This paper presents the transformation method of the system of algebraic equations describing the symmetric cipher into the QUBO problem. After transformation of given equations
f0,f1,…,fn−1f0,f1,...,fn-1to equations over integers
f0′,f1′,…,fn−1′f0',f1',...,fn-1', one can linearize each, obtaining
flini′=lin(fi′)flini'=lin(fi'), for
i=0,n−1i=0,n-1¯, where
linlindenotes linearization operation. Finally, one can obtain problem in the QUBO form as
(flin0′)2+⋯+(flinn−1′)2+Pen−C(flin0')2+⋯+(flinn-1')2+Pen-C, where
PenPendenotes penalties obtained during linearization of equations,
nnis the number of equations and
CCis constant appearing in the polynomial
(flin0′)2+⋯+(flinn−1′)2+Pen(flin0')2+⋯+(flinn-1')2+Pen. This paper presents the transformation method of SPN block ciphers to the QUBO problem. What is more, we present the results of the transformation of the complete AES-128 cipher to the QUBO problem, where the number of variables of the equivalent QUBO problem equals approximately 30,026. It is worth noting that AES-128 is much easier to solve using quantum annealing than the factorization problem and the discrete logarithm problem of a similar level of security. For example, factorizing a 3072 bit long RSA integer using quantum annealing requires a QUBO problem of about 2,360,000 variables.
This work focuses on quantum methods for cryptanalysis of schemes based on the integer factorization problem and the discrete logarithm problem. We demonstrate how to practically solve the largest instances of the factorization problem by improving an approach that combines quantum and classical computations, assuming the use of the best publicly available special-class quantum computer: the quantum annealer. We achieve new computational experiment results by solving the largest instance of the factorization problem ever announced as solved using quantum annealing, with a size of 29 bits. The core idea of the improved approach is to leverage known subexponential classical method to break the problem down into many smaller computations and perform the most critical ones on a quantum computer. This approach does not reduce the complexity class, but it assesses the pragmatic capabilities of an attacker. It also marks a step forward in the development of hybrid methods, which in practice may surpass classical methods in terms of efficiency sooner than purely quantum computations will.
We aim to advance the state-of-the-art in Quadratic Unconstrained Binary Optimization formulation with a focus on cryptography algorithms. As the minimal QUBO encoding of the linear constraints of optimization problems emerges as the solution of integer linear programming (ILP) problems, by solving special boolean logic formulas (like ANF and DNF) for their integer coefficients it is straightforward to handle any normal form, or any substitution for multi-input AND, OR or XOR operations in a QUBO form. To showcase the efficiency of the proposed approach we considered the most widespread cryptography algorithms including AES-128/192/256, MD5, SHA1 and SHA256. For each of these, we achieved QUBO instances reduced by thousands of logical variables compared to previously published results, while keeping the QUBO matrix sparse and the magnitude of the coefficients low. In the particular case of AES-256 cryptography function we obtained more than 8x reduction in variable count compared to previous results. The demonstrated reduction in QUBO sizes notably increases the vulnerability of cryptography algorithms against future quantum annealers, capable of embedding around
30 thousands of logical variables.
The application in cryptography of quantum algorithms for prime factorization fostered the interest in quantum computing. However, quantum computers, and particularly quantum annealers, can also be helpful to construct secure cryptographic keys. Indeed, finding robust Boolean functions for cryptography is an important problem in sequence ciphers, block ciphers, and hash functions, among others. Due to the superexponential size
O(22n) of the associated space, finding
n-variable Boolean functions with global cryptographic constraints is computationally hard. This problem has already been addressed employing generic low-connected incoherent D-Wave quantum annealers. However, the limited connectivity of the Chimera graph, together with the exponential growth in the complexity of the Boolean-function design problem, limit the problem scalability. Here, we propose a special-purpose coherent quantum-annealing architecture with three couplers per qubit, designed to optimally encode the bent-function design problem. A coherent quantum annealer with this tree-type architecture has the potential to solve the eight-variable bent-function design problem, which is classically unsolved, with only
127 physical qubits and
126 couplers. This paves the way to reach useful quantum supremacy within the framework of quantum annealing for cryptographic purposes.
Quantum annealing is a metaheuristic method aimed at finding the global minimum of a pseudo-Boolean quadratic function. To solve a problem by a quantum annealer, it must be restated as a QUBO (quadratic unconstrained binary optimization) problem, the standard format acceptable to an annealer. However, there are many natural problems that turn out be of cubic or higher order, yet promising solutions can be obtained for many instances using quantum annealers, after converting to the QUBO format, using order reduction criteria. Accordingly, quadratization becomes a crucial preprocessing step in order to make use of present-day quantum annealing hardware that relies on coupling technology, which can only process interactions of at most degree two. A critical challenge is that quadratization methods often require many additional variables, degrading the performance of the annealers. A number of algorithmic and heuristic quadratization techniques have been proposed and used for annealing in the literature, affecting the performance of the annealers differently. We examine a number of such quadratization techniques, focusing on their efficiency, compactness, and applicability. We analyze the approaches motivated by diverse goals and highlight the transition from foundational approaches to more recent compact approaches. An evaluation of the current usage of different quadratization methods in the literature is carried out at the end of the survey. Further, we comment on the limitations on usage and show possible opportunities to enhance the performance of annealers.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.