Paper 2023/1708
Algebraic properties of the maps $\chi_n$
Abstract
The Boolean map $\chi_n \colon \mathbb{F}_2^n \to \mathbb{F}_2^n,\ x \mapsto y$ 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., Keccakf (the SHA3permutation), 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 $\mathbb{F}_2$.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint.
 Keywords
 Boolean mapschicryptographypolynomial mapspower functionssymmetric cryptography
 Contact author(s)
 jan schoone @ ru nl
 History
 20231106: approved
 20231103: received
 See all versions
 Short URL
 https://ia.cr/2023/1708
 License

CC BYSA
BibTeX
@misc{cryptoeprint:2023/1708, author = {Jan Schoone and Joan Daemen}, title = {Algebraic properties of the maps $\chi_n$}, howpublished = {Cryptology ePrint Archive, Paper 2023/1708}, year = {2023}, note = {\url{https://eprint.iacr.org/2023/1708}}, url = {https://eprint.iacr.org/2023/1708} }