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Improved Quantum Information Set Decoding

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Post-Quantum Cryptography (PQCrypto 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10786))

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Abstract

In this paper we present quantum information set decoding (ISD) algorithms for binary linear codes. First, we refine the analysis of the quantum walk based algorithms proposed by Kachigar and Tillich (PQCrypto’17). This refinement allows us to improve the running time of quantum decoding in the leading order term: for an n-dimensional binary linear code the complexity of May-Meurer-Thomae ISD algorithm (Asiacrypt’11) drops down from \(2^{0.05904n + o(n)}\) to \(2^{0.05806n+o(n)}\). Similar improvement is achieved for our quantum version of Becker-Jeux-May-Meurer (Eurocrypt’12) decoding algorithm. Second, we translate May-Ozerov Near Neighbour technique (Eurocrypt’15) to an ‘update-and-query’ language more common in a similarity search literature. This re-interpretation allows us to combine Near Neighbour search with the quantum walk framework and use both techniques to improve a quantum version of Dumer’s ISD algorithm: the running time goes down from \(2^{0.059962n+o(n)}\) to \(2^{0.059450+o(n)}\).

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Notes

  1. 1.

    We omit sub-exponential in n factors throughout, because we are only interested in the constant \(\mathsf {c}\). Furthermore, our analysis is for an average case and we sometimes omit the word ‘expected’.

  2. 2.

    The (dimensionless) distances we consider here, denoted further \(\gamma , \alpha , \beta \), are all \(\le 1/2\), since we can flip the bits of the query point and search for ‘close’ rather than ‘far apart’ vectors.

  3. 3.

    By ‘error’ we mean missing a vector which is \(\gamma \)-close to \({\mathbf {q}}\).

  4. 4.

    One could also run Grover inside each bucket during the Query phase, when the buckets are larger than 1. This, however, does not seem to bring an improvement.

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Acknowledgements

The author thanks Alexander May for enlightening discussions and suggestions. This work is supported by ERC Starting Grant ERC-2013-StG-335086-LATTAC.

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  1. Laboratoire LIP, ENS de Lyon, Lyon, France

    Elena Kirshanova

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  1. Elena Kirshanova
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Correspondence to Elena Kirshanova .

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Editors and Affiliations

  1. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Tanja Lange

  2. Florida Atlantic University, Boca Raton, Florida, USA

    Rainer Steinwandt

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Kirshanova, E. (2018). Improved Quantum Information Set Decoding. In: Lange, T., Steinwandt, R. (eds) Post-Quantum Cryptography. PQCrypto 2018. Lecture Notes in Computer Science(), vol 10786. Springer, Cham. https://doi.org/10.1007/978-3-319-79063-3_24

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  • DOI: https://doi.org/10.1007/978-3-319-79063-3_24

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