The Boolean map \(\chi _n :\mathbb {F}_2^n \rightarrow \mathbb {F}_2^n,\ x \mapsto y\) defined by \(y_i = x_i + (x_{i+1}+1)x_{i+2}\) (where \(i\in \mathbb {Z}/n\mathbb {Z}\)) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider \(\chi _n\) (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if \(n=1,3\). We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of \(\chi _n\) (if it exists). This number coincides with binomial coefficients. Lastly, we consider \(\chi _n\) as a polynomial map, to study whether the same rule (\(y_i = x_i + (x_{i+1}+1)x_{i+2}\)) gives a bijection on field extensions of \(\mathbb {F}_2\). We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of \(\mathbb {F}_2\).

布尔映射\(\chi _n :\mathbb {F}_2^n \rightarrow \mathbb {F}_2^n,\ x \mapsto y\)定义为\(y_i = x_i + (x_{i+1}) +1)x_{i+2}\)（其中\(i\in \mathbb {Z}/n\mathbb {Z}\)）用于作为加密方案一部分的各种排列，例如Keccak -f （SHA-3 排列）、ASCON（NIST 轻量级竞赛的获胜者）、Xoodoo、Rasta 和 Subterranean (2.0)。在本文中，我们研究了该映射的各种代数性质。我们将\(\chi _n\)（通过向量同构）视为单变量多项式。我们证明它是一个幂函数当且仅当\(n=1,3\)。我们还计算这些单变量多项式的稀疏性和次数的界限，以及不同单变量表示的数量。其次，我们计算\(\chi _n\)的倒数中给定次数的单项式的数量（如果存在）。该数字与二项式系数一致。最后，我们将\(\chi _n\)视为多项式映射，以研究相同的规则 ( \(y_i = x_i + (x_{i+1}+1)x_{i+2}\) )是否给出\(\mathbb {F}_2\)的字段扩展上的双射。我们证明，对于次数可被二或三整除的扩展，情况并非如此。基于这些结果，我们推测该规则不会在\(\mathbb {F}_2\)的任何扩展域上给出双射。