Algebraic Structure of the Iterates of $\chi$

Björn Kriepke; Gohar Kyureghyan

Paper 2024/801

Algebraic Structure of the Iterates of $\chi$

Björn Kriepke

, University of Rostock

Gohar Kyureghyan

, University of Rostock

Abstract

We consider the map $\chi:\mathbb{F}_2^n\to\mathbb{F}_2^n$ for $n$ odd given by $y=\chi(x)$ with $y_i=x_i+x_{i+2}(1+x_{i+1})$, where the indices are computed modulo $n$. We suggest a generalization of the map $\chi$ which we call generalized $\chi$-maps. We show that these maps form an Abelian group which is isomorphic to the group of units in $\mathbb{F}_2[X]/(X^{(n+1)/2})$. Using this isomorphism we easily obtain closed-form expressions for iterates of $\chi$ and explain their properties.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published by the IACR in CRYPTO 2024
Keywords: \chi-map shift-invariant functions iterates sha3
Contact author(s): bjoern kriepke @ uni-rostock de
gohar kyureghyan @ uni-rostock de
History: 2024-05-24: approved; 2024-05-23: received; See all versions
Short URL: https://ia.cr/2024/801
License: CC BY

BibTeX

@misc{cryptoeprint:2024/801,
      author = {Björn Kriepke and Gohar Kyureghyan},
      title = {Algebraic Structure of the Iterates of $\chi$},
      howpublished = {Cryptology ePrint Archive, Paper 2024/801},
      year = {2024},
      note = {\url{https://eprint.iacr.org/2024/801}},
      url = {https://eprint.iacr.org/2024/801}
}