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International Conference on Platform Technology and Service

Quantum Gauss-Jordan Elimination for Code in Quantum

Kyungbae Jang, Hyunji Kim, Hwajeong Seo

Published Aug 22, 2022• 0 citations
Publisher

Abstract

Quantum computers can efficiently model and solve certain problems on their own. Already in various fields, quantum computing is expected to outperform classical computers. In this flow, it is important to quantize classical arithmetic to achieve the benefits of quantum computing. In this paper, we efficiently implement quantum Gauss-Jordan elimination for binary matrix. Quantum Gauss-Jordan is required to accelerate Information Set Decoding, a cryptanalysis algorithm for code-based post-quantum ciphers, with Grover’s algorithm. In our understanding, the most important module when implementing ISD as a quantum version is quantum Gauss-Jordan elimination. We implement quantum Gauss-Jordan using only quantum gates (e.g., X, CX, CCX, and Swap) that replace classical operations. Finally, we analyze the quantum resources required for our implementation. For the simulation and resource estimation of our work, the quantum programming tool ProjectQ is used.

DOI
10.1109/platcon55845.2022.9932108

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Quantum computers can efficiently model and solve certain problems on their own. Already in various fields, quantum computing is expected to outperform classical computers. In this flow, it is important to quantize classical arithmetic to achieve the benefits of quantum computing. In this paper, we efficiently implement quantum Gauss-Jordan elimination for binary matrix. Quantum Gauss-Jordan is required to accelerate Information Set Decoding, a cryptanalysis algorithm for code-based post-quantum ciphers, with Grover’s algorithm. In our understanding, the most important module when implementing ISD as a quantum version is quantum Gauss-Jordan elimination. We implement quantum Gauss-Jordan using only quantum gates (e.g., X, CX, CCX, and Swap) that replace classical operations. Finally, we analyze the quantum resources required for our implementation. For the simulation and resource estimation of our work, the quantum programming tool ProjectQ is used.
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If large-scale quantum computers capable of running quantum algorithms are developed, it is expected that powerful quantum cryptanalysis will become possible. To establish secure quantum-resistant encryption systems, it is necessary to evaluate the quantum security of cryptographic algorithms. As a result, various quantum cryptanalysis research studies have been published, and implementation techniques that minimize the resources required for analyzing encryption have been proposed. In this paper, we present an efficient quantum circuit implementation of Classic McEliece, which is one of the candidate algorithms in the NIST Post-Quantum Cryptography Standardization Round 4. We optimize the encoding, and decoding operations of Classic McEliece on quantum circuits, with a specific focus on binary field arithmetic, linear operations, and the Berlekamp-Massey decoding quantum circuit.
Abstract In the coding theory, it is one of the most important research problems to find an efficient algorithm for the decoding process. A quantum computer is an innovative computing device that performs calculations using the principles of quantum mechanics. In this paper, we find a decoding algorithm for binary linear codes using quantum circuits; this is the first time that quantum circuits are used as a main tool for decoding binary linear codes. Our decoding algorithm is mainly based on Grover’s search algorithm. In detail, we study the process of implementing the exhaustive search using quantum circuits for decoding binary linear codes. For our purpose, we design a modified Grover’s algorithm (algorithm 1) using three tools: the matrix multiplier, the Hamming weight counter, and the state indicator. We present algorithm 2, which is a quantum decoding algorithm for binary linear codes. We illustrate algorithm 1 using the [7, 4, 3] Hamming code, and algorithm 2 is implemented by IBM Quantum Composer.
A quantum computer capable of running the Grover search algorithm, which reduces the complexity of brute-force attacks by a square root, has the potential to undermine the security strength of symmetric-key cryptography and hash functions. Recently, studies on quantum approaches have proposed analyzing potential quantum attacks using the Grover search algorithm in conjunction with optimized quantum circuit implementations for symmetric-key cryptography and hash functions. Analyzing quantum attacks on a cipher (i.e., quantum cryptanalysis) and estimating the necessary quantum resources are related to evaluating post-quantum security for the target cryptography algorithms. In this paper, we revisit quantum implementations of CHAM block cipher, an ultra lightweight cipher, with a focus on optimizing the linear operations in its key schedule. We optimized the linear equations of CHAM as matrices by applying novel optimized decomposition techniques. Using the improved CHAM quantum circuits, we estimate the cost of Grover’s key search and evaluate the post-quantum security strength with further reduced costs.
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Abstract Grover’s algorithm reduces the security level of the target cipher from n-bit to n/2-bit. To implement Grover’s algorithm, a quantum circuit capable of performing the target cipher operations must be constructed. In this paper, we propose a quantum circuit implementation scheme for the involutive lightweight block cipher (IVLBC) block cipher. We utilized the DORCIS tool to implement the quantum circuit for the S-box and applied Gaussian elimination to construct the quantum circuit for matrix multiplication. Subsequently, we evaluated the overall quantum resource consumption required for the implementation. Based on this, we performed a quantum attack on IVLBC using Grover’s algorithm, calculating the attack cost using the NCT gate library, and further analyzed its security. Finally, we conducted a comparative analysis of the classical attack complexity and quantum attack complexity of IVLBC, highlighting the potential of using quantum complexity as a standard for future security assessments.