance one.
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BibTeX
@misc{koo2026classes,
title = {{New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra}},
author = {Namhun Koo and Soonhak Kwon},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10766},
eprint = {2611.10766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10766},
}
RIS
TY - UNPB
TI - New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra
AU - Namhun Koo
AU - Soonhak Kwon
PY - 2026
DA - 2026/11/07
PB - arXiv
DO - 10.48550/arxiv.2611.10766
UR - https://arxiv.org/abs/2611.10766
AB - Let q=p^n>3 be an odd prime power and let g:F_q→F_q be an even perfect nonlinear (PN) function with g(0)=0. We construct almost perfect nonlinear (APN) functions G_λ,M(x)=g(x)+M(x)+λ(x^q−1−1), where λ∈F_q^*, −λ∉ g(F_q), and M is an F_p-linearized permutation polynomial. These one-point modifications of g+M have boomerang uniformity one, the smallest possible value for APN functions in odd characteristic. We give complete differential and boomerang spectrum formulas and obtain closed-form evaluations for selected constructions based on Coulter--Matthews monomials and Dembowski--Ostrom polynomials. For q>7, comparison of differential spectra establishes CCZ-inequivalence to all APN monomial families listed in the paper and every APN binomial in the original Ness--Helleseth family or its extension by Zeng et al. We characterize CCZ-equivalence within our construction, showing that any two linearly inequivalent even PN functions over the same field yield CCZ-inequivalent APN functions. We obtain three pairwise CCZ-inequivalent APN functions with boomerang uniformity one over F_p^15ℓ for every ℓ≥1, six over F_p^120r for every r≥1, and unbounded numbers of such classes in characteristic three. We further prove that CCZ-equivalence and EA-equivalence coincide for functions at Hamming distance at most m from a PN function, provided that m(m+1)<q−q/p. We also show that all our APN functions have the maximal possible algebraic degree n(p−1). Finally, we exhibit both CCZ-equivalent and CCZ-inequivalent APN pairs with boomerang uniformity one at Hamming distance one.
ER -