International Association
for Cryptologic Research

The cube attack extracts the information of secret key bits by recovering the coefficient called superpoly in the output bit with respect to a subset of plaintexts/IV, which is called a cube. While the division property provides an efficient way to detect the structure of the superpoly, superpoly recovery could still be prohibitively costly if the number of rounds is sufficiently high. In particular, Core Monomial Prediction (CMP) was proposed at ASIACRYPT 2022 as a scaled-down version of Monomial Prediction (MP), which sacrifices accuracy for efficiency but ultimately gets stuck at 848 rounds of \trivium. In this paper, we provide new insights into CMP by elucidating the algebraic meaning to the core monomial trails. We prove that it is sufficient to recover the superpoly by extracting all the core monomial trails, an approach based solely on CMP, thus demonstrating that CMP can achieve perfect accuracy as MP does. We further reveal that CMP is still MP in essence, but with variable substitutions on the target function. Inspired by the divide-and-conquer strategy that has been widely used in previous literature, we design a meet-in-the-middle (MITM) framework, in which the CMP-based approach can be embedded to achieve a speedup. To illustrate the power of these new techniques, we apply the MITM framework to \trivium, \grain and \kreyvium. As a result, not only can the previous computational cost of superpoly recovery be reduced (e.g., 5x faster for superpoly recovery on 192-round \grain), but we also succeed in recovering superpolies for up to 851 rounds of \trivium and up to 899 rounds of \kreyvium. This surpasses the previous best results by respectively 3 and 4 rounds. Using the memory-efficient M\"obius transform proposed at EUROCRYPT 2021, we can perform key recovery attacks on target ciphers, even though the superpoly may contain over $2^{40}$ monomials. This leads to the best key recovery attacks on the target ciphers.

The Higher-order Differential-Linear (HDL) attack was introduced by Biham \textit{et al.} at FSE 2005, where a linear approximation was appended to a Higher-order Differential (HD) transition. It is a natural generalization of the Differential-Linear (DL) attack. Due to some practical restrictions, however, HDL cryptanalysis has unfortunately attracted much less attention compared to its DL counterpart since its proposal. In this paper, we revisit HD/HDL cryptanalysis from an algebraic perspective and provide two novel tools for detecting possible HD/HDL distinguishers, including: (a) Higher-order Algebraic Transitional Form (HATF) for probabilistic HD/HDL attacks; (b) Differential Supporting Function (\DSF) for deterministic HD attacks. In general, the HATF can estimate the biases of $\ell^{th}$-order HDL approximations with complexity $\mathcal{O}(2^{\ell+d2^\ell})$ where $d$ is the algebraic degree of the function studied. If the function is quadratic, the complexity can be further reduced to $\mathcal{O}(2^{3.8\ell})$. HATF is therefore very useful in HDL cryptanalysis for ciphers with quadratic round functions, such as \ascon and \xoodyak. \DSF provides a convenient way to find good linearizations on the input of a permutation, which facilitates the search for HD distinguishers. Unsurprisingly, HD/HDL attacks have the potential to be more effective than their simpler differential/DL counterparts. Using HATF, we found many HDL approximations for round-reduced \ascon and \xoodyak initializations, with significantly larger biases than DL ones. For instance, there are deterministic 2$^{nd}$-order/4$^{th}$-order HDL approximations for \ascon/\xoodyak initializations, respectively (which is believed to be impossible in the simple DL case). We derived highly biased HDL approximations for 5-round \ascon up to 8$^{th}$ order, which improves the complexity of the distinguishing attack on 5-round \ascon from $2^{16}$ to $2^{12}$ calls. We also proposed HDL approximations for 6-round \ascon and 5-round \xoodyak (under the single-key model), which couldn't be reached with simple DL so far. For key recovery, HDL attacks are also more efficient than DL attacks, thanks to the larger biases of HDL approximations. Additionally, HATF works well for DL (1$^{st}$-order HDL) attacks and some well-known DL biases of \ascon and \xoodyak that could only be obtained experimentally before can now be predicted theoretically. With \DSF, we propose a new distinguishing attack on 8-round \ascon permutation, with a complexity of $2^{48}$. Also, we provide a new zero-sum distinguisher for the full 12-round \ascon permutation with $2^{55}$ time/data complexity. We highlight that our cryptanalyses do not threaten the security of \ascon or \xoodyak.

Recent practical applications using advanced cryptographic protocols such as multi-party computation (MPC) and zero-knowledge proofs (ZKP) have prompted a range of novel symmetric primitives described over large finite fields, characterized as arithmetization-oriented (AO) ciphers. Such designs, aiming to minimize the number of multiplications over fields, have a high risk of being vulnerable to algebraic attacks, especially to the higher-order differential attack. Thus, it is significant to carefully evaluate the growth of the algebraic degree. However, degree estimation for AO ciphers has been a challenge for cryptanalysts due to the lack of general and accurate methods. In this paper, we extend the division property, a state-of-the-art frame- work for finding the upper bound of the algebraic degree over binary fields, to the scope of F2n, called general monomial prediction. It is a generic method to detect the algebraic degree for AO ciphers, even applicable to Feistel ciphers which have no better bounds than the trivial exponential one. In the general monomial prediction, our idea is to evaluate whether the polynomial representation of a block cipher contains some specific monomials. With a deep investigation of the arithmetical feature, we introduce the propagation rules of monomials for field-based operations, which can be efficiently modeled using the bit-vector theory of SMT. Then the new searching tool for degree estimation can be constructed due to the relationship between the algebraic degree and the exponents of monomials. We apply our new framework to some important AO ciphers, including Feistel MiMC, GMiMC, and MiMC. For Feistel MiMC, we show that the algebraic degree grows significantly slower than the native exponential bound. For the first time, we present a secret-key higher-order differential distinguisher for up to 124 rounds, much better than the 83-round distinguisher for Feistel MiMC permutation proposed at CRYPTO 2020. We also exhibit a full-round zero-sum distinguisher with a data complexity of 2^{251}. Our method can be further extended for the general Feistel structure with more branches and exhibit higher-order differential distinguishers against the practical instance of GMiMC for up to 50 rounds. For MiMC in SP-networks, our results correspond to the exact algebraic degree proved by Bouvier. We also point out that the number of rounds in MiMC specification is not necessary to guarantee the security against the higher-order differential attack for MiMC-like schemes with different exponents. The investigation of different exponents provides some guidance on the cipher design.

Cube attacks exploit the algebraic properties of symmetric ciphers by recovering a special polynomial, the superpoly, and subsequently the secret key. When the algebraic normal forms of the corresponding Boolean functions are not available, the division property based approach allows to recover the exact superpoly in a clever way. However, the computational cost to recover the superpoly becomes prohibitive as the number of rounds of the cipher increases. For example, the nested monomial predictions (NMP) proposed at ASIACRYPT 2021 stuck at round 845 for \trivium. To alleviate the bottleneck of the NMP technique, i.e., the unsolvable model due to the excessive number of monomial trails, we shift our focus to the so-called valuable terms of a specific middle round that contribute to the superpoly. Two new techniques are introduced, namely, Non-zero Bit-based Division Property (NBDP) and Core Monomial Prediction (CMP), both of which result in a simpler MILP model compared to the MILP model of MP. It can be shown that the CMP technique offers a substantial improvement over the monomial prediction technique in terms of computational complexity of recovering valuable terms. Combining the divide-and-conquer strategy with these two new techniques, we catch the valuable terms more effectively and thus avoid wasting computational resources on intermediate terms contributing nothing to the superpoly. As an illustration of the power of our techniques, we apply our framework to \trivium, \grain, \kreyvium and \acorn. As a result, the computational cost of earlier attacks can be significantly reduced and the exact ANFs of the superpolies for 846-, 847- and 848-round \trivium, 192-round \grain, 895-round \kreyvium and 776-round \acorn can be recovered in practical time, even though the superpoly of 848-round \trivium contains over 500 million terms; this corresponds to respectively 3, 1, 1 and 1 rounds more than the previous best results. Moreover, by investigating the internal properties of M\"obius transformation, we show how to perform key recovery using superpolies involving full key bits, which leads to the best key recovery attacks on the targeted ciphers.

Being one of the winning algorithms of the CAESAR competition and currently a second round candidate of the NIST lightweight cryptography standardization project, the authenticated encryption scheme Ascon (designed by Dobraunig, Eichlseder, Mendel, and Schläffer) has withstood extensive self and third-party cryptanalysis. The best known attack on Ascon could only penetrate up to 7 (out of 12) rounds due to Li et al. (ToSC Vol I, 2017). However, it violates the data limit of 264 blocks per key specified by the designers. Moreover, the best known distinguishers of Ascon in the AEAD context reach only 6 rounds. To fill these gaps, we revisit the security of 7-round Ascon in the nonce-respecting setting without violating the data limit as specified in the design. First, we introduce a new superpoly-recovery technique named as partial polynomial multiplication for which computations take place between the so-called degree-d homogeneous parts of the involved Boolean functions for a 2d-dimensional cube. We apply this method to 7-round Ascon and present several key recovery attacks. Our best attack can recover the 128-bit secret key with a time complexity of about 2123 7-round Ascon permutations and requires 264 data and 2101 bits memory. Also, based on division properties, we identify several 60 dimensional cubes whose superpolies are constant zero after 7 rounds. We further improve the cube distinguishers for 4, 5 and 6 rounds. Although our results are far from threatening the security of full 12-round Ascon, they provide new insights in the security analysis of Ascon.

Determining the exact algebraic structure or some partial information of the superpoly for a given cube is a necessary step in the cube attack -- a generic cryptanalytic technique for symmetric-key primitives with some secret and public tweakable inputs. Currently, the division property based approach is the most powerful tool for exact superpoly recovery. However, as the algebraic normal form (ANF) of the targeted output bit gets increasingly complicated as the number of rounds grows, existing methods for superpoly recovery quickly hit their bottlenecks. For example, previous method stuck at round 842, 190, and 892 for \trivium, \grain, and \kreyvium, respectively. In this paper, we propose a new framework for recovering the exact ANFs of massive superpolies based on the monomial prediction technique (ASIACRYPT 2020, an alternative language for the division property). In this framework, the targeted output bit is first expressed as a polynomial of the bits of some intermediate states. For each term appearing in the polynomial, the monomial prediction technique is applied to determine its superpoly if the corresponding MILP model can be solved within a preset time limit. Terms unresolved within the time limit are further expanded as polynomials of the bits of some deeper intermediate states with symbolic computation, whose terms are again processed with monomial predictions. The above procedure is iterated until all terms are resolved. Finally, all the sub-superpolies are collected and assembled into the superpoly of the targeted bit. We apply the new framework to \trivium, \grain, and \kreyvium. As a result, the exact ANFs of the superpolies for 843-, 844- and 845-round \trivium, 191-round \grain and 894-round \kreyvium are recovered. Moreover, with help of the M\"{o}bius transform, we present a novel key-recovery technique based on superpolies involving \textit{all} key bits by exploiting the sparse structures, which leads to the best key-recovery attacks on the targets considered.

The bit-based division property (BDP) is the most effective technique for finding integral characteristics of symmetric ciphers. Recently, automatic search tools have become one of the most popular approaches to evaluating the security of designs against many attacks. Constraint-aided automatic tools for the BDP have been applied to many ciphers with simple linear layers like bit-permutation. Constructing models of complex linear layers accurately and efficiently remains hard. A straightforward method proposed by Sun et al. (called the S method), decomposes a complex linear layer into basic operations like COPY and XOR, then models them one by one. However, this method can easily insert invalid division trails into the solution pool, which results in a quicker loss of the balanced property than the cipher itself would. In order to solve this problem, Zhang and Rijmen propose the ZR method to link every valid trail with an invertible sub-matrix of the matrix corresponding to the linear layer, and then generate linear inequalities to represent all the invertible sub-matrices. Unfortunately, the ZR method is only applicable to invertible binary matrices (defined in Definition 3).To avoid generating a huge number of inequalities for all the sub-matrices, we build a new model that only includes that the sub-matrix corresponding to a valid trail should be invertible. The computing scale of our model can be tackled by most of SMT/SAT solvers, which makes our method practical. For applications, we improve the previous BDP for LED and MISTY1. We also give the 7-round BDP results for Camellia with FL/FL−1, which is the longest to date.Furthermore, we remove the restriction of the ZR method that the matrix has to be invertible, which provides more choices for future designs. Thanks to this, we also reproduce 5-round key-dependent integral distinguishers proposed at Crypto 2016 which cannot be obtained by either the S or ZR methods.

Since it was proposed in 2015 as a generalization of integral properties, the division property has evolved into a powerful tool for probing the structures of Boolean functions whose algebraic normal forms are not available. We capture the most essential elements for the detection of division properties from a pure algebraic perspective, proposing a technique named as {\it monomial prediction}, which can be employed to determine the presence or absence of a monomial in the product of the coordinate functions of a vectorial Boolean function $\bs f$ by counting the number of the so-called {\it monomial trails} across a sequence of simpler functions whose composition is $\bs f$. Under the framework of the monomial prediction, we formally prove that most algorithms for detecting division properties in previous literature raise no false alarms but may miss. We also establish the equivalence between the monomial prediction and the three-subset bit-based division property without unknown subset presented at EUROCRYPT 2020, and show that these two techniques are perfectly accurate. This algebraic formulation gives more insights into division properties and inspires new search strategies. With the monomial prediction, we obtain the {\it exact} algebraic degrees of \TRIVIUM up to 834 rounds for the first time. In the context of cube attacks, we are able to explore a larger search space in limited time and recover the exact algebraic normal forms of complex superpolies with the help of a divide-and-conquer strategy. As a result, we identify more cubes with smaller dimensions, leading to improvements of some near-optimal attacks against 840-, 841- and 842-round \TRIVIUM.

Statistical saturation attack takes advantage of a set of plaintext with some bits fixed while the others vary randomly, and then track the evolution of a non-uniform plaintext distribution through the cipher. Previous statistical saturation attacks are all implemented under single-key setting, and there is no public attack models under related-key/tweak setting. In this paper, we propose a new cryptanalytic method which can be seen as related-key/tweak statistical saturation attack by revealing the link between the related-key/tweak statistical saturation distinguishers and KDIB (Key Difference Invariant Bias) / TDIB (Tweak Difference Invariant Bias) ones. KDIB cryptanalysis was proposed by Bogdanov et al. at ASIACRYPT’13 and utilizes the property that there can exist linear trails such that their biases are deterministically invariant under key difference. And this method can be easily extended to TDIB distinguishers if the tweak is also alternated. The link between them provides a new and more efficient way to find related-key/tweak statistical saturation distinguishers in ciphers. Thereafter, an automatic searching algorithm for KDIB/TDIB distinguishers is also given in this paper, which can be implemented to find word-level KDIB distinguishers for S-box based key-alternating ciphers. We apply this algorithm to QARMA-64 and give related-tweak statistical saturation attack for 10-round QARMA-64 with outer whitening key. Besides, an 11-round attack on QARMA-128 is also given based on the TDIB technique. Compared with previous public attacks on QARMA including outer whitening key, all attacks presented in this paper are the best ones in terms of the number of rounds.