In the present paper we describe the technology for translating algorithmic descriptions of discrete functions to SAT. The proposed technology is aimed at applications in algebraic cryptanalysis. We describe how cryptanalysis instances are reduced to SAT in such a way that it should be perceived as natural by the cryptographic community. Therefore, in the theoretical part of the paper we justify the main principles of general reduction to SAT for discrete functions from a class containing the majority of functions employed in cryptography. Based on these principles we describe the Transalg software system, developed with SAT-based cryptanalysis specifics in mind. We show the results of applications of Transalg to construction of a number of attacks on various cryptographic functions. Some of the corresponding attacks are state of the art. We also compare the functional capabilities of the proposed system with that of other software systems that can be used to reduce cryptanalysis instances to SAT, and also with the CBMC system widely employed in symbolic verification. In the paper we also present vast experimental data, obtained using the SAT-solvers that took first places at the SAT-competitions in the recent several years.
As we already noted, @cite_25 was the first paper to demonstrate the applicability of SAT-based cryptanalysis to relevant cryptographic algorithms. In that paper using the solver @cite_95 it was possible to quite effectively find single-block collisions for MD4. Using new propositional encoding methods (in particular, the system) and state-of-the-art SAT solvers it is possible to find preimages for MD4 and MD5 several hundred times faster than it was done in @cite_25 . Nevertheless, on the current stage SAT-based cryptanalysis is less effective than specialized methods (see, for example @cite_58 @cite_99 @cite_82 ) on problems of finding collisions of cryptographic hash functions. However, as far as we know, it is the SAT-based approach that yields best known preimage attacks on truncated variants of hash functions @cite_32 @cite_78 @cite_54 . In application to MD4-39 for a long time the SAT-based preimage attack from @cite_32 was considered to be the best. In @cite_15 we significantly improved the results from @cite_32 . It was possible for the large part thanks to functional capabilities of the system.
@cite_99 @cite_78 @cite_54 @cite_32 @cite_95 @cite_15 @cite_58 @cite_25 @cite_82
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In this paper we construct preimage attack on the truncated variant of the MD4 hash function. Specifically, we study the MD4-39 function defined by the first 39 steps of the MD4 algorithm. We suggest a new attack on MD4-39, which develops the ideas proposed by H. Dobbertin in 1998. Namely, the special relaxation constraints are introduced in order to simplify the equations corresponding to the problem of finding a preimage for an arbitrary MD4-39 hash value. The equations supplemented with the relaxation constraints are then reduced to the Boolean Satisfiability Problem (SAT) and solved using the state-of-the-art SAT solvers. We show that the effectiveness of a set of relaxation constraints can be evaluated using the black-box function of a special kind. Thus, we suggest automatic method of relaxation constraints generation by applying the black-box optimization to this function. The proposed method made it possible to find new relaxation constraints that contribute to a SAT-based preimage attack on MD4-39 which significantly outperforms the competition. Several standard cryptographic hash functions were broken in 2005. Some essential building blocks of these attacks lend themselves well to automation by encoding them as CNF formulas, which are within reach of modern SAT solvers. In this paper we demonstrate effectiveness of this approach. In particular, we are able to generate full collisions for MD4 and MD5 given only the differential path and applying a (minimally modified) off-the-shelf SAT solver. To the best of our knowledge, this is the first example of a SAT-solver-aided cryptanalysis of a non-trivial cryptographic primitive. We expect SAT solvers to find new applications as a validation and testing tool of practicing cryptanalysts. In this paper, we propose an algorithm for constructing guess-and-determine attacks on keystream generators and apply it to the cryptanalysis of the alternating step generator (ASG) and two its modifications (MASG and MASG0). In a guess-and-determine attack, we first “guess” some part of an initial state and then apply some procedure to determine, if the guess was correct and we can use the guessed information to solve the problem, thus performing an exhaustive search over all possible assignments of bits forming a chosen part of an initial state. We propose to use in the “determine” part the algorithms for solving Boolean satisfiability problem (SAT). It allows us to consider sets of bits with nontrivial structure. For each such set it is possible to estimate the runtime of a corresponding guess-and-determine attack via the Monte-Carlo method, so we can search for a set of bits yielding the best attack via a black-box optimization algorithm augmented with several SAT-specific features. We constructed and implemented such attacks on ASG, MASG, and MASG0 to prove that the constructed runtime estimations are reliable. We show, that the constructed attacks are better than the trivial ones, which imply exhaustive search over all possible states of the control register, and present the results of experiments on cryptanalysis of ASG and MASG MASG0 with total registers length of 72 and 96, which have not been previously published in the literature. Cryptography ensures the confidentiality and authenticity of information but often relies on unproven assumptions. SAT solvers are a powerful tool to test the hardness of certain problems and have successfully been used to test hardness assumptions. This paper extends a SAT solver to efficiently work on cryptographic problems. The paper further illustrates how SAT solvers process cryptographic functions using automatically generated visualizations, introduces techniques for simplifying the solving process by modifying cipher representations, and demonstrates the feasibility of the approach by solving three stream ciphers. To optimize a SAT solver for cryptographic problems, we extended the solver's input language to support the XOR operation that is common in cryptography. To better understand the inner workings of the adapted solver and to identify bottlenecks, we visualize its execution. Finally, to improve the solving time significantly, we remove these bottlenecks by altering the function representation and by pre-parsing the resulting system of equations. The main contribution of this paper is a new approach to solving cryptographic problems by adapting both the problem description and the solver synchronously instead of tweaking just one of them. Using these techniques, we were able to solve a well-researched stream cipher 26 times faster than was previously possible. In 2010, Tao Xie and Dengguo Feng [XF10] constructed the rst single-block collision for MD5 consisting of two 64-byte messages that have the same MD5 hash. Details of their attack, developed using what they call an evolutionary approach, has not been disclosed security reasons". Instead they have posted a challenge to the cryptology community to nd a new dierent single-block collision attack for MD5. This paper answers that challenge by presenting a single-block collision attack based on other message dierences together with an example colliding message pair. The attack is based on a new collision nding algorithm that exploits the low number of bitconditions in the rst round. It uses a new way to choose message blocks that satisfy bitconditions up to step 22 and additionally uses three known tunnels to correct bitconditions up to step 25. The attack has an average runtime complexity equivalent to 2 49:8 calls to MD5’s compression function. SAT solvers are increasingly being used for cryptanalysis of hash functions and symmetric encryption schemes. Inspired by this trend, we present MapleCrypt which is a SAT solver-based cryptanalysis tool for inverting hash functions. We reduce the hash function inversion problem for fixed targets into the satisfiability problem for Boolean logic, and use MapleCrypt to construct preimages for these targets. MapleCrypt has two key features, namely, a multi-armed bandit based adaptive restart (MABR) policy and a counterexample-guided abstraction refinement (CEGAR) technique. The MABR technique uses reinforcement learning to adaptively choose between different restart policies during the run of the solver. The CEGAR technique abstracts away certain steps of the input hash function, replacing them with the identity function, and verifies whether the solution constructed by MapleCrypt indeed hashes to the previously fixed targets. If it is determined that the solution produced is spurious, the abstraction is refined until a correct inversion to the input hash target is produced. We show that the resultant system is faster for inverting the SHA-1 hash function than state-of-the-art inversion tools. The stream cipher Salsa20 was introduced by Bernstein in 2005 as a candidate in the eSTREAM project, accompanied by the reduced versions Salsa20 8 and Salsa20 12. ChaCha is a variant of Salsa20 aiming at bringing better diffusion for similar performance. Variants of Salsa20 with up to 7 rounds (instead of 20) have been broken by differential cryptanalysis, while ChaCha has not been analyzed yet. We introduce a novel method for differential cryptanalysis of Salsa20 and ChaCha, inspired by correlation attacks and related to the notion of neutral bits. This is the first application of neutral bits in stream cipher cryptanalysis. It allows us to break the 256-bit version of Salsa20 8, to bring faster attacks on the 7-round variant, and to break 6- and 7-round ChaCha. In a second part, we analyze the compression function Rumba, built as the XOR of four Salsa20 instances and returning a 512-bit output. We find collision and preimage attacks for two simplified variants, then we discuss differential attacks on the original version, and exploit a high-probability differential to reduce complexity of collision search from 2256to 279for 3-round Rumba. To prove the correctness of our approach we provide examples of collisions and near-collisions on simplified versions. Polar codes, as the first provable capacity-achieving error-correcting codes, have received much attention in recent years. However, the decoding performance of polar codes with traditional successive-cancellation (SC) algorithm cannot match that of the low-density parity-check or Turbo codes. Because SC list (SCL) decoding algorithm can significantly improve the error-correcting performance of polar codes, design of SCL decoders is important for polar codes to be deployed in practical applications. However, because the prior latency reduction approaches for SC decoders are not applicable for SCL decoders, these list decoders suffer from the long-latency bottleneck. In this paper, we propose a multibit-decision approach that can significantly reduce latency of SCL decoders. First, we present a reformulated SCL algorithm that can perform intermediate decoding of 2 b together. The proposed approach, referred as 2-bit reformulated SCL ( 2b-rSCL ) algorithm , can reduce the latency of SCL decoder from ( @math ) to ( @math ) clock cycles without any performance loss. Then, we extend the idea of 2-b-decision to general case, and propose a general decoding scheme that can perform intermediate decoding of any @math bits simultaneously. This general approach, referred as @math -bit reformulated SCL ( @math b-rSCL ) algorithm , can reduce the overall decoding latency to as short as @math cycles. Furthermore, on the basis of the proposed algorithms, very large-scale integration architectures for 2b-rSCL and 4b-rSCL decoders are synthesized. Compared with a prior SCL decoder, the proposed (1024, 512) 2b-rSCL and 4b-rSCL decoders can achieve 21 and 60 reduction in latency, 1.66 and 2.77 times increase in coded throughput with list size 2, and 2.11 and 3.23 times increase in coded throughput with list size 4, respectively. In logical cryptanalysis a problem of search of a secret key of a cryptographic system is formulated as a SAT problem, i.e. a problem of search of a satisfying assignment for some CNF. In this paper we consider some natural strategies for parallelization of these SAT problems. We apply coarse-grained approach which makes it possible to use distributed computing environments with slow interconnect. The main practical result of this paper is successful logical cryptanalysis of keystream generator A5 1 in BNB-Grid system.