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Quantum Supremacy in Cryptography with a Low-Connectivity Quantum Annealer

Feng Hu, Lucas Lamata, Chao Wang, Xi Chen, E. Solano, Mikel Sanz

Published Jun 19, 2019• 1 citation
OpenAlex

Abstract

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 super-exponential size $\mathcal{O}(2^{2^n})$ 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 $8$-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.

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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 super-exponential size O(22n)\mathcal{O}(2^{2^n})O(22n) of the associated space, finding nnn-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 888-variable bent function design problem, which is classically unsolved, with only 127127127 physical qubits and 126126126 couplers. This paves the way to reach useful quantum supremacy within the framework of quantum annealing for cryptographic purposes.
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)\mathcal{O}({2}^{{2}^{n}})O(22n) of the associated space, finding nnn-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 127127127 physical qubits and 126126126 couplers. This paves the way to reach useful quantum supremacy within the framework of quantum annealing for cryptographic purposes.
As the building block in symmetric cryptography, designing Boolean functions satisfying multiple properties is an important problem in sequence ciphers, block ciphers, and hash functions. However, the search of nnn-variable Boolean functions fulfilling global cryptographic constraints is computationally hard due to the super-exponential size O(22n)\mathcal{O}(2^{2^n})O(22n) of the space. Here, we introduce a codification of the cryptographically relevant constraints in the ground state of an Ising Hamiltonian, allowing us to naturally encode it in a quantum annealer, which seems to provide a quantum speedup. Additionally, we benchmark small nnn cases in a D-Wave machine, showing its capacity of devising bent functions, the most relevant set of cryptographic Boolean functions. We have complemented it with local search and chain repair to improve the D-Wave quantum annealer performance related to the low connectivity. This work shows how to codify super-exponential cryptographic problems into quantum annealers and paves the way for reaching quantum supremacy with an adequately designed chip.
Universal quantum computers are far from achieving practical applications. The D-Wave quantum computer is initially designed for combinatorial optimizations. Therefore, exploring the potential applications of the D-Wave device in the field of cryptography is of great importance. First, although we optimize the general quantum Hamiltonian on the basis of the structure of the multiplication table (factor up to 1 005 973), this study attempts to explore the simplification of Hamiltonian derived from the binary structure of the integers to be factored. A simple factorization on 143 with four qubits is provided to verify the potential of further advancing the integer-factoring ability of the D-Wave device. Second, by using the quantum computing cryptography based on the D-Wave 2000Q system, this research further constructs a simple version of quantum-classical computing architecture and a Quantum-Inspired Simulated Annealing (QISA) framework. Good functions and a high-performance platform are introduced, and additional balanced Boolean functions with high nonlinearity and optimal algebraic immunity can be found. Further comparison between QISA and Quantum Annealing (QA) on six-variable bent functions not only shows the potential speedup of QA, but also suggests the potential of architecture to be a scalable way of D-Wave annealer toward a practical cryptography design.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
This paper presents a new method to reduce the optimization of a pseudo-Boolean function to QUBO problem which can be solved by quantum annealer. The new method has two aspects, one is coefficient optimization and the other is variable optimization. The former is an improvement on the existing algorithm in a special case. The latter is realized by means of the maximal independent point set in graph theory. We apply this new method in integer factorization on quantum annealers and achieve the largest integer factorization (4137131) with 93 variables, the range of coefficients is [-1024,1024] which is much smaller than the previous results. We also focus on the quantum attacks on block ciphers and present an efficient method with smaller coefficients to transform Boolean equation systems into QUBO problems.
To date, conventional computers have never been able to efficiently handle certain tasks, where the number of computation steps is likely to blow up as the problem size increases. As an emerging technology and new computing paradigm, quantum computing has a great potential to tackle those hard tasks efficiently. Among all the existing quantum computation models, quantum annealing has drawn significant attention in recent years due to the realization of the commercialized quantum annealer, sparking research interests in developing applications to solve problems that are intractable for classical computers.However, designing and implementing algorithms that manage to harness the enormous computation power from quantum annealer remains a challenging task. Generally, it requires mapping of the given optimization problem into quadratic unconstrained binary optimization(QUBO) problem and embedding the subsequent QUBO onto the physical architecture of quantum annealer. Additionally, practical quantum annealers are susceptible to errors leading to low probability of the correct solution.In this study, we focus on solving Boolean satisfiability (SAT) problem using quantum annealer while addressing practical limitations. We have proposed a mapping technique that maps SAT problem to QUBO, and we have further devised a tool flow that embeds the QUBO onto the architecture of a quantum annealing device. Additionally, We have optimized the proposed embedding flow to reduce run-time in addition to shortening the qubit chain length, leading to robust quantum annealing. To further improve the reliability of quantum annealing, we have also developed a post processing embedding technique that enlarges the energy gap between ground state and the first excited state. To demonstrate the effectiveness of proposed methods, we have conducted experiments on real quantum annealing devices manufactured by D-Wave Systems, showing compelling result of using quantum annealer to solve SAT problem.