关于Kasami APN函数的一个猜想：归约、结构定理、k mod n∈{1,2,n−2,n−1}时的证明，以及n≤13时的穷举验证

中文摘要 AI 辅助

我们研究GF(2ⁿ)上Kasami几乎完美非线性（APN）函数F(x)=x^(4ᵏ−2ᵏ+1)的一个猜想，其中gcd(k,n)=1：对于大小为2ⁿ⁻¹的集合Δ={F(b)+F(b+1)+1: b∈GF(2ⁿ)}，以及GF(2ⁿ)中所有互不相同的非零元素v₁、v₂，满足v₁x+v₂y+(v₁+v₂)z=0的三元组(x,y,z)∈Δ³的数量等于2^(2n−3)。该猜想由NSUCRYPTO~2019密码学奥林匹克竞赛提出（问题提出者未公开）。我们证明了当k mod n∈{1,2,n−2,n−1}时该猜想成立，其中尤其通过二次型理论和精确根计数归约给出了k=2（对应d=13）时的完整证明，并对所有满足n≤13的合法(n,k)，通过计算机进行了穷举验证。

英文摘要

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 $Δ=\{F(b)+F(b+1)+1: b\in GF(2^n)\}$ and all distinct nonzero $v_1,v_2\in GF(2^n)$, \[ \bigl|\{(x,y,z)\inΔ^3 : v_1x+v_2y+(v_1+v_2)z=0\}\bigr| \;=\; 2^{2n-3}. \] 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 $k\bmod n\in\{1,2,n-2,n-1\}$, 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 $n\le13$.

发表机构

• Bolyai Institute, University of Szeged（塞格德大学博莱约研究所）

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