We study the problem of how to find the inverse of shift invariant (SI) transformations proposed in Daemen’s thesis. In particular, two of them have been used in practice: \(y_i=x_i\oplus \overline{x_{i+1}}x_{i+2}\) and \(y_i=x_i\oplus \overline{x_{i+1}}x_{i+2}x_{i+3}\) . The first one is the well-known \(\chi \) transformation used in SHA-3, Subterranean 2.0 and Rasta, while the second one is used in a recently proposed ZK-friendly hash function called Monolith. While the concrete formula of the inverse of \(\chi \) of arbitrary size has been given and proved by Liu et al. at JoC 2022, it remains unknown how to deduce such a formula and how to systematically study other SI transformations. In this work, we aim to provide a general method and flow to find the inverse of SI transformations, though it is still limited to some specific types and it may not work for all such transformations. However, such a general method does shed new insight on how to find their inverse, as we can apply this method to several different SI transformations, including the one used in Monolith. We expect that this method can be further generalized and applied to more SI transformations.
