Quadratic Binomial APN Functions and Absolutely Irreducible Polynomials

Progress Towards the Conjecture on APN Functions and Absolutely Irreducible Polynomials

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami…

具有低爆破均匀性的局部APN二项式

Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic

本文研究了在奇特征下，局部APN二项式函数的爆破均匀性，通过条件分析扩展了相关结果，并探讨了特定函数的差分谱和爆破谱。

On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in $\Bbb{P}^3$ and the conjecture on exceptional APN functions: the Kasami-Welch degree case

Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McG…

一类二项式的差分与回旋镖性质

The Differential and Boomerang Properties of a Class of Binomials

本文研究了有限域上一类二项式函数的差分与回旋镖性质，确定了其差分均匀度、差分谱及回旋镖均匀度，并推翻了关于存在无穷多 APN 函数实例的猜想。

关于Dillon的APN六项式分类

On the Classification of Dillon's APN Hexanomials

本文研究了Dillon六项式在有限域上的分类，通过代数方法推导APN条件并排除大量六项式，发现其代数障碍限制了差分均匀性，并通过计算验证了APN六项式的CCZ等价性。

伽罗瓦环上APN置换的约化多项式提升与有效非APN界

Reduced APN permutations and a disproof of the Galois-ring lifting conjecture

本文阐明了伽罗瓦环上的APN提升猜想，证明了有效非APN界，给出了偶次数推论，并证明三次置换多项式满足修正后的提升猜想。