Extended Algorithm for Solving Underdefined Multivariate Quadratic Equations

• MIURA Hiroyuki

  Graduate School of Mathematics, Kyushu University

• HASHIMOTO Yasufumi

  Department of Mathematical Sciences, University of the Ryukyus

• TAKAGI Tsuyoshi

  Institute of Mathematics for Industry, Kyushu University

抄録

It is well known that solving randomly chosen Multivariate Quadratic equations over a finite field (MQ-Problem) is NP-hard, and the security of Multivariate Public Key Cryptosystems (MPKCs) is based on the MQ-Problem. However, this problem can be solved efficiently when the number of unknowns n is sufficiently greater than that of equations m (This is called “Underdefined”). Indeed, the algorithm by Kipnis et al. (Eurocrypt'99) can solve the MQ-Problem over a finite field of even characteristic in a polynomial-time of n when n ≥ m(m+1). Therefore, it is important to estimate the hardness of the MQ-Problem to evaluate the security of Multivariate Public Key Cryptosystems. We propose an algorithm in this paper that can solve the MQ-Problem in a polynomial-time of n when n ≥ m(m+3)/2, which has a wider applicable range than that by Kipnis et al. We will also compare our proposed algorithm with other known algorithms. Moreover, we implemented this algorithm with Magma and solved the MQ-Problem of m=28 and n=504, and it takes 78.7 seconds on a common PC.

収録刊行物

• IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

  IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E97.A (6), 1418-1425, 2014

  一般社団法人 電子情報通信学会

キーワード

• multivariate public key cryptosystems (MPKCs)
• multivariate quadratic equations
• MQ-problem

詳細情報

• CRID: 1390282681286985344
• NII論文ID: 130004770873
• DOI: 10.1587/transfun.e97.a.1418
• ISSN: 17451337
• ISSN: 09168508
• Web Site: https://www.jstage.jst.go.jp/article/transfun/E97.A/6/E97.A_1418/_pdf
• 本文言語コード: en
• データソース種別:
  • JaLC
  • Crossref
  • CiNii Articles
  • KAKEN
• 抄録ライセンスフラグ: 使用不可