Boolean Satisfiability Research Articles

Novel Optimized Implementations of Lightweight Cryptographic S-Boxes via SAT Solvers

Novel Optimized Implementations of Lightweight Cryptographic S-Boxes via SAT Solvers

Automatic Search of Differential Characteristics and Improved Differential Cryptanalysis for PRINCE, QARMA, and MANTIS

Reflection structure has a significant advantage that realizing decryption and encryption results in minimum additional costs, and many block ciphers tend to adopt such structure to achieve the requirement of low overhead. PRINCE, MANTIS, QARMA, and PRINCEv2 are lightweight block ciphers with reflection feature proposed in recent years. In this paper, we consider the automatic differential cryptanalysis of reflection block ciphers based on Boolean satisfiability (SAT) method. Since reflection block ciphers have different round functions, we extend forward and backward from the middle structure and achieve to accelerate the search of the optimal differential characteristics for such block ciphers with the Matsui’s bounding conditions. As a result, we present the optimal differential characteristics for PRINCE up to 12 rounds (full round), and they are also the optimal characteristics for PRINCEv2. We also find the optimal differential characteristics for MANTIS, QARMA‐64, and QARMA‐128 up to 10, 12, and 8 rounds, respectively. To mount an efficient differential attack on such block ciphers, we present a uniform SAT model by combining the differential characteristic searching process and the key recovery process. With this model, we find two sets of 7‐round differential characteristics for PRINCE with less guessed key bits and use them to present a multiple differential attack against 11‐round PRINCE, which improves the known single‐key attack on PRINCE by one round to our knowledge.

Quantum computing with time-travelling quantum gates

Quantum computing can solve problems that are difficult to solve in classical computing, expanding the range of problems that can be effectively computed within the allowable range of classical physical principles, and posing a challenge to the extended Church-Turing thesis in classical computing. Here, we discuss an interesting question: how to achieve more powerful computers by breaking through the limitations of physical principles, further enhancing the capabilities of quantum computers. To extend quantum computing, novel operations related to relativistic physics are a crucial candidate. Among them, the concept of closed time-like curve has aroused widespread interest, and it introduces the ability for time travel. Mathematically, quantum state along the closed time-like curve is determined through self-consistent equations, which has been demonstrated in simulations. Here, we consider a novel manipulation capability that allows quantum computing to achieve time-travelling quantum control gate. This is an intuitive extension of the graphical language of quantum circuits. Explaining quantum circuits as tensor networks, we first explain how to experimentally simulate the output of such a circuit in a system without time-travel capability. Then, we take an example to demonstrate an extended quantum algorithm that can efficiently solve SAT problems, indicating that with the involvement of time-travelling quantum gates, the computational complexity class P = NP. We also anticipate that the time-travelling quantum gates will play a facilitating role in accomplishing other quantum tasks, including achieving deterministic non-orthogonal quantum state discrimination, and quantum state cloning. Our results contribute to a more in-depth understanding of the relationship between computation and physical principles.

Augmenting an electronic Ising machine to effectively solve boolean satisfiability

With the slowdown of improvement in conventional von Neumann systems, increasing attention is paid to novel paradigms such as Ising machines. They have very different approach to solving combinatorial optimization problems. Ising machines have shown great potential in solving binary optimization problems like MaxCut. In this paper, we present an analysis of these systems in boolean satisfiability (SAT) problems. We demonstrate that, in the case of 3-SAT, a basic architecture fails to produce meaningful acceleration, largely due to the relentless progress made in conventional SAT solvers. Nevertheless, careful analysis attributes part of the failure to the lack of two important components: cubic interactions and efficient randomization heuristics. To overcome these limitations, we add proper architectural support for cubic interaction on a state-of-the-art Ising machine. More importantly, we propose a novel semantic-aware annealing schedule that makes the search-space navigation much more efficient than existing annealing heuristics. Using numerical simulations, we show that such an “Augmented” Ising Machine for SAT is projected to outperform state-of-the-art software-based, GPU-based and conventional hardware SAT solvers by orders of magnitude.

On the upper bounds of (1,0)-super solutions for the regular balanced random (k,2s)-SAT problem

On the upper bounds of (1,0)-super solutions for the regular balanced random (k,2s)-SAT problem

A Dynamically Programmable Quantum Photonic Microprocessor for Graph Computation

AbstractQuantum computing has grown extensively, especially in system design and development, and the current research focus has gradually evolved from validating quantum advantage to practical applications. In particular, nondeterministic‐polynomial‐time (NP)‐complete problems are central in numerous important application areas. Still, in practice, it is difficult to solved efficiently with conventional computers, limited by the exponential jump in hardness. Here, a quantum photonic microprocessor based on Gaussian boson sampling (GBS) that offers dynamic programmability to solve various graph‐related NP‐complete problems is demonstrated. The system with optical, electrical, and thermal packaging implements a GBS with 16 modes of single‐mode squeezed vacuum states, a universal programmable 16‐mode interferometer, and a single photon readout on all outputs with high accuracy, generality, and controllability. The developed system is applied to demonstrate applications in solving NP‐complete problems, manifesting the ability of photonic quantum computing to realize practical applications for conventionally intractable computations. The GBS‐based quantum photonic microprocessor is applied to solve task assignment, Boolean satisfiability, graph clique, max cut, and vertex cover. These demonstrations suggest an excellent benchmarking platform, paving the way toward large‐scale combinatorial optimization.

Design and Implementation of a New Framework for Post-Synthesis Obfuscation with a Mixture of Multiple Cells with an Integrated Anti-SAT Block

This paper proposes a new framework for post-synthesis obfuscation of digital circuits using a mixture of cells combined with an Anti-SAT block. Furthermore, a novel integrated framework has been established wherein obfuscation, along with Anti-SAT and validation of the benchmarks, progress through MATLAB®, Python, Cadence RTL Encounter® and Cadence LEC® to implement the proposed methodology. Area, delay, leakage power and total power are adopted as elements of the evaluation matrix. These parameters are compared between the original circuit, the circuit after obfuscation, the circuit after integration with Anti-SAT and the circuit after implementation of the proposed method of multiple-cell obfuscation with Anti-SAT. The probability of breaking a circuit is taken as the security criterion. It is mathematically proven that as the number of types of obfuscated cells used increases, then the probability of breaking the circuit decreases. The results obtained accord with the mathematical proof. The framework minimizes the delay by inserting obfuscation cells (OCs) in the non-critical paths, strengthens the security by using several types of OCs and allows the user to select a design based on justified area, leakage power and total power. However, against a Boolean SATisfiability (SAT) attack, obfuscation with multiple cells is not a sufficient defense. An Anti-SAT block performs better than obfuscation but has its own limitations. Thus, use of an Anti-SAT block in combination with multiple-cell obfuscation is proposed and implemented, giving better results against an efficient SAT attack. The number of iterations, as well as runtime to obtain the correct keys, increase significantly for the Anti-SAT block combined with multiple-cell obfuscation compared to the Anti-SAT or obfuscation block alone.

Using SAT Solvers to Reverse-Engineer FSM Models of Digital Devices

Inferring a functional specification from an existing digital design is a challenge that is suitable with reverse-engineering methods. One of the most widely used functional specification formats is a finite state machine (FSM). This article studies the possibility of blind passive specification mining for a digital device, where the device is treated as a “black box”. The presented approach treats an input and output signal waveform as the transition graph of an incomplete deterministic FSM and learns the FSM through FSM minimization. It employs a Boolean satisfiability problem (SAT) solver to find a minimal FSM that complies with observed object behavior. The known approach to identifying state machines in discrete event systems is adapted to operate with variables in the form of coloring and transition tables. The developed implementation produces a synthesizable specification in hardware description language (HDL) and a state diagram in unified modeling language (UML). The proposed approach for inferring an FSM from a waveform trace can serve as a supplementary tool during reverse engineering to provide developers with meaningful insight regarding the analyzed device. The presented case study defines metrics of successful FSM inference and applies them to a synthetic FSM and a real-world example FSM to demonstrate the applicability of the approach.

Effectively computing high strength mixed covering arrays with constraints

Covering arrays have become a key piece in Combinatorial Testing. In particular, we focus on the efficient construction of Covering Arrays with Constraints of high strength. SAT solving technology has been proven to be well suited when solving Covering Arrays with Constraints. However, the size of the SAT reformulations rapidly grows up with higher strengths. To this end, we present a new incomplete algorithm that mitigates substantially memory blow-ups and its parallel version that allows reducing run-time consumption. Thanks to these new developments we provide a tool for Combinatorial Testing in practical environments. The experimental results confirm the goodness of the approach, opening avenues for new practical applications.

On the Security of Sequential Logic Locking Against Oracle-Guided Attacks

The Boolean satisfiability (SAT) attack is an oracle-guided attack that can break most combinational logic locking schemes by efficiently pruning out all the wrong keys from the search space. Extending such an attack to sequential logic locking requires multiple time-consuming rounds of SAT solving, performed using an “unrolled” version of the sequential circuit, and model checking, used to determine the successful termination of the attack. This paper addresses these challenges by formally characterizing the relation between the minimum unrolling depth required to prune out the wrong keys of a SAT-based attack and a notion of functional corruptibility (FC) for sequential circuits, which can be efficiently estimated from a locked circuit to indicate the progress of a SAT-based attack. Based on this analysis, we present an FC-guided SAT-based attack that can significantly reduce unnecessary SAT and model checking tasks. We present two versions of the attack, namely, Fun-SAT and Fun-SAT+, based on whether the attacker has a priori knowledge of the key length. Fun-SAT aims to find the correct key sequence, while Fun-SAT+ aims to retrieve the correct initial state of the circuit. Numerical evaluation shows that Fun-SAT can be, on average, 90× faster than previous attacks against state-of-the-art locking methods. On the other hand, when using an approximate termination condition, Fun-SAT+ can find an initial state that leads to at most 0.1% FC in 76.9% instances that would otherwise time out after one day.

An efficient circuit-based SAT solver and its application in logic equivalence checking

In this paper, we propose an integrated approach that combines a circuit-based SAT solver (CirSAT) with a technique based on XOR-Majority Graph (XMG) rewriting for logic equivalence checking. CirSAT harnesses circuit information to implement a variety of efficient SAT algorithms on the And-Inverter Graph (AIG). Specifically, we introduce the “AIG-J-frontier” algorithm to minimize unnecessary search space during SAT solving. This algorithm employs the fanout count of logic gates as a heuristic to guide conflict resolution decisions. Furthermore, we present a flexible and efficient two-pointer watching scheme for rapid Boolean constraint propagation. The XMG-based rewriting technique is used to simplify logic networks and accommodate a larger number of high fanout nodes. In experiments using ISCAS’85 and EPFL benchmark suites, our method outperforms CNF-based solvers such as Minisat and Lingeling with a 3.66 speedup. It also exhibits an 8.26 speedup compared to the circuit-based solver QuteSAT and a 5.99 speedup when compared to using the CirSAT solver independently.

Ambient-aware continuous aid for mountain rescue activities

Ambient-awareness in conjunction with pervasive computing is a significant challenge for system designers. It follows the necessity of gathering raw, massive and heterogeneous environmental data which we obtained, while middleware processes must merge context modelling and reasoning seamlessly. We proposed a system supporting mountain rescuers which is demanding due to the large number of environmental objects interacting, as well as high data variability. We presented complex context processing embedded in the proposed context life cycle and implemented it in a difficult mountain environment. We introduced five weather scenarios which are a basis for contextual and perceptual processing during the validation of our model. The system merges a message streaming broker for massive data transport, low and high-level processing algorithms, repositories and a logical SAT solver. It constitutes a Context-Aware-as-a-Service (CAaaS) system, offering advanced support for mountain rescue operations. The provided software model defines middleware components which act on a predicted context and transform in situ sensor data into smart decisions, and which could operate as a platform-based cloud computing model. It is an enabler yielding a synergy effect with different software components orchestration when providing pro-activeness and non-intrusiveness concerning smart decisions.

Complexity Analysis of the SAT Attack on Logic Locking

Due to the adoption of horizontal business models following the globalization of semiconductor manufacturing, the overproduction of integrated circuits (ICs) and the piracy of intellectual properties (IPs) can lead to significant damage to the integrity of the semiconductor supply chain. Logic locking emerges as a primary design-for-security measure to counter these threats, where ICs become fully functional only when unlocked with a secret key. However, Boolean satisfiability-based attacks have rendered most locking schemes ineffective. This gives rise to numerous defenses and new locking methods to achieve SAT resiliency. This paper provides a unique perspective on the SAT attack efficiency based on conjunctive normal form (CNF) stored in SAT solver. First, we show how the attack learns new relations between keys in every iteration using distinguishing input patterns and the corresponding oracle responses. The input-output pairs result in new CNF clauses of unknown keys to be appended to the SAT solver, which leads to an exponential reduction in incorrect key values. Second, we demonstrate that the SAT attack can break any locking scheme within linear iteration complexity of key size. Moreover, we show how key constraints on point functions affect the SAT attack complexity. We explain why proper key constraint on AntiSAT reduces the complexity effectively to constant 1. The same constraint helps the breaking of CAS-Lock down to linear iteration complexity. Our analysis provides a new perspective on the capabilities of SAT attack against multiplier benchmark c6288, and we provide new directions to achieve SAT resiliency.

IntSat: integer linear programming by conflict-driven constraint learning

State-of-the-art SAT solvers are nowadays able to handle huge real-world instances. The key to this success is the Conflict-Driven Clause-Learning (CDCL) scheme, which encompasses a number of techniques that exploit the conflicts that are encountered during the search for a solution. In this article, we extend these techniques to Integer Linear Programming (ILP), where variables may take general integer values instead of purely binary ones, constraints are more expressive than just propositional clauses, and there may be an objective function to optimize. We explain how these methods can be implemented efficiently and discuss possible improvements. Our work is backed with a basic implementation showing that, even in this far less mature stage, our techniques are already a useful complement to the state of the art in ILP.

Self-averaging of digital memcomputing machines.

Digital memcomputing machines (DMMs) are a new class of computing machines that employ nonquantum dynamical systems with memory to solve combinatorial optimization problems. Here, we show that the time to solution (TTS) of DMMs follows an inverse Gaussian distribution, with the TTS self-averaging with increasing problem size, irrespective of the problem they solve. We provide both an analytical understanding of this phenomenon and numerical evidence by solving instances of the 3-SAT (satisfiability) problem. The self-averaging property of DMMs with problem size implies that they are increasingly insensitive to the detailed features of the instances they solve. This is in sharp contrast to traditional algorithms applied to the same problems, illustrating another advantage of this physics-based approach to computation.

Universal Quantum Optimization with Cold Atoms in an Optical Cavity.

Cold atoms in an optical cavity have been widely used for quantum simulations of many-body physics, where the quantum control capability has been advancing rapidly in recent years. Here, we show the atom cavity system is universal for quantum optimization with arbitrary connectivity. We consider a single-mode cavity and develop a Raman coupling scheme by which the engineered quantum Hamiltonian for atoms directly encodes number partition problems. The programmability is introduced by placing the atoms at different positions in the cavity with optical tweezers. The number partition problem solution is encoded in the ground state of atomic qubits coupled through a photonic cavity mode, which can be reached by adiabatic quantum computing. We construct an explicit mapping for the 3-SAT and vertex cover problems to be efficiently encoded by the cavity system, which costs linear overhead in the number of atomic qubits. The atom cavity encoding is further extended to quadratic unconstrained binary optimization problems. The encoding protocol is optimal in the cost of atom number scaling with the number of binary degrees of freedom of the computation problem. Our theory implies the atom cavity system is a promising quantum optimization platform searching for practical quantum advantage.

Learning to select SAT encodings for pseudo-Boolean and linear integer constraints

Many constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT). However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.

Symmetry Breaking for Voting Mechanisms.

Recently, Rowe and Aishwaryaprajna (2019) introduced a simple majority vote technique that efficiently solves Jump with large gaps, OneMax with large noise, and any monotone function with a polynomial-size image. In this paper, we identify a pathological condition for this algorithm: the presence of spin-flip symmetry in the problem instance. Spin-flip symmetry is the invariance of a pseudo-Boolean function to complementation. Many important combinatorial optimization problems admit objective functions that exhibit this pathology, such as graph problems, Ising models, and variants of propositional satisfiability. We prove that no population size exists that allows the majority vote technique to solve spin-flip symmetric functions of unitation with reasonable probability. To remedy this, we introduce a symmetry-breaking technique that allows the majority vote algorithm to overcome this issue for many landscapes. This technique requires only a minor modification to the original majority vote algorithm to force it to sample strings in {0,1}n from a dimension n-1 hyperplane. We prove a sufficient condition for a spin-flip symmetric function to possess in order for the symmetry-breaking voting algorithm to succeed, and prove its efficiency on generalized TwoMax, a spin-flip symmetric variant of Jump, and families of constructed 3-NAE-SAT and 2-XOR-SAT formulas. We also prove that the algorithm fails on the one-dimensional Ising model, and suggest different techniques for overcoming this. Finally, we present empirical results that explore the tightness of the runtime bounds and the performance of the technique on randomized satisfiability variants.

Robust transshipment problem under consistent flow constraints

AbstractIn this article, we study robust transshipment under consistent flow constraints. We consider demand uncertainty represented by a finite set of scenarios and characterize a subset of arcs as so‐called fixed arcs. In each scenario, we require an integral flow that satisfies the respective flow balance constraints. In addition, on each fixed arc, we require equal flow for all scenarios. The objective is to minimize the maximum cost occurring among all scenarios. We show that the problem is strongly ‐complete on acyclic digraphs by a reduction from the ‐Sat problem. Furthermore, we prove that the problem is weakly ‐complete on series‐parallel digraphs by a reduction from a special case of the Partition problem. If in addition the number of scenarios is constant, we observe the pseudo‐polynomial‐time solvability of the problem. We provide poly‐nomial‐time algorithms for three special cases on series‐parallel digraphs. Finally, we present a polynomial‐time algorithm for pearl digraphs.

Solving Restricted Preemptive Scheduling on Parallel Machines with SAT and PMS

Restricted preemption plays a crucial role in reducing total completion time while controlling preemption overhead. A typical version of restricted preemptive models is k-restricted preemptive scheduling, where preemption is only allowed after a task has been continuously processed for at least k units of time. Though solving this problem of minimizing the makespan on parallel machines is NP-hard in general, it is of vital importance to obtain the optimal solution for small-sized problems, as well as for evaluation of heuristics. This paper proposes optimal strategies to the aforementioned problem. Motivated by the dramatic speed-up of Boolean Satisfiability (SAT) solvers, we make the first step towards a study of applying a SAT solver to the k-restricted scheduling problem. We set out to encode the scheduling problem into propositional Boolean logic and determine the optimal makespan by repeatedly calling an off-the-shelf SAT solver. Moreover, we move one step further by encoding the problem into Partial Maximum Satisfiability (PMS), which is an optimized version of SAT, so that the explicit successive calls of the solver can be eliminated. The optimal makespan of the problem and the performance of the proposed methods are studied experimentally. Furthermore, an existing heuristic algorithm is evaluated by the optimization methods.