论文标题
关于逆函数的CCZ-等效性
On CCZ-equivalence of the inverse function
论文作者
论文摘要
$ \ mathbb {f} _ {2^n} $上的倒数函数$ x \ mapsto x^{ - 1} $是加密术中研究最多的功能之一，因为它的广泛使用用作block ciphers的s-box。在本文中，我们表明，如果$ n \ geq 5 $，则与逆函数相当的每个功能已经与之相等。这证实了Budaghyan，Calderini和Villa的猜想。我们还证明，与逆函数等效的CCZ等效的每个排列已经相当于它。本文的大部分都致力于证明$ l_1（x^{ - 1}）+l_2（x）$ over $ \ mathbb {f} _ {2^n} $，如果$ n \ geq 5 $，$ l_1，l_1，l_2 $是非Zero linear linear linear functions。在证明中，我们结合了Kloosterman总和，辅助组合学中的二次形式和工具。
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.
