Links between Division Property and Other Cube Attack Variants

A theoretically reliable key-recovery attack should evaluate not only the non-randomness for the correct key guess but also the randomness for the wrong ones as well. The former has always been the main focus but the absence of the latter can also cause self-contradicted results. In fact, the theoretic discussion of wrong key guesses is overlooked in quite some existing key-recovery attacks, especially the previous cube attack variants based on pure experiments. In this paper, we draw links between the division property and several variants of the cube attack. In addition to the zero-sum property, we further prove that the bias phenomenon, the non-randomness widely utilized in dynamic cube attacks and cube testers, can also be reflected by the division property. Based on such links, we are able to provide several results: Firstly, we give a dynamic cube key-recovery attack on full Grain-128. Compared with Dinur et al.’s original one, this attack is supported by a theoretical analysis of the bias based on a more elaborate assumption. Our attack can recover 3 key bits with a complexity 297.86 and evaluated success probability 99.83%. Thus, the overall complexity for recovering full 128 key bits is 2125. Secondly, now that the bias phenomenon can be efficiently and elaborately evaluated, we further derive new secure bounds for Grain-like primitives (namely Grain-128, Grain-128a, Grain-V1, Plantlet) against both the zero-sum and bias cube testers. Our secure bounds indicate that 256 initialization rounds are not able to guarantee Grain-128 to resist bias-based cube testers. This is an efficient tool for newly designed stream ciphers for determining the number of initialization rounds. Thirdly, we improve Wang et al.’s relaxed term enumeration technique proposed in CRYPTO 2018 and extend their results on Kreyvium and ACORN by 1 and 13 rounds (reaching 892 and 763 rounds) with complexities 2121.19 and 2125.54 respectively. To our knowledge, our results are the current best key-recovery attacks on these two primitives

SoK: Security Evaluation of SBox-Based Block Ciphers

Cryptanalysis of block ciphers is an active and important research area with an extensive volume of literature. For this work, we focus on SBox-based ciphers, as they are widely used and cover a large class of block ciphers. While there have been prior works that have consolidated attacks on block ciphers, they usually focus on describing and listing the attacks. Moreover, the methods for evaluating a cipher\u27s security are often ad hoc, differing from cipher to cipher, as attacks and evaluation techniques are developed along the way. As such, we aim to organise the attack literature, as well as the work on security evaluation. In this work, we present a systematization of cryptanalysis of SBox-based block ciphers focusing on three main areas: (1) Evaluation of block ciphers against standard cryptanalytic attacks; (2) Organisation and relationships between various attacks; (3) Comparison of the evaluation and attacks on existing ciphers

Finding Bit-Based Division Property for Ciphers with Complex Linear Layers

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

Observations on the Dynamic Cube Attack of 855-Round TRIVIUM from Crypto\u2718

Recently, another kind of dynamic cube attack is proposed by Fu et al. With some key guesses and a transformation in the output bit, they claim that, when the key guesses are correct, the degree of the transformed output bit can drop so significantly that the cubes of lower dimension can not exist, making the output bit vulnerable to the zero-sum cube tester using slightly higher dimensional cubes. They applied their method to 855-round TRIVIUM. In order to verify the correctness of their result, they even proposed a practical attack on 721-round TRIVIUM claiming that the transformed output bit after 721-rounds of initialization does not contain cubes of dimensions 31 and below. However, the degree evaluation algorithm used by Fu et al. is innovative and complicated, and its complexity is not given. Their algorithm can only be implemented on huge clusters and cannot be verified by existing theoretic tools. In this paper, we theoretically analyze the dynamic cube attack method given by Fu et al. using the division property and MILP modeling technique. Firstly, we draw links between the division property and Fu et al.\u27s dynamic cube attack so that their method can be described as a theoretically well founded and computationally economic MILP-aided division-property-based cube attack. With the MILP model drawn according to the division property, we analyzed the 721-round TRIVIUM in detail and find some interesting results: \begin​{enumerate} \item The degree evaluation using our MILP method is more accurate than that of Fu et al.\u27s. Fu et al. prove that the degree of pure z721z721 is 40 while our method gives 29. We practically proved the correctness of our method by trying thousands of random keys, random 30-dimensional cubes and random assignments to non-cube IVs finding that the summations are constantly 0. \item For the transformed output bit (1+s2901)⋅z721(1+s1290)⋅z721, we proved the same degree 31 as Fu et al. and we also find 32-dimensional cubes have zero-sum property for correct key guesses. But since the degree of pure z721z721 is only 29, the 721-round practical attack on TRIVIUM is violating the principle of Fu et al.\u27s work: after the transformation in the output bit, when the key guesses are correct, the degree of the transformed output bit has not dropped but risen. \item Now that the degree theoretic foundation of the 721-round attack has been violated, we also find out that the key-recovery attack cannot be carried out either. We theoretically proved and practically verified that no matter the key guesses are correct or incorrect, the summation over 32-dimensional cube are always 0. So, no key bit can be recovered at all. \end{enumerate} All these analysis on 721-round TRIVIUM can be verified practically and we open our C++ source code for implementation as well. Secondly, we revisit their 855-round result. Our MILP model reveal that the 855-round result suffers from the same problems with its 721-round counterpart. We provide theoretic evidence that, after their transformation, the degree of the output bit is more likely to rise rather than drop. Furthermore, since Fu \etal\u27s degree evaluation is written in an unclear manner and no complexity analysis is given, we rewrite the algorithm according to their main ideas and supplement a detailed complexity analysis. Our analysis indicates that a precise evaluation to the degree requires complexities far beyond practical reach. We also demonstrate that further abbreviation to our rewritten algorithm can result in wrong evaluation. This might be the reason why Fu \etal give such a degree evaluation. This is also an additional argument against Fu \etal\u27s dynamic cube attack method. Thirdly, the selection of Fu \etal\u27s cube dimension is also questionable. According to our experiments and existing theoretic results, there is high risk that the correct key guesses and wrong ones share the same zero-sum property using Fu \etal\u27s cube testers. As a remedy, we suggest that concrete cubes satisfying particular conditions should be identified rather than relying on the IV-degree drop hypothesis. To conclude, Fu \etal\u27s dynamic cube attack on 855-round TRIVIUM is questionable. 855-round as well as 840-and-up-round TRIVIUM should still be open for further convincible cryptanalysis

Ten years of cube attacks

In 2009, Dinur and Shamir proposed the cube attack, an algebraic cryptanalysis technique that only requires black box access to a target cipher. Since then, this attack has received both many criticisms and endorsements from crypto community; this work aims at revising and collecting the many attacks that have been proposed starting from it. We categorise all of these attacks in five classes; for each class, we provide a brief summary description along with the state-of-the-art references and the most recent cryptanalysis results. Furthermore, we extend and refine the new notation we proposed in 2021 and we use it to provide a consistent definition for each attack family. Finally, in the appendix, we provide an in-depth description of the kite attack framework, a cipher independent tool we firstly proposed in 2018 that implements the kite attack on GPUs. To prove its effectiveness, we use Mickey2.0 as a use case, showing how to embed it in the framework

Contributions to Confidentiality and Integrity Algorithms for 5G

The confidentiality and integrity algorithms in cellular networks protect the transmission of user and signaling data over the air between users and the network, e.g., the base stations. There are three standardised cryptographic suites for confidentiality and integrity protection in 4G, which are based on the AES, SNOW 3G, and ZUC primitives, respectively. These primitives are used for providing a 128-bit security level and are usually implemented in hardware, e.g., using IP (intellectual property) cores, thus can be quite efficient. When we come to 5G, the innovative network architecture and high-performance demands pose new challenges to security. For the confidentiality and integrity protection, there are some new requirements on the underlying cryptographic algorithms. Specifically, these algorithms should: 1) provide 256 bits of security to protect against attackers equipped with quantum computing capabilities; and 2) provide at least 20 Gbps (Gigabits per second) speed in pure software environments, which is the downlink peak data rate in 5G. The reason for considering software environments is that the encryption in 5G will likely be moved to the cloud and implemented in software. Therefore, it is crucial to investigate existing algorithms in 4G, checking if they can satisfy the 5G requirements in terms of security and speed, and possibly propose new dedicated algorithms targeting these goals. This is the motivation of this thesis, which focuses on the confidentiality and integrity algorithms for 5G. The results can be summarised as follows.1. We investigate the security of SNOW 3G under 256-bit keys and propose two linear attacks against it with complexities 2172 and 2177, respectively. These cryptanalysis results indicate that SNOW 3G cannot provide the full 256-bit security level. 2. We design some spectral tools for linear cryptanalysis and apply these tools to investigate the security of ZUC-256, the 256-bit version of ZUC. We propose a distinguishing attack against ZUC-256 with complexity 2236, which is 220 faster than exhaustive key search. 3. We design a new stream cipher called SNOW-V in response to the new requirements for 5G confidentiality and integrity protection, in terms of security and speed. SNOW-V can provide a 256-bit security level and achieve a speed as high as 58 Gbps in software based on our extensive evaluation. The cipher is currently under evaluation in ETSI SAGE (Security Algorithms Group of Experts) as a promising candidate for 5G confidentiality and integrity algorithms. 4. We perform deeper cryptanalysis of SNOW-V to ensure that two common cryptanalysis techniques, guess-and-determine attacks and linear cryptanalysis, do not apply to SNOW-V faster than exhaustive key search. 5. We introduce two minor modifications in SNOW-V and propose an extreme performance variant, called SNOW-Vi, in response to the feedback about SNOW-V that some use cases are not fully covered. SNOW-Vi covers more use cases, especially some platforms with less capabilities. The speeds in software are increased by 50% in average over SNOW-V and can be up to 92 Gbps.Besides these works on 5G confidentiality and integrity algorithms, the thesis is also devoted to local pseudorandom generators (PRGs). 6. We investigate the security of local PRGs and propose two attacks against some constructions instantiated on the P5 predicate. The attacks improve existing results with a large gap and narrow down the secure parameter regime. We also extend the attacks to other local PRGs instantiated on general XOR-AND and XOR-MAJ predicates and provide some insight in the choice of safe parameters

Community Detection Boosts Network Dismantling on Real-World Networks

Network dismantling techniques have gained increasing interest during the last years caused by the need for protecting and strengthening critical infrastructure systems in our society. We show that communities play a critical role in dismantling, given their inherent property of separating a network into strongly and weakly connected parts. The process of community-based dismantling depends on several design factors, including the choice of community detection method, community cut strategy, and inter-community node selection. We formalize the problem of community attacks to networks, identify critical design decisions for such methods, and perform a comprehensive empirical evaluation with respect to effectiveness and efficiency criteria on a set of more than 40 community-based network dismantling methods. We compare our results to state-of-the-art network dismantling, including collective influence, articulation points, as well as network decycling. We show that community-based network dismantling significantly outperforms existing techniques in terms of solution quality and computation time in the vast majority of real-world networks, while existing techniques mainly excel on model networks (ER, BA) mostly. We additionally show that the scalability of community-based dismantling opens new doors towards the efficient analysis of large real-world networks.We acknowledge financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación/ SuMaEco Project (RTI2018-095441-B-C22) and the María de Maeztu Program for Units of Excellence in R&D (No. MDM-2017-0711). D.R.-R. also acknowledges the Fellowship No. BES-2016-076264 under the FPI program of MINECO, Spain.Peer reviewe

New MILP Modeling: Improved Conditional Cube Attacks on Keccak-based Constructions

In this paper, we propose a new MILP modeling to find better or even optimal choices of conditional cubes, under the general framework of conditional cube attacks. These choices generally find new or improved attacks against the keyed constructions based on Keccak permutation and its variants, including Keccak-MAC, KMAC, Keyak, and Ketje, in terms of attack complexities or the number of attacked rounds. Interestingly, conditional cube attacks were applied to round-reduced Keccak-MAC, but not to KMAC despite the great similarity between Keccak-MAC and KMAC, and the fact that KMAC is the NIST standard way of constructing MAC from SHA-3. As examples to demonstrate the effectiveness of our new modeling, we report key recovery attacks against KMAC128 and KMAC256 reduced to 7 and 9 rounds, respectively; the best attack against Lake Keyak with 128-bit key is improved from 6 to 8 rounds in the nonce-respected setting and 9 rounds of Lake Keyak can be attacked if the key size is of 256 bits; attack complexity improvements are found generally on other constructions. Our new model is also applied to Keccak-based full-state keyed sponge and gives a positive answer to the open question proposed by Bertoni et al. whether cube attacks can be extended to more rounds by exploiting full-state absorbing. To verify the correctness of our attacks, reduced-variants of the attacks are implemented and verified on a PC practically. It is remarked that this work does not threaten the security of any full version of the instances analyzed in this paper