International Association
for Cryptologic Research

<p>Restricted syndrome decoding problems (R-SDP and R-SDP($G$)) provide an interesting basis for post-quantum cryptography. Indeed, they feature in CROSS, a submission in the ongoing process for standardizing post-quantum signatures.</p><p>This work improves our understanding of the security of both problems. Firstly, we propose and implement a novel collision attack on R-SDP($G$) that provides the best attack under realistic restrictions on memory. Secondly, we derive precise complexity estimates for algebraic attacks on R-SDP that are shown to be accurate by our experiments. We note that neither of these improvements threatens the updated parameters of CROSS. </p>

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).

The Regular Syndrome Decoding (RSD) problem, a variant of the Syndrome Decoding problem with a particular error distribution, was introduced almost 20 years ago by Augot \emph{et al.}. In this problem, the error vector is divided into equally sized blocks, each containing a single noisy coordinate. More recently, the last five years have seen increased interest in this assumption due to its use in MPC and ZK applications. Generally referred to as ``LPN with regular noise" in this context, the assumption allows to achieve better efficiency when compared to plain LPN. In all previous works of cryptanalysis, it has not been shown how to exploit the special feature of this problem in an attack. We present the first algebraic attack on RSD. Based on a careful theoretical analysis of the underlying polynomial system, we propose concrete attacks that are able to take advantage of the regular noise distribution. In particular, we can identify several examples of concrete parameters where our techniques outperform other algorithms.

The area of multi-party computation (MPC) has recently increased in popularity and number of use cases. At the current state of the art, Ciminion, a Farfalle-like cryptographic function, achieves the best performance in MPC applications involving symmetric primitives. However, it has a critical weakness. Its security highly relies on the independence of its subkeys, which is achieved by using an expensive key schedule. Many MPC use cases involving symmetric pseudo-random functions (PRFs) rely on secretly shared symmetric keys, and hence the expensive key schedule must also be computed in MPC. As a result, Ciminion's performance is significantly reduced in these use cases. In this paper we solve this problem. Following the approach introduced by Ciminion's designers, we present a novel primitive in symmetric cryptography called Megafono. Megafono is a keyed extendable PRF, expanding a fixed-length input to an arbitrary-length output. Similar to Farfalle, an initial keyed permutation is applied to the input, followed by an expansion layer, involving the parallel application of keyed ciphers. The main novelty regards the expansion of the intermediate/internal state for "free", by appending the sum of the internal states of the first permutation to its output. The combination of this and other modifications, together with the impossibility for the attacker to have access to the input state of the expansion layer, make Megafono very efficient in the target application. As a concrete example, we present the PRF Hydra, an instance of Megafono based on the Hades strategy and on generalized versions of the Lai--Massey scheme. Based on an extensive security analysis, we implement Hydra in an MPC framework. The results show that it outperforms all MPC-friendly schemes currently published in the literature.

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.

Designing novel symmetric-key primitives for advanced protocols like secure multiparty computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proof systems (ZK), has been an important research topic in recent years. Many such existing primitives adopt quite different design strategies from conventional block ciphers. Notable features include that many of these ciphers are defined over a large finite field, and that a power map is commonly used to construct the nonlinear component due to its efficiency in these applications as well as its strong resistance against the differential and linear cryptanalysis. In this paper, we target the MPC-friendly ciphers AIM and RAIN used for the post-quantum signature schemes AIMer (CCS 2023 and NIST PQC Round 1 Additional Signatures) and Rainier (CCS 2022), respectively. Specifically, we can find equivalent representations of 2-round RAIN and full-round AIM, respectively, which make them vulnerable to either the polynomial method, or the crossbred algorithm, or the fast exhaustive search attack. Consequently, we can break 2-round RAIN with the 128/192/256-bit key in only 2111/2170/2225 bit operations. For full-round AIM with the 128/192/256-bit key, we could break them in 2136.2/2200.7/2265 bit operations, which are equivalent to about 2115/2178/2241 calls of the underlying primitives. In particular, our analysis indicates that AIM does not reach the required security levels by the NIST competition.

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.

Algebraically simple PRFs, ciphers, or cryptographic hash functions are becoming increasingly popular, for example due to their attractive properties for MPC and new proof systems (SNARKs, STARKs, among many others). In this paper, we focus on the algebraically simple construction MiMC, which became an attractive cryptanalytic target due to its simplicity, but also due to its use as a baseline in a competition for more recent algorithms exploring this design space. For the first time, we are able to describe key-recovery attacks on all full-round versions of MiMC over GF(2^n), requiring half the code book. In the chosen-ciphertext scenario, recovering the key from this data for the n-bit full version of MiMC takes the equivalent of less than 2^(n - log_2(n) + 1) calls to MiMC and negligible amounts of memory. The attack procedure is a generalization of higher-order differential cryptanalysis, and it is based on two main ingredients. First, we present a higher-order distinguisher which exploits the fact that the algebraic degree of MiMC grows significantly slower than originally believed. Secondly, we describe an approach to turn this distinguisher into a key-recovery attack without guessing the full subkey. Finally, we show that approximately ceil(log_3(2 * R)) more rounds (where R = ceil(n * log_3(2)) is the current number of rounds of MiMC-n/n) can be necessary and sufficient to restore the security against the key-recovery attack presented here. The attack has been practically verified on toy versions of MiMC. Note that our attack does not affect the security of MiMC over prime fields.