On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for kbmod n∈{1,2,n−2,n−1}, and exhaustive verification for n≤ 13
Abstract
We study Carlet's cyclic-additive conjecture for the Kasami almost perfect nonlinear (APN) function F(x)=x4k−2k+1 on GF(2n), gcd(k,n)=1: for the 2n−1-element set Δ={F(b)+F(b+1)+1: b∈ GF(2n)} and all distinct nonzero v1,v2∈ GF(2n), |{(x,y,z)∈Δ3 : v1x+v2y+(v1+v2)z=0}| = 22n−3. This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for kbmod n∈{1,2,n−2,n−1}, in particular a complete proof for k=2 (d=13) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible (n,k) with n≤13.
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BibTeX
@misc{nagy2026conjecture,
title = {{On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for $k\bmod n\in\1,2,n-2,n-1\$, and exhaustive verification for $n\le 13$}},
author = {Gábor P. Nagy and Attila Vajda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18584},
eprint = {2608.18584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18584},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for kbmod n∈{1,2,n−2,n−1}, and exhaustive verification for n≤ 13
AU - Gábor P. Nagy
AU - Attila Vajda
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2608.18584
UR - https://arxiv.org/abs/2608.18584
AB - We study Carlet's cyclic-additive conjecture for the Kasami almost perfect nonlinear (APN) function F(x)=x^4^k−2^k+1 on GF(2^n), gcd(k,n)=1: for the 2^n−1-element set Δ={F(b)+F(b+1)+1: b∈ GF(2^n)} and all distinct nonzero v_1,v_2∈ GF(2^n), |{(x,y,z)∈Δ^3 : v_1x+v_2y+(v_1+v_2)z=0}| = 2^2n−3. This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for kbmod n∈{1,2,n−2,n−1}, in particular a complete proof for k=2 (d=13) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible (n,k) with n≤13.
ER -