Let \(\mathbbm {1}\) be the all-one vector and \(\odot \) denote the component-wise multiplication of two vectors in \(\mathbb {F}_2^n\) . We study the vector space \(\Gamma _n\) over \(\mathbb {F}_2\) generated by the functions \(\gamma _{2k}:\mathbb {F}_2^n \rightarrow \mathbb {F}_2^n, k\ge 0\) , where \( \gamma _{2k}(x) = S^{2k}(x)\odot (\mathbbm {1}+S^{2k-1}(x))\odot (\mathbbm {1}+S^{2k-3}(x))\odot \cdots \odot (\mathbbm {1}+S(x)) \) and \(S:\mathbb {F}_2^n\rightarrow \mathbb {F}_2^n\) is the cyclic left shift function. The functions in \(\Gamma _n\) are shift invariant and the well-known \(\chi \) function used in several cryptographic primitives is contained in \(\Gamma _n\) . For even n, we show that the permutations from \(\Gamma _n\) , with respect to composition, form an abelian group, which is isomorphic to the unit group of the ring \(\mathbb {F}_2[X]/(X^n +X^{n/2})\) . This isomorphism yields an efficient theoretic and algorithmic method for constructing and studying a rich family of shift-invariant permutations on \(\mathbb {F}_2^n\) which are natural generalizations of \(\chi \) . To demonstrate it, we apply the obtained results to investigate the function \(\gamma _0 +\gamma _2+\gamma _4\) on \(\mathbb {F}_2^n\) .
