The Legendre Symbol and the Modulo-2 Operator in Symmetric Schemes over Fnp: Preimage Attack on Full Grendel 📺
Motivated by modern cryptographic use cases such as multi-party computation (MPC), homomorphic encryption (HE), and zero-knowledge (ZK) protocols, several symmetric schemes that are efficient in these scenarios have recently been proposed in the literature. Some of these schemes are instantiated with low-degree nonlinear functions, for example low-degree power maps (e.g., MiMC, HadesMiMC, Poseidon) or the Toffoli gate (e.g., Ciminion). Others (e.g., Rescue, Vision, Grendel) are instead instantiated via high-degree functions which are easy to evaluate in the target application. A recent example for the latter case is the hash function Grendel, whose nonlinear layer is constructed using the Legendre symbol. In this paper, we analyze high-degree functions such as the Legendre symbol or the modulo-2 operation as building blocks for the nonlinear layer of a cryptographic scheme over Fnp.Our focus regards the security analysis rather than the efficiency in the mentioned use cases. For this purpose, we present several new invertible functions that make use of the Legendre symbol or of the modulo-2 operation.Even though these functions often provide strong statistical properties and ensure a high degree after a few rounds, the main problem regards their small number of possible outputs, that is, only three for the Legendre symbol and only two for the modulo-2 operation. By fixing them, it is possible to reduce the overall degree of the function significantly. We exploit this behavior by describing the first preimage attack on full Grendel, and we verify it in practice.
Influence of the Linear Layer on the Algebraic Degree in SP-Networks 📺
We consider SPN schemes, i.e., schemes whose non-linear layer is defined as the parallel application of t ≥ 1 independent S-Boxes over F2n and whose linear layer is defined by the multiplication with a (n · t) × (n · t) matrix over F2. Even if the algebraic representation of a scheme depends on all its components, upper bounds on the growth of the algebraic degree in the literature usually only consider the details of the non-linear layer. Hence a natural question arises: (how) do the details of the linear layer influence the growth of the algebraic degree? We show that the linear layer plays a crucial role in the growth of the algebraic degree and present a new upper bound on the algebraic degree in SP-networks. As main results, we prove that in the case of low-degree round functions with large S-Boxes: (a) an initial exponential growth of the algebraic degree can be followed by a linear growth until the maximum algebraic degree is reached; (b) the rate of the linear growth is proportional to the degree of the linear layer over Ft2n. Besides providing a theoretical insight, our analysis is particularly relevant for assessing the security of the security of cryptographic permutations designed to be competitive in applications like MPC, FHE, SNARKs, and STARKs, including permutations based on the Hades design strategy. We have verified our findings on small-scale instances and we have compared them against the currently best results in the literature, showing a substantial improvement of upper bounds on the algebraic degree in case of low-degree round functions with large S-Boxes.
Proving Resistance Against Infinitely Long Subspace Trails: How to Choose the Linear Layer 📺
Designing cryptographic permutations and block ciphers using a substitutionpermutation network (SPN) approach where the nonlinear part does not cover the entire state has recently gained attention due to favorable implementation characteristics in various scenarios.For word-oriented partial SPN (P-SPN) schemes with a fixed linear layer, our goal is to better understand how the details of the linear layer affect the security of the construction. In this paper, we derive conditions that allow us to either set up or prevent attacks based on infinitely long truncated differentials with probability 1. Our analysis is rather broad compared to earlier independent work on this problem since we consider (1) both invariant and non-invariant/iterative trails, and (2) trails with and without active S-boxes.For these cases, we provide rigorous sufficient and necessary conditions for the matrix that defines the linear layer to prevent the analyzed attacks. On the practical side, we present a tool that can determine whether a given linear layer is vulnerable based on these results. Furthermore, we propose a sufficient condition for the linear layer that, if satisfied, ensures that no infinitely long truncated differential exists. This condition is related to the degree and the irreducibility of the minimal polynomial of the matrix that defines the linear layer. Besides P-SPN schemes, our observations may also have a crucial impact on the Hades design strategy, which mixes rounds with full S-box layers and rounds with partial S-box layers.
On a Generalization of Substitution-Permutation Networks: The HADES Design Strategy 📺
Keyed and unkeyed cryptographic permutations often iterate simple round functions. Substitution-permutation networks (SPNs) are an approach that is popular since the mid 1990s. One of the new directions in the design of these round functions is to reduce the substitution (S-Box) layer from a full one to a partial one, uniformly distributed over all the rounds. LowMC and Zorro are examples of this approach. A relevant freedom in the design space is to allow for a highly non-uniform distribution of S-Boxes. However, choosing rounds that are so different from each other is very rarely done, as it makes security analysis and implementation much harder. We develop the design strategy HADES and an analysis framework for it, which despite this increased complexity allows for security arguments against many classes of attacks, similar to earlier simpler SPNs. The framework builds upon the wide trail design strategy, and it additionally allows for security arguments against algebraic attacks, which are much more of a concern when algebraically simple S-Boxes are used. Subsequently, this is put into practice by concrete instances and benchmarks for a use case that generally benefits from a smaller number of S-Boxes and showcases the diversity of design options we support: A candidate cipher natively working with objects in GF(p), for securing data transfers with distributed databases using secure multiparty computation (MPC). Compared to the currently fastest design MiMC, we observe significant improvements in online bandwidth requirements and throughput with a simultaneous reduction of preprocessing effort, while having a comparable online latency.
An Algebraic Attack on Ciphers with Low-Degree Round Functions: Application to Full MiMC 📺
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.
Algebraic Cryptanalysis of STARK-Friendly Designs: Application to MARVELlous and MiMC
The block cipher Jarvis and the hash function Friday, both members of the MARVELlous family of cryptographic primitives, are among the first proposed solutions to the problem of designing symmetric-key algorithms suitable for transparent, post-quantum secure zero-knowledge proof systems such as ZK-STARKs. In this paper we describe an algebraic cryptanalysis of Jarvis and Friday and show that the proposed number of rounds is not sufficient to provide adequate security. In Jarvis, the round function is obtained by combining a finite field inversion, a full-degree affine permutation polynomial and a key addition. Yet we show that even though the high degree of the affine polynomial may prevent some algebraic attacks (as claimed by the designers), the particular algebraic properties of the round function make both Jarvis and Friday vulnerable to Gröbner basis attacks. We also consider MiMC, a block cipher similar in structure to Jarvis. However, this cipher proves to be resistant against our proposed attack strategy. Still, our successful cryptanalysis of Jarvis and Friday does illustrate that block cipher designs for “algebraic platforms” such as STARKs, FHE or MPC may be particularly vulnerable to algebraic attacks.