关于反函数的CCZ-等价性
On CCZ-equivalence of the inverse function
摘要：
$\mathbb{F}_{2^n}$上的反函数$x \mapsto x^{-1}$是密码学中研究最多的函数之一，因为它广泛用作AES等分组密码中的S盒。本文证明了，若$n\geq 5$，则每一个与反函数CCZ-等价的函数都是EA-等价的，从而证实了Budaghyan，Calderini和Villa的一个猜想.本文主要证明了如果$n\geq 5$，则在$\mathbb{F}_{2^n}$上不存在$L_1(x^{-1})+L_2(x)$形式的置换多项式，其中$L_1,L_2$是非零线性函数。在证明中，我们结合Kloosterman和，二次型和工具添加剂组合。
英文摘要：
The inverse function $x \mapsto x^{-1}$ on $\mathbb{F}_{2^n}$ is one of the most studied functions in cryptography due to its widespread use as an S-box in block ciphers like AES. In this paper, we show that, if $n\geq 5$, every function that is CCZ-equivalent to the inverse function is already EA-equivalent to it. This confirms a conjecture by Budaghyan, Calderini and Villa. We also prove that every permutation that is CCZ-equivalent to the inverse function is already affine equivalent to it. The majority of the paper is devoted to proving that there are no permutation polynomials of the form $L_1(x^{-1})+L_2(x)$ over $\mathbb{F}_{2^n}$ if $n\geq 5$, where $L_1,L_2$ are nonzero linear functions. In the proof, we combine Kloosterman sums, quadratic forms and tools from additive combinatorics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
