Welcome to the resource topic for 2025/1780
Title:
There are siblings of \chi which are permutations for n even
Authors: Björn Kriepke, Gohar Kyureghyan
Abstract:Let 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 \to \mathbb F_2^n, k\geq 0, where $$ \gamma_{2k} =
S^{2k}\odot(1+S^{2k-1})\odot(1+S^{2k-3})\odot\ldots\odot(1+S) $$ and S:\mathbb F_2^n\to\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 residue 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.
ePrint: https://eprint.iacr.org/2025/1780
See all topics related to this paper.
Feel free to post resources that are related to this paper below.
Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.
For more information, see the rules for Resource Topics .