Paper 2023/328
The state diagram of $\chi$
Abstract
In symmetric cryptography, block ciphers, stream ciphers and permutations often make use of a round function and many round functions consist of a linear and a nonlinear layer. One that is often used is based on the cellular automaton that is denoted by $\chi$ as a Boolean map on biinfinite sequences, $\mathbb{F}^{\mathbb{Z}}$. It is defined by $\sigma \mapsto \nu$ where each $\nu_i = \sigma_i + (\sigma_{i+1}+1)\sigma_{i+2}$. A map $\chi_n$ is a map that operatos on $n$bit arrays with periodic boundary conditions. This corresponds with $\chi$ restricted to periodic infinite sequences with period that divides $n$. This map $\chi_n$ is used in various permutations, e.g., Keccakf (the permutation in SHA3), ASCON (the NIST standard for lightweight cryptography), Xoodoo, Rasta and Subterranean (2.0). In this paper, we characterize the graph of $\chi$ on periodic sequences. It turns out that $\chi$ is surjective on the set of \emph{all} periodic sequences. We will show what sequences will give collisions after one application of $\chi$. We prove that, for odd $n$, the order of $\chi_n$ (in the group of bijective maps on $\mathbb{F}^n$) is $2^{\lceil \operatorname{lg}(\frac{n+1}2)\rceil}$. A given periodic sequence lies on a cycle in the graph of $\chi$, or it can be represented as a polynomial. By regarding the divisors of such a polynomial one can see whether it lies in a cycle, or after how many iterations of $\chi$ it will. Furthermore, we can see, for a given $\sigma$, the length of the cycle in its component in the state diagram. Finally, we extend the surjectivity of $\chi$ to $\mathbb{F}^{\mathbb{Z}}$, thus to include nonperiodic sequences.
Metadata
 Available format(s)
 Category
 Secretkey cryptography
 Publication info
 Published elsewhere. Minor revision. Designs, Codes and Cryptography
 DOI
 10.1007/s10623023013498
 Keywords
 boolean mapscellular automatachicryptographystate diagramsymmetric cryptography
 Contact author(s)
 jan schoone @ ru nl
 History
 20240118: revised
 20230306: received
 See all versions
 Short URL
 https://ia.cr/2023/328
 License

CC BYSA
BibTeX
@misc{cryptoeprint:2023/328, author = {Jan Schoone and Joan Daemen}, title = {The state diagram of $\chi$}, howpublished = {Cryptology ePrint Archive, Paper 2023/328}, year = {2023}, doi = {10.1007/s10623023013498}, note = {\url{https://eprint.iacr.org/2023/328}}, url = {https://eprint.iacr.org/2023/328} }