Using SAT solvers to finding short cycles in cryptographic algorithms

A desirable property of iterated cryptographic algorithms, such as stream ciphers or pseudo-random generators, is the lack of short cycles. Many of the previously mentioned algorithms are based on the use of linear feedback shift registers (LFSR) and nonlinear feedback shift registers (NLFSR) and their combination. It is currently known how to construct LFSR to generate a bit sequence with a maximum period, but there is no such knowledge in the case of NLFSR. The latter would be useful in cryptography application (to have a few taps and relatively low algebraic degree). In this article, we propose a simple method based on the generation of algebraic equations to describe iterated cryptographic algorithms and find their solutions using an SAT solver to exclude short cycles in algorithms such as stream ciphers or nonlinear feedback shift register (NLFSR). Thanks to the use of AIG graphs, it is also possible to fully automate our algorithm, and the results of its operation are comparable to the results obtained by manual generation of equations. We present also the results of experiments in which we successfully found short cycles in the NLFSRs used in KSG, Grain-80, Grain-128 and Grain-128a stream ciphers and also in stream ciphers Bivium and Trivium (without constants used in the initialization step)

Improved algebraic cryptanalysis of QUAD, Bivium and Trivium via graph partitioning on equation systems

We present a novel approach for preprocessing systems of polynomial equations via graph partitioning. The variable-sharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the corresponding system of equations can be split into smaller ones that can be solved individually. This can provide a tremendous speed-up in computing the solution to the system, but is unlikely to occur either randomly or in applications. However, by deleting certain vertices on the graph, the variable-sharing graph could be disconnected in a balanced fashion, and in turn the system of polynomial equations would be separated into smaller systems of near-equal sizes. In graph theory terms, this process is equivalent to finding balanced vertex partitions with minimum-weight vertex separators. The techniques of finding these vertex partitions are discussed, and experiments are performed to evaluate its practicality for general graphs and systems of polynomial equations. Applications of this approach in algebraic cryptanalysis on symmetric ciphers are presented: For the QUAD family of stream ciphers, we show how a malicious party can manufacture conforming systems that can be easily broken. For the stream ciphers Bivium and Trivium, we nachieve significant speedups in algebraic attacks against them, mainly in a partial key guess scenario. In each of these cases, the systems of polynomial equations involved are well-suited to our graph partitioning method. These results may open a new avenue for evaluating the security of symmetric ciphers against algebraic attacks

Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if FS does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of FS and the condition number of FS. As a consequence, we give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are small, say polynomial in the size of FS. We apply our quantum algorithm for solving Boolean equations to the cryptanalysis of several important cryptosystems: the stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, and the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the condition numbers of the corresponding equation systems are large. This leads to a new criterion for designing cryptosystems that can against the attack of quantum computers: their corresponding equation systems must have large condition numbers

Cryptanalysis of Bivium using a Boolean all solution solver

Cryptanalysis of Bivium is presented with the help of a new Boolean system solver algorithm. This algorithm uses a Boolean equation model of Bivium for a known keystream. The Boolean solver uses implicant based computation of satisfying assignments and is distinct from well known CNF-satisfiability solvers or algebraic cryptanalysis methods. The solver is also inherently parallel and returns all satisfying assignments of the system of equations in terms of implicants. Cryptanalysis of Bivium is classified in four categories of increasing strength and it is shown that the solver algorithm is able to complete the key recovery in category 2 in 48 hours by a Python code. (This benchmark is improved to 3 hours by a C++ code). Computational algorithms for formation of equations and symbolic operations are also developed afresh for handling Boolean functions and presented. Limitations of these implementations are determined with respect to Bivium model and its cryptanalysis which shall be useful for cryptanalysis of general stream ciphers

On Finding Short Cycles in Cryptographic Algorithms

We show how short cycles in the state space of a cryptographic algorithm can be used to mount a fault attack on its implementation which results in a full secret key recovery. The attack is based on the assumption that an attacker can inject a transient fault at a precise location and time of his/her choice and more than once. We present an algorithm which uses a SAT-based bounded model checking for finding all short cycles of a given length. The existing Boolean Decision Diagram (BDD) based algorithms for finding cycles have limited capacity due to the excessive memory requirements of BDDs. The simulation-based algorithms can be applied to larger problem instances, however, they cannot guarantee the detection of all cycles of a given length. The same holds for general-purpose SAT-based model checkers. The presented algorithm can find all short cycles in cryptographic algorithms with very large state spaces. We evaluate it by analyzing Trivium, Bivium, Grain-80 and Grain-128 stream ciphers. The analysis shows these ciphers have short cycles whose existence, to our best knowledge, was previously unknown