[Resource Topic] 2024/801: Algebraic Structure of the Iterates of $\chi$

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Title:
Algebraic Structure of the Iterates of \chi

Authors: Björn Kriepke, Gohar Kyureghyan

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.

ePrint: https://eprint.iacr.org/2024/801

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