MILP Modeling for (Large) S-boxes to Optimize Probability of Differential Characteristics

Current Mixed Integer Linear Programming (MILP)-based search against symmetric-key primitives with 8-bit S-boxes can only build word-wise model to search for truncated differential characteristics. In such a model, the properties of the Differential Distribution Table (DDT) are not considered. To take these properties into account, a bit-wise model is necessary, which can be generated by the H-representation of the convex hull or the logical condition modeling. However, the complexity of both approaches becomes impractical when the size of the S-box exceeds 5 bits. In this paper, we propose a new modeling for large (8-bit or more) S-boxes. In particular, we first propose an algorithm to generate a bit-wise model of the DDT for large S-boxes. We observe that the problem of generating constraints in logical condition modeling can be converted into the problem of minimizing the product-of-sum of Boolean functions, which is a well-studied problem. Hence, classical off-the-shelf solutions such as the Quine-McCluskey algorithm or the Espresso algorithm can be utilized, which makes building a bit-wise model, for 8-bit or larger S-boxes, practical. Then this model is further extended to search for the best differential characteristic by considering the probabilities of each propagation in the DDT, which is a much harder problem than searching for the lower bound on the number of active S-boxes. Our idea is to separate the DDT into multiple tables for each probability and add conditional constraints to control the behavior of these multiple tables. The proposed modeling is first applied to SKINNY-128 to find that there is no differential characteristic having probability higher than 2−128 for 14 rounds, while the designers originally expected that 15 rounds were required. We also applied the proposed modeling to two, arbitrarily selected, constructions of the seven AES round function based constructions proposed in FSE 2016 and managed to improve the lower bound on the number of the active S-boxes in one construction and the upper bound on the differential characteristic for the other

Modeling Large S-box in MILP and a (Related-key) Differential Attack on Full Round PIPO-64/128

Mixed integer linear programming (MILP) based tools are used to estimate the strength of block ciphers against the cryptanalytic attacks. The existing tools use partial difference distribution table (p-DDT) approach to optimize the probability of differential characteristics for large (≥8-bit) S-box based ciphers. We propose to use the full difference distribution table (DDT) with the probability of each possible propagation for MILP modeling of large S-boxes. This requires more than 16 variables to represent the linear inequalities of each propagation and corresponding probabilities. The existing tools (viz. Logic Friday) cannot handle the linear inequalities in more than 16 variables. In this paper, we present a new tool (namely MILES) to minimize the linear inequalities in more than 16 variables. This tool reduces the number of inequalities by minimizing the truth table corresponding to the DDT of S-box. We use our tool to minimize the linear inequalities for 8-bit S-boxes (AES and SKINNY) and get better results than existing tools. We show the application of MILES on 8-bit S-box based lightweight block cipher PIPO. There are 20621 inequalities in 23 variables corresponding to the possible propagations in DDT and these are minimized to 6035 inequalities using MILES. MILP model based on these linear inequalities is used to optimizethe probability of differential characteristics for round-reduced PIPO. MILP model based on these inequalities is used to optimize the probability of differential and impossible differential characteristics for PIPO-64/128 reduced to 9 and 4 rounds respectively. We present an iterative 2-round related-key differential characteristic with the probability of 2^{-4} and that is used to construct a full round related-key differential distinguisher with the probability of 2^{-24}. We present a major collision in PIPO-64/128 which produces the same ciphertext (C) by encrypting the plaintext (P) under two different keys

Full-Round Differential Attack on ULC and LICID Block Ciphers Designed for IoT

The lightweight block ciphers ULC and LICID are introduced by Sliman et al. (2021) and Omrani et al. (2019) respectively. These ciphers are based on substitution permutation network structure. ULC is designed using the ULM method to increase efficiency, memory usage, and security. On the other hand, LICID is specifically designed for image data. In the ULC paper, the authors have given a full-round differential characteristic with a probability of $2^{-80}$. In the LICID paper, the authors have presented an 8-round differential characteristic with a probability of $2^{-112.66}$. In this paper, we present the 15-round ULC and the 14-round LICID differential characteristics of probabilities $2^{-45}$ and $2^{-40}$ respectively using the MILP model

New Automatic search method for Truncated-differential characteristics: Application to Midori, SKINNY and CRAFT

In this paper, using Mixed Integer Linear Programming, a new automatic search tool for truncated differential characteristic is presented. Our method models the problem of finding a maximal probability truncated differential characteristic, which is able to distinguish the cipher from a pseudo random permutation. Using this method, we analyse Midori64, SKINNY64/X and CRAFT block ciphers, for all of which the existing results are improved. In all cases, the truncated differential characteristic is much more efficient than the (upper bound of) bit-wise differential characteristic proven by the designers, for any number of rounds. More specifically, the highest possible rounds, for which an efficient differential characteristic can exist for Midori64, SKINNY64/X and CRAFT are 6, 7 and 10 rounds respectively, for which differential characteristics with maximum probabilities of $2^{-60}$, $2^{-52}$ and $2^{-62.61}$ (may) exist. Using our new method, we introduce new truncated differential characteristics for these ciphers with respective probabilities $2^{-54}$, $2^{-4}$ and $2^{-24}$ at the same number of rounds. Moreover, the longest truncated differential characteristics found for SKINNY64/X and CRAFT have 10 and 12 rounds, respectively. This method can be used as a new tool for differential analysis of SPN block ciphers

MILP-aided Cryptanalysis of the FUTURE Block Cipher

FUTURE is a recently proposed, lightweight block cipher. It has an AES-like, SP-based, 10-round encryption function, where, unlike most other lightweight constructions, the diffusion layer is based on an MDS matrix. Despite its relative complexity, it has a remarkable hardware performance due to careful design decisions. In this paper, we conducted a MILP-based analysis of the cipher, where we incorporated exact probabilities rather than just the number of active S-boxes into the model. Through the MILP analysis, we were able to find differential and linear distinguishers for up to 5 rounds of FUTURE, extending the known distinguishers of the cipher by one round

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

Proposing an MILP-based method for the experimental verification of difference-based trails: application to SPECK, SIMECK

Under embargo until: 2022-07-08Searching for the right pairs of inputs in difference-based distinguishers is an important task for the experimental verification of the distinguishers in symmetric-key ciphers. In this paper, we develop an MILP-based approach to verify the possibility of difference-based distinguishers and extract the right pairs. We apply the proposed method to some published difference-based trails (Related-Key Differentials (RKD), Rotational-XOR (RX)) of block ciphers SIMECK, and SPECK. As a result, we show that some of the reported RX-trails of SIMECK and SPECK are incompatible, i.e. there are no right pairs that follow the expected propagation of the differences for the trail. Also, for compatible trails, the proposed approach can efficiently speed up the search process of finding the exact value of a weak key from the target weak key space. For example, in one of the reported 14-round RX trails of SPECK, the probability of a key pair to be a weak key is 2−94.91 when the whole key space is 296; our method can find a key pair for it in a comparatively short time. It is worth noting that it was impossible to find this key pair using a traditional search. As another result, we apply the proposed method to SPECK block cipher, to construct longer related-key differential trails of SPECK which we could reach 15, 16, 17, and 19 rounds for SPECK32/64, SPECK48/96, SPECK64/128, and SPECK128/256, respectively. It should be compared with the best previous results which are 12, 15, 15, and 20 rounds, respectively, that both attacks work for a certain weak key class. It should be also considered as an improvement over the reported result of rotational-XOR cryptanalysis on SPECK.acceptedVersio

SKINNY with Scalpel - Comparing Tools for Differential Analysis

Evaluating resistance of ciphers against differential cryptanalysis is essential to define the number of rounds of new designs and to mount attacks derived from differential cryptanalysis. In this paper, we compare existing automatic tools to find the best differential characteristic on the SKINNY block cipher. As usually done in the literature, we split this search in two stages denoted by Step 1 and Step 2. In Step 1, each difference variable is abstracted with a Boolean variable and we search for the value that minimizes the trail weight, whereas Step 2 tries to instantiate each difference value while maximizing the overall differential characteristic probability. We model Step 1 using a MILP tool, a SAT tool, an ad-hoc method and a CP tool based on the Choco-solver library and provide performance results. Step 2 is modeled using the Choco-solver as it seems to outperform all previous methods on this stage. Notably, for SKINNY-128 in the SK model and for 13 rounds, we retrieve the results of Abdelkhalek et al. within a few seconds (to compare with 16 days) and we provide, for the first time, the best differential related-tweakey characteristic up to respectively 14 and 12 rounds for the TK1 and TK2 models

SoK: Modeling for Large S-boxes Oriented to Differential Probabilities and Linear Correlations (Long Paper)

Automatic methods for differential and linear characteristic search are well-established at the moment. Typically, the designers of novel ciphers also give preliminary analytical findings for analysing the differential and linear properties using automatic techniques. However, neither MILP-based nor SAT/SMT-based approaches have fully resolved the problem of searching for actual differential and linear characteristics of ciphers with large S-boxes. To tackle the issue, we present three strategies for developing SAT models for 8-bit S-boxes that are geared toward differential probabilities and linear correlations. While these approaches cannot guarantee a minimum model size, the time needed to obtain models is drastically reduced. The newly proposed SAT model for large S-boxes enables us to establish that the upper bound on the differential probability for 14 rounds of SKINNY-128 is 2^{-131}, thereby completing the unsuccessful work of Abdelkhalek et al. We also analyse the seven AES-based constructions C1 - C7 designed by Jean and Nikolic and compute the minimum number of active S-boxes necessary to cause an internal collision using the SAT method. For two constructions C3 and C5, the current lower bound on the number of active S-boxes is increased, resulting in a more precise security analysis for these two structures

On Two Factors Affecting the Efficiency of MILP Models in Automated Cryptanalyses

In recent years, mixed integer linear programming (MILP, in short) gradually becomes a popular tool of automated cryptanalyses in symmetric ciphers, which can be used to search differential characteristics and linear approximations with high probability/correlation. A key problem in the MILP method is how to build a proper model that can be solved efficiently in the MILP solvers like Gurobi or Cplex. It is known that a MILP problem is NP-hard, and the numbers of variables and inequalities are two important measures of its scale and time complexity. Whilst the solution space and the variables in many MILP models built for symmetric cryptanalyses are fixed without introducing dummy variables, the cardinality, i.e., the number of inequalities, is a main factor that might affect the runtime of MILP models. We notice that the norm of a MILP model, i.e., the maximal absolute value of all coefficients in its inequalities, is also an important factor affecting its runtime. In this work we will illustrate the effects of two parameters cardinality and norm of inequalities on the runtime of Gurobi by a large number of cryptanalysis experiments. Here we choose the popular MILP solver Gurobi and view it a black box, construct a large number of MILP models with different cardinalities or norms by means of differential analyses and impossible differential analyses for some classic block ciphers with SPN structure, and observe their runtimes in Gurobi. As a result, our experiments show that although minimizing the number of inequalities and the norm of coefficients might not always minimize the runtime, it is still a better choice in most situations