The Algorithm for Solving Quantum Linear Systems of Equations With Coherent Superposition and Its Extended Applications
Abstract: Many quantum algorithms for attacking symmetric cryptography involve the rank problem of quantum linear equations. In this paper, we first propose two quantum algorithms for solving quantum linear systems of equations with coherent superposition and construct their specific quantum circuits. Unlike previous related works, our quantum algorithms are universal. Specifically, the two quantum algorithms can both compute the rank and general solution by one measurement. The difference between them is whether the data register containing the quantum coefficient matrix can be disentangled with other registers and keep the data qubits unchanged. On this basis, we apply the two quantum algorithms as a subroutine to parallel Simon's algorithm (with multiple periods), Grover Meets Simon algorithm, and Alg-PolyQ2 algorithm, respectively. Afterwards, we construct a quantum classifier within Grover Meets Simon algorithm and the test oracle within Alg-PolyQ2 algorithm in detail, including their respective quantum circuits. To our knowledge, no such specific analysis has been done before. We rigorously analyze the success probability of those algorithms to ensure that the success probability based on the proposed quantum algorithms will not be lower than that of those original algorithms. Finally, we discuss the lower bound of the number of CNOT gates for solving quantum linear systems of equations with coherent superposition, and our quantum algorithms reach the optimum in terms of minimizing the number of CNOT gates. Furthermore, our analysis indicates that the proposed algorithms are mainly suitable for conducting attacks against lightweight symmetric ciphers, within the effective working time of an ion trap quantum computer.
- Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994.
- Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996.
- Lov K Grover. Quantum mechanics helps in searching for a needle in a haystack. Physical review letters, 79(2):325, 1997.
- Quantum distinguisher between the 3-round feistel cipher and the random permutation. In 2010 IEEE International Symposium on Information Theory, pages 2682–2685. IEEE, 2010.
- Breaking symmetric cryptosystems using quantum period finding. In Advances in Cryptology–CRYPTO 2016: 36th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 14-18, 2016, Proceedings, Part II 36, pages 207–237. Springer, 2016.
- Xavier Bonnetain. Quantum key-recovery on full aez. In Selected Areas in Cryptography–SAC 2017: 24th International Conference, Ottawa, ON, Canada, August 16-18, 2017, Revised Selected Papers 24, pages 394–406. Springer, 2018.
- Quantum attacks without superposition queries: the offline simon’s algorithm. In Advances in Cryptology–ASIACRYPT 2019: 25th International Conference on the Theory and Application of Cryptology and Information Security, Kobe, Japan, December 8–12, 2019, Proceedings, Part I, pages 552–583. Springer, 2019.
- Daniel R Simon. On the power of quantum computation. SIAM journal on computing, 26(5):1474–1483, 1997.
- Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778–797, 2001.
- Security on the quantum-type even-mansour cipher. In 2012 International Symposium on Information Theory and its Applications, pages 312–316. IEEE, 2012.
- Using simon’s algorithm to attack symmetric-key cryptographic primitives. Quantum Information and Computation, 17(1&2):0065–0078, 2017.
- On quantum slide attacks. In Selected Areas in Cryptography–SAC 2019: 26th International Conference, Waterloo, ON, Canada, August 12–16, 2019, Revised Selected Papers, pages 492–519. Springer, 2020.
- Grover meets simon–quantumly attacking the fx-construction. In Advances in Cryptology–ASIACRYPT 2017: 23rd International Conference on the Theory and Applications of Cryptology and Information Security, Hong Kong, China, December 3-7, 2017, Proceedings, Part II 23, pages 161–178. Springer, 2017.
- Quantum algorithm for linear systems of equations. Physical review letters, 103(15):150502, 2009.
- Exponential improvement in precision for simulating sparse hamiltonians. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283–292, 2014.
- Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing, 46(6):1920–1950, 2017.
- Quantum linear system algorithm for dense matrices. Physical review letters, 120(5):050502, 2018.
- Quantum recommendation systems. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 49:1–49:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017.
- Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Physical review letters, 122(6):060504, 2019.
- A complete quantum circuit to solve the information set decoding problem. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 366–377. IEEE, 2021.
- An optimized quantum implementation of ISD on scalable quantum resources. CoRR, abs/2112.06157, 2021.
- Quantum cryptography: Public key distribution and coin tossing. Theoretical Computer Science, 560:7–11, 2014.
- Quantum authentication of classical messages. Physical Review A, 64(6):062309, 2001.
- Quantum secure direct communication with high-dimension quantum superdense coding. Physical Review A, 71(4):044305, 2005.
- Quantum period finding against symmetric primitives in practice. IACR Transactions on Cryptographic Hardware and Embedded Systems, 2022(1):1–27, 2021.
- Li Yang and Rui-Rui Zhou. On the post-quantum security of encrypted key exchange protocols. arXiv preprint arXiv:1305.5640, 2013.
- On the cnot-cost of toffoli gates. Quantum Information & Computation, 9(5):461–486, 2009.
- Amit Saha and Om Khanna. Intermediate-qudit assisted improved quantum algorithm for string matching with an advanced decomposition of fredkin gate. CoRR, abs/2304.03050, 2023.
- Quantum computation and quantum information. Cambridge university press, 2010.
- Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53–74, 2002.
- Single-qubit quantum memory exceeding ten-minute coherence time. Nature Photonics, 11(10):646–650, 2017.
- Biyao Yang and Li Yang. Effect on ion-trap quantum computers from the quantum nature of the driving field. Science China Information Sciences, 63:1–15, 2020.
- How to protect des against exhaustive key search. In Advances in Cryptology—CRYPTO’96: 16th Annual International Cryptology Conference Santa Barbara, California, USA August 18–22, 1996 Proceedings 16, pages 252–267. Springer, 1996.
- Prince–a low-latency block cipher for pervasive computing applications. In Advances in Cryptology–ASIACRYPT 2012: 18th International Conference on the Theory and Application of Cryptology and Information Security, Beijing, China, December 2-6, 2012. Proceedings 18, pages 208–225. Springer, 2012.
- Block ciphers–focus on the linear layer (feat. pride). In Advances in Cryptology–CRYPTO 2014: 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17-21, 2014, Proceedings, Part I 34, pages 57–76. Springer, 2014.
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