MILP-Aided Bit-Based Division Property for Primitives with Non-Bit-Permutation Linear Layers

Division property is a general integral property introduced by Todo at EUROCRYPT 2015. Recently, at ASIACRYPT 2016, Xiang et al. applied the Mixed Integer Linear Programming (MILP) method to search bit-based division property, and handled the complexity which restricted the application of bit-based division property proposed by Todo and Morii at FSE 2016. However, their MILP-aided search was only applied to some lightweight block ciphers whose linear layers were limited to bit-permutations, and the feasibility of MILP-aided bit-based division property for ciphers with non-bit-permutation linear layers was an open problem. This paper comes out with the affirmative answer. First, we transform the complicated linear layers to their primitive representations, which only involves Copy and XOR operations. Then, the original Copy and XOR models are respectively generalized to deal with more output branches and input elements, and these generalized models are adopted to depict the primitive representations. Accordingly, the MILP-aided bit-based division property can be applied to much more primitives with complicated linear layers. As an illustration, we first evaluate the bit-based division propertyies of some word-oriented block ciphers including Midori64, LED, Joltik-BC, and AES. For Midori64, we obtain a 7-round integral distinguisher, which achieves one more round than the previous results. At the same time, the data requirements of some existing distinguishers are also reduced. We decrease the number of required chosen plaintexts of 4-round and 5-round integral distinguishers for LED and Joltik-BC by half. As to AES, our searching experiments show that integral distinguishers, which are based on the bit-based division property, covering more than four rounds probably do not exist. Then, the bit-based division properties of some bit-oriented block ciphers, such as Serpent and Noekeon, are considered. The data complexities of their distinguishers for short rounds are improved. Moreover, we evaluate the bit-based division properties of the internal permutations involved in some hash functions, e.g., SPONGENT and PHOTON. An 18-round zero-sum distinguisher for SPONGENT-88 is proposed, which achieves four more rounds than the previous ones. We also provide 20-round and 21-round zero-sum distinguishers for SPONGENT-128 and SPONGENT-160, respectively. For most PHOTON permutations $P_{t}$ with 4-bit cell, the data requirements for the 4-round distinguishers are reduced by half. Besides, the length of $P_{256}$\u27s distinguisher is extended by one round. Furthermore, for $P_{288}$ using 8-bit S-boxes, we improve the data complexities of their integral distinguishers significantly

MILP-Aided Bit-Based Division Property for ARX-Based Block Cipher

The huge time and memory complexities of utilizing bit-based division property, which was first presented by Todo and Morri at FSE 2016, bothered cryptographers for quite some time and it had been solved by Xiang \textit{et al.} at ASIACRYPT 2016. They applied MILP method to search integral distinguisher based on division property, and used it to analyze six lightweight block ciphers. Later on, Sun \textit{et al.} handled the feasibility of MILP-aided bit-based division property for primitives with non-bit-permutation linear layers. Although MILP-aided bit-based division property has gave many perfect results since its appearance, there still are many left problems when we want to develop its further applications. In this paper, we focus on the feasibility of MILP-aided bit-based division property for ARX-based primitive. More specifically, we consider the construction of MILP models for some components of ARX-based structure. Firstly, the \texttt{Modulo} model is proposed by using its iterated expression and introducing some auxiliary variables. Then, to propagate the operations of \texttt{AND} and \texttt{OR} with a constant (or a subkey), we prove that the known-region deduced by the input division property is always included in the known-region derived from the output division property, which allows us to ignore these operations. Furthermore, with its help, we also handle the \texttt{Modulo} operation with a constant (or a subkey). As a result, these new models are exploited to search integral distinguishers for some ARX-based block ciphers. For HIGHT and LEA, the lengths of the distinguishers both are improved by one round. Some 15-round integral distinguishers for TEA/XTEA are presented. Comparing with the existing one transformed by utilizing the equivalence between zero-correlation and integral cryptanalysis, our newly obtained distinguishers either reduces the data requirement or increases the number of zero-sum bits. Moreover, the bit-based division properties for KATAN and KTANTAN families of block ciphers are also provided

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

MILP-aided Cryptanalysis of Some Block Ciphers

Symmetric-key cryptographic primitives, such as block ciphers, play a pivotal role in achieving confidentiality, integrity, and authentication – which are the core services of information security. Since symmetric-key primitives do not rely on well-defined hard mathematical problems, unlike public-key primitives, there are no formal mathematical proofs for the security of symmetric-key primitives. Consequently, their security is guaranteed only by measuring their immunity against a set of predefined cryptanalysis techniques, e.g., differential, linear, impossible differential, and integral cryptanalysis. The attacks based on cryptanalysis techniques usually include searching in an exponential space of patterns, and for a long time, cryptanalysts have performed this task manually. As a result, it has been hard, time-consuming, and an error-prone task. Indeed, the need for automatic tools becomes more pressing. This thesis is dedicated to investigating the security of symmetric-key cryptographic primitives, precisely block ciphers. One of our main goals is to utilize Mixed Integer Linear Programming (MILP) to automate the evaluation and the validation of block cipher security against a wide range of cryptanalysis techniques. Our contributions can be summarized as follows. First, we investigate the security of two recently proposed block ciphers, CRAFT and SPARX-128/256 against two variants of differential cryptanalysis. We utilize the simple key schedule of CRAFT to construct several repeatable 2-round related-key differential characteristics with the maximum differential probability. Consequently, we are able to mount a practical key recovery attack on full-round CRAFT in the related-key setting. In addition, we use impossible differential cryptanalysis to assess SPARX-128/256 that is provable secure against single-trail differential and linear cryptanalysis. As a result, we can attack 24 rounds similar to the internal attack presented by the designers. However, our attack is better than the integral attack regarding the time and memory complexities. Next, we tackle the limitation of the current Mixed Integer Linear Programming (MILP) model to automate the search for differential distinguishers through modular additions. The current model assumes that the inputs to the modular addition and the consecutive rounds are independent. However, we show that this assumption does not necessarily hold and the current model might lead to invalid attacks. Accordingly, we propose a more accurate MILP model that takes into account the dependency between consecutive modular additions. As a proof of the validity and efficiency of our model, we use it to analyze the security of Bel-T cipher—the standard of the Republic of Belarus. Afterwards, we shift focus to another equally important cryptanalysis technique, i.e., integral cryptanalysis using the bit-based division property (BDP). We present MILP models to automate the search for the BDP through modular additions with a constant and modular subtractions. Consequently, we assess the security of Bel-T block cipher against the integral attacks. Next, we analyze the security of the tweakable block cipher T-TWINE. We present key recovery attacks on 27 and 28 rounds of T-TWINE-80 and T-TWINE-128, respectively. Finally, we address the limitation of the current MILP model for the propagation of the bit-based division property through large non-bit-permutation linear layers. The current models are either inaccurate, which might lead to missing some balanced bits, or inefficient in terms of the number of constraints. As a proof of the effectiveness of our approach, we improve the previous 3- and 4-round integral distinguishers of the Russian encryption standard—Kuznyechik, and the 4-round one of PHOTON’s internal permutation (P288). We also report a 4-round integral distinguisher for the Ukrainian standard Kalyna and a 5-round integral distinguisher for PHOTON’s internal permutation (P288)

MILP-aided Cryptanalysis of Round Reduced ChaCha

The inclusion of ChaCha20 and Poly1305 into the list of supported ciphers in TLS 1.3 necessitates a security evaluation of those ciphers with all the state-of-the-art tools and innovative cryptanalysis methodologies. Mixed Integer Linear Programming (MILP) has been successfully applied to find more accurate characteristics of several ciphers such as SIMON and SPECK. In our research, we use MILP-aided cryptanalysis to search for differential characteristics, linear approximations and integral properties of ChaCha. We are able to find differential trails up to 2 rounds and linear trails up to 1 round. However, no integral distinguisher has been found, even for 1 round

A cautionary note on the use of Gurobi for cryptanalysis

Mixed Integer Linear Programming (MILP) is a powerful tool that helps to automate several cryptanalysis techniques for symmetric key primitives. $\textsf{Gurobi}$ is one of the most popular solvers used by researchers to obtain useful results from the MILP models corresponding to these cryptanalysis techniques. In this report, we provide a cautionary note on the use of $\textsf{Gurobi}$ in the context of bit-based division property integral attacks. In particular, we report four different examples in which $\textsf{Gurobi}$ gives contradictory results when solving the same MILP model by just changing the number of used threads or reordering some constraints

Cryptanalysis of Some Block Cipher Constructions

When the public-key cryptography was introduced in the 1970s, symmetric-key cryptography was believed to soon become outdated. Nevertheless, we still heavily rely on symmetric-key primitives as they give high-speed performance. They are used to secure mobile communication, e-commerce transactions, communication through virtual private networks and sending electronic tax returns, among many other everyday activities. However, the security of symmetric-key primitives does not depend on a well-known hard mathematical problem such as the factoring problem, which is the basis of the RSA public-key cryptosystem. Instead, the security of symmetric-key primitives is evaluated against known cryptanalytic techniques. Accordingly, the topic of furthering the state-of-the-art of cryptanalysis of symmetric-key primitives is an ever-evolving topic. Therefore, this thesis is dedicated to the cryptanalysis of symmetric-key cryptographic primitives. Our focus is on block ciphers as well as hash functions that are built using block ciphers. Our contributions can be summarized as follows: First, we tackle the limitation of the current Mixed Integer Linear Programming (MILP) approaches to represent the differential propagation through large S-boxes. Indeed, we present a novel approach that can efficiently model the Difference Distribution Table (DDT) of large S-boxes, i.e., 8-bit S-boxes. As a proof of the validity and efficiency of our approach, we apply it on two out of the seven AES-round based constructions that were recently proposed in FSE 2016. Using our approach, we improve the lower bound on the number of active S-boxes of one construction and the upper bound on the best differential characteristic of the other. Then, we propose meet-in-the-middle attacks using the idea of efficient differential enumeration against two Japanese block ciphers, i.e., Hierocrypt-L1 and Hierocrypt-3. Both block ciphers were submitted to the New European Schemes for Signatures, Integrity, and Encryption (NESSIE) project, selected as one of the Japanese e-Government recommended ciphers in 2003 and reselected in the candidate recommended ciphers list in 2013. We construct five S-box layer distinguishers that we use to recover the master keys of reduced 8 S-box layer versions of both block ciphers. In addition, we present another meet-in-the-middle attack on Hierocrypt-3 with slightly higher time and memory complexities but with much less data complexity. Afterwards, we shift focus to another equally important cryptanalytic attack, i.e., impossible differential attack. SPARX-64/128 is selected among the SPARX family that was recently proposed to provide ARX based block cipher whose security against differential and linear cryptanalysis can be proven. We assess the security of SPARX-64/128 against impossible differential attack and show that it can reach the same number of rounds the division-based integral attack, proposed by the designers, can reach. Then, we pick Kiasu-BC as an example of a tweakable block cipher and prove that, on contrary to its designers’ claim, the freedom in choosing the publicly known tweak decreases its security margin. Lastly, we study the impossible differential properties of the underlying block cipher of the Russian hash standard Streebog and point out the potential risk in using it as a MAC scheme in the secret-IV mode

Slid Pairs of the Fruit-80 Stream Cipher

Fruit is a small-state stream cipher designed for securing communications among resource-constrained devices. The design of Fruit was first known to the public in 2016. It was later improved as Fruit-80 in 2018 and becomes the latest and final version among all versions of the Fruit stream ciphers. In this paper, we analyze the Fruit-80 stream cipher. We found that Fruit-80 generates identical keystreams from certain two distinct pairs of key and IV. Such pair of key and IV pairs is known as a slid pair. Moreover, we discover that when two pairs of key and IV fulfill specific characteristics, they will generate identical keystreams. This shows that slid pairs do not always exist arbitrarily in Fruit-80. We define specific rules which are equivalent to the characteristics. Using the defined rules, we are able to automate the searching process using an MILP solver, which makes searching of the slid pairs trivial

Design of Efficient Symmetric-Key Cryptographic Algorithms

兵庫県立大学大学院202