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A Polynomial-Time Algorithm for Solving a Class of Underdetermined Multivariate Quadratic Equations over Fields of Odd Characteristics

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Post-Quantum Cryptography (PQCrypto 2014)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8772))

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Abstract

Following up a series of works by Kipnis-Patarin-Goubin, Courtois-Goubin-Meier-Tacier, and Thomae-Wolf, in PQCrypto 2013 Miura, Hashimoto, and Takagi proposed an efficient algorithm for solving a class of underdetermined multivariate quadratic equations. Their algorithm does not use any generic Gröbner-basis solving techniques and asymptotically requires the least degree of underdeterminedness among all similar algorithms in the current literature. Building on top of their work, in this paper we focus on solving polynomially underdetermined multivariate quadratic equations over fields of odd characteristics. We show that we can further improve the applicable range of the Miura-Hashimoto-Takagi algorithm essentially for free. Furthermore, we show how to allow a certain degree of trade-off between applicable range and running time. Last but not least, we show that the running time of the improved algorithm is actually polynomial in number of equations and variables. To the best of our knowledge, this is the first result showing that this class of polynomially underdetermined multivariate quadratic equations over fields of odd characteristics can be solved in polynomial time.

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Author information

Authors and Affiliations

  1. Kyushu University, Fukuoka-shi, 819-0395, Japan

    Chen-Mou Cheng, Hiroyuki Miura & Tsuyoshi Takagi

  2. University of the Ryukyus, Okinawa-ken, 903-0213, Japan

    Yasufumi Hashimoto

Authors
  1. Chen-Mou Cheng
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  2. Yasufumi Hashimoto
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  3. Hiroyuki Miura
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  4. Tsuyoshi Takagi
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Editor information

Editors and Affiliations

  1. Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, N2L 3G1, Waterloo, ON, Canada

    Michele Mosca

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Cheng, CM., Hashimoto, Y., Miura, H., Takagi, T. (2014). A Polynomial-Time Algorithm for Solving a Class of Underdetermined Multivariate Quadratic Equations over Fields of Odd Characteristics. In: Mosca, M. (eds) Post-Quantum Cryptography. PQCrypto 2014. Lecture Notes in Computer Science, vol 8772. Springer, Cham. https://doi.org/10.1007/978-3-319-11659-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-11659-4_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11658-7

  • Online ISBN: 978-3-319-11659-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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