International Association
for Cryptologic Research

In this paper, we present a new type of algebraic attack that applies to many recent arithmetization-oriented families of permutations, such as those used in Griffin, Anemoi, ArionHash, and XHash8, whose security relies on the hardness of the constrained-input constrained-output (CICO) problem. We introduce the FreeLunch approach: the monomial ordering is chosen so that the natural polynomial system encoding the CICO problem already is a Gröbner basis. In addition, we present a new dedicated resolution algorithm for FreeLunch systems, of complexity lower than applicable state-of-the-art FGLM algorithms. We show that the FreeLunch approach challenges the security of full-round instances of Anemoi, Arion and Griffin. We confirm these theoretical results with experimental results on those three permutations. In particular, using the FreeLunch attack combined with a new technique to bypass 3 rounds of Griffin, we recover a CICO solution for 7 out of 10 rounds of Griffin in less than four hours on one core of AMD EPYC 7352 (2.3GHz).

Symmetric primitives are a cornerstone of cryptography, and have traditionally been defined over fields, where cryptanalysis is now well understood. However, a few symmetric primitives defined over rings Z _q for a composite number q have recently been proposed, a setting where security is much less studied. In this paper we focus on studying established algebraic attacks typically defined over fields and the extent of their applicability to symmetric primitives defined over the ring of integers modulo a composite q. Based on our analysis, we present an attack on full Rubato, a family of symmetric ciphers proposed by Ha et al. at Eurocrypt 2022 designed to be used in a transciphering framework for approximate fully homomorphic encryption. We show that at least 25% of the possible choices for q satisfy certain conditions that lead to a successful key recovery attack with complexity significantly lower than the claimed security level for five of the six ciphers in the Rubato family.

In this paper, we study the effect of two modifications to multivariate public key encryption schemes: internal perturbation (ip), and Q_+. Focusing on the Dob encryption scheme, a construction utilising these modifications, we accurately predict the number of degree fall polynomials produced in a Gröbner basis attack, up to and including degree five. The predictions remain accurate even when fixing variables. Based on this new theory we design a novel attack on the Dob encryption scheme, which breaks Dob using the parameters suggested by its designers. While our work primarily focuses on the Dob encryption scheme, we also believe that the presented techniques will be of particular interest to the analysis of other big-field schemes.