Carlet's cyclic-additive conjecture for the Kasami monomials
Abstract
Let K be a finite field of characteristic two with |K| = 2n, let gcd(k,n) = 1, let dk = 4k − 2k + 1 be the Kasami exponent, and let Δk = {(b+1)dk + bdk + 1 : b ∈ K} be the image of the normalised derivative of the Kasami monomial in the direction $1$. We show that, for all distinct nonzero v1,v2 ∈ K, |{(x,y,z) ∈ Δk3 : v1x + v2y + (v1+v2)z = 0}| = 22n−3. This establishes the cyclic-additive difference-set condition introduced by Carlet and later posed for the Kasami functions at NSUCRYPTO~2019. Starting from the known half-size property of the derivative image, we express the Fourier correction as twisted root counts and prove their required nonnegativity by an incidence argument on the Fermat cubic. An exact average over the slopes then forces equality pointwise. The argument covers every admissible pair (n,k) and has been formalised and machine-checked in Lean~4 with Mathlib.
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BibTeX
@misc{nagy2026carlets,
title = {{Carlet's cyclic-additive conjecture for the Kasami monomials}},
author = {Gábor P. Nagy and Douglas S. McNeil and Attila Vajda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07905},
eprint = {2610.07905},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07905},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Carlet's cyclic-additive conjecture for the Kasami monomials
AU - Gábor P. Nagy
AU - Douglas S. McNeil
AU - Attila Vajda
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07905
UR - https://arxiv.org/abs/2610.07905
AB - Let K be a finite field of characteristic two with |K| = 2^n, let gcd(k,n) = 1, let d_k = 4^k − 2^k + 1 be the Kasami exponent, and let Δ_k = {(b+1)^d_k + b^d_k + 1 : b ∈ K} be the image of the normalised derivative of the Kasami monomial in the direction $1$. We show that, for all distinct nonzero v_1,v_2 ∈ K, |{(x,y,z) ∈ Δ_k^3 : v_1x + v_2y + (v_1+v_2)z = 0}| = 2^2n−3. This establishes the cyclic-additive difference-set condition introduced by Carlet and later posed for the Kasami functions at NSUCRYPTO~2019. Starting from the known half-size property of the derivative image, we express the Fourier correction as twisted root counts and prove their required nonnegativity by an incidence argument on the Fermat cubic. An exact average over the slopes then forces equality pointwise. The argument covers every admissible pair (n,k) and has been formalised and machine-checked in Lean~4 with Mathlib.
ER -