### 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.

Original language | English |
---|---|

Pages (from-to) | 40-58 |

Number of pages | 19 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 8772 |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*8772*, 40-58.

**A polynomial-time algorithm for solving a class of underdetermined multivariate quadratic equations over fields of odd characteristics.** / Cheng, Chen Mou; Hashimoto, Yasufumi; Miura, Hiroyuki; Takagi, Tsuyoshi.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 8772, pp. 40-58.

}

TY - JOUR

T1 - A polynomial-time algorithm for solving a class of underdetermined multivariate quadratic equations over fields of odd characteristics

AU - Cheng, Chen Mou

AU - Hashimoto, Yasufumi

AU - Miura, Hiroyuki

AU - Takagi, Tsuyoshi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84921645550&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921645550&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84921645550

VL - 8772

SP - 40

EP - 58

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -