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Memory-Efficient Quantum Information Set Decoding Algorithm

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Information Security and Privacy (ACISP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13915))

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Abstract

Code-based cryptography is a candidate for post-quantum cryptography and the security of code-based cryptosystems relate to the hardness of the syndrome decoding problem. The Information Set Decoding (ISD) algorithm initiated by Prange is a typical method for solving the syndrome decoding problem. Various methods have been proposed that make use of exponentially large lists to accelerate the ISD algorithm. Furthermore, Bernstein (PQCrypto 2010) and Kachigar and Tillich (PQCrypto 2017) applied Grover’s algorithm and quantum walks to obtain quantum ISD algorithms that are much faster than their classical ones. These quantum ISD algorithms also require exponentially large lists as the classical algorithms, and they must be kept in quantum states. In this paper, we propose a new quantum ISD algorithm by combining Both and May’s classical ISD algorithm (PQcrypto 2018), Grover’s algorithm, and Kirshanova’s quantum walk (PQCrypto 2018). The proposed algorithm keeps an exponentially large list in the quantum state just like the existing quantum ISD algorithms, but the list size is much smaller. Although the proposed algorithm is slower than the existing algorithms when there is sufficient quantum memory, it is fastest when the amount of quantum memory is limited. Due to the property, we believe that our algorithm will be the fastest ISD algorithm in actual quantum computing since large-scale quantum computers seem hard to realize.

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Notes

  1. 1.

    See also [9, 11] as the follow-up works of the algorithms.

  2. 2.

    We note that the syndrome decoding problem which we study in this paper is full distance decoding.

  3. 3.

    The depth \(d = 2\) is not an optimized value. If we use the larger d, we may be able to obtain the faster quantum ISD algorithms. However, we cannot analyze the time complexity of larger \(d \ge 3\) since the numerical analysis of the computational complexity requires huge time due to more parameters that should be optimized.

  4. 4.

    To be precise, the original \(\texttt{Prange}\) fixes the parameter to \(p = 0\).

  5. 5.

    To be precise, the condition of \(\textbf{P}\) is not the same as that of \(\texttt{Prange}\).

  6. 6.

    To be precise, \(\texttt {MO}\)’s proposed \(\gamma \)-nearest neighbor search algorithm [20] does not work for arbitrary parameters. In such cases, we use other algorithms for solving the problem. See [4] for the detail.

  7. 7.

    Kirshanova proposed other quantum variants of \(\texttt {Stern}\) (Sect. 4 of [16]) that run in time \(\widetilde{\mathcal {O}}(2^{0.059922n})\) (resp. \(\widetilde{\mathcal {O}}(2^{0.059922n})\)) with space \(\widetilde{\mathcal {O}}(2^{0.00897n})\) (resp. \(\widetilde{\mathcal {O}}(2^{0.00808n})\)). Although we tried to find the optimized parameters to obtain the computational complexity via brute force search, we could not find them. Thus, we do not compare our result with these algorithms.

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Acknowledgement

This work was partially supported by JSPS KAKENHI Grant Number 19K20267 and 21H03440, Japan, and JST CREST Grant Number JPMJCR2113, Japan.

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    Naoto Kimura, Atsushi Takayasu & Tsuyoshi Takagi

  2. National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

    Atsushi Takayasu

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  1. Naoto Kimura
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  2. Atsushi Takayasu
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Correspondence to Atsushi Takayasu .

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  1. Queensland University of Technology, Brisbane, QLD, Australia

    Leonie Simpson

  2. Queensland University of Technology, Brisbane, QLD, Australia

    Mir Ali Rezazadeh Baee

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Kimura, N., Takayasu, A., Takagi, T. (2023). Memory-Efficient Quantum Information Set Decoding Algorithm. In: Simpson, L., Rezazadeh Baee, M.A. (eds) Information Security and Privacy. ACISP 2023. Lecture Notes in Computer Science, vol 13915. Springer, Cham. https://doi.org/10.1007/978-3-031-35486-1_20

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  • DOI: https://doi.org/10.1007/978-3-031-35486-1_20

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