Paper 2022/399

The Inverse of $\chi$ and Its Applications to Rasta-like Ciphers

Fukang Liu, Santanu Sarkar, Willi Meier, and Takanori Isobe

Abstract

At ASIACRYPT 2021, Liu et al. pointed out a weakness of the Rasta-like ciphers neglected by the designers. The main strategy is to construct exploitable equations of the $n$-bit $\chi$ operation denoted by ${\chi }_n$. However, these equations are all obtained by first studying ${\chi }_n$ for small $n$. In this note, we demonstrate that if the explicit formula of the inverse of ${\chi }_n$ denoted by ${\chi }_n^{-1}$ is known, all these exploitable equations would have been quite obvious and the weakness of the Rasta-like ciphers could have been avoided at the design phase. However, the explicit formula of ${\chi }_n^{-1}$ seems to be not well-known and the most relevant work was published by Biryukov et al. at ASIACRYPT 2014. In this work, we give a very simple formula of ${\chi }_n^{-1}$ that can be written down in only one line and we prove its correctness in a rigorous way. Based on its formula, the formula of exploitable equations for Rasta-like ciphers can be easily derived and therefore more exploitable equations are found.