Security Analysis of Elliptic Curve Cryptography using Groebner Basis

• Project/Area Number: 25540047
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Information security
• Research Institution: Kyushu University
• Principal Investigator: Takagi Tsuyoshi 九州大学, マス・フォア・インダストリ研究所, 教授 (60404802)
• Co-Investigator(Kenkyū-buntansha): HAKUTA KEISUKE 島根大学, 総合理工学研究科, 助教 (90587099)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 暗号・認証等 / 公開鍵暗号 / 楕円曲線暗号 / 離散対数問題 / グレブナ基底

Outline of Final Research Achievements

In this research, we have investigated some algorithms using Groebner basis for solving the discrete logarithm problem over elliptic curve of characteristic 2. From the symmetric structure of Semaev polynomial we proposed an efficient algorithm that reduces the complexity and memory during the computation of Groebner basis. The proposed algorithm enables us to solve the discrete logarithm problem over elliptic curve of finite field of extension degree 29 in about 34 days using computer algebra software Magma on AMD Opteron 6276 with 512GB memory. From this cryptanalysis data we are able to estimate the computational over-limit of the expected attackers more precisely.

Research Products

• [Journal Article] On tameness of Matsumoto-Imai central maps in three variables over the finite field F2 (2016)
  • Author(s): Keisuke Hakuta, Hisayoshi Sato, Tsuyoshi Takagi
  • Journal Title: Advances in Mathematics of Communications Volume: Vol.10 Issue: 2 Pages: 221-228
  • DOI: 10.3934/amc.2016002
  • Peer Reviewed / Open Access
• [Journal Article] ポスト量子暗号 (2015)
  • Author(s): 高木 剛
  • Journal Title: 数学セミナー Volume: 7月号 Pages: 22-26
• [Journal Article] Improvement of FPPR method to solve ECDLP (2015)
  • Author(s): Yun-Ju Huang, Christophe Petit, Naoyuki Shinohara, Tsuyoshi Takagi
  • Journal Title: Pacific Journal of Mathematics for Industry Volume: 7-1 Issue: 1 Pages: 1-9
  • DOI: 10.1186/s40736-015-0012-6
  • Peer Reviewed / Open Access
• [Journal Article] A polynomial-time algorithm for solving a class of underdetermined multivariate quadratic equations (2014)
  • Author(s): C.-M. Cheng, Y. Hashimoto, H. Miura, and T. Takagi
  • Journal Title: 6th International Workshop on Post-Quantum Cryptography, PQCrypto 2014 Volume: LNCS 8772 Pages: 40-58
  • DOI: 10.1007/978-3-319-11659-4_3
  • ISBN: 9783319116587, 9783319116594
  • Peer Reviewed / Open Access
• [Journal Article] Web Workersを用いた多変数公開鍵暗号Rainbowの並列実装 (2014)
  • Author(s): 鷲見拓哉, 石黒司, 清本晋作, 三宅優, 小林透, 高木剛
  • Journal Title: 情報処理学会論文誌 Volume: Vol.55, No.9 Pages: 2061-2071
  • NAID: 110009822846
  • Peer Reviewed / Open Access
• [Journal Article] Improved Attacks on Multi-Prime RSA with Small Prime Difference (2014)
  • Author(s): Hui Zhang, Tsuyoshi Takagi
  • Journal Title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences Volume: E97.A Issue: 7 Pages: 1533-1541
  • DOI: 10.1587/transfun.E97.A.1533
  • NAID: 130004519181
  • ISSN: 0916-8508, 1745-1337
  • Peer Reviewed / Open Access
• [Journal Article] Extended Algorithm for Solving Underdefined Multivariate Quadratic Equations (2014)
  • Author(s): Hiroyuki Miura, Yasufumi Hashimoto, Tsuyoshi Takagi,
  • Journal Title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences Volume: E97.A Issue: 6 Pages: 1418-1425
  • DOI: 10.1587/transfun.E97.A.1418
  • NAID: 130004770873
  • ISSN: 0916-8508, 1745-1337
  • Peer Reviewed / Open Access
• [Journal Article] Improvement of Faugere et al.'s method to solve ECDLP (2014)
  • Author(s): Yun-Ju Huang, Christophe Petit, Naoyuki Shinohara, Tsuyoshi Takagi
  • Journal Title: 2014年暗号と情報セキュリティシンポジウム, SCIS2014 Volume: 3F5-4 Pages: 118-118
• [Journal Article] 奇標数の有限体上の多変数連立2 次方程式の求解アルゴリズムの改良について (2014)
  • Author(s): 三浦浩幸, 橋本康史, 高木剛
  • Journal Title: 2014年暗号と情報セキュリティシンポジウム, SCIS2014 Volume: 1C2-2 Pages: 12-12
• [Journal Article] 拡大体上の楕円曲線暗号へのGHS攻撃に対する安全性 (2014)
  • Author(s): 安田貴徳, 齋藤恆和, 小林鉄太郎, 高木剛
  • Journal Title: 2014年暗号と情報セキュリティシンポジウム, SCIS2014 Volume: 4F2-3 Pages: 138-138
• [Journal Article] Improvement of Faugere et al.'s Method to Solve ECDLP (2013)
  • Author(s): Yun-Ju Huang, Christophe Petit, Naoyuki Shinohara, Tsuyoshi Takagi
  • Journal Title: The 8th International Workshop on Security, IWSEC 2013 Volume: LNCS 8231 Pages: 115-132
  • DOI: 10.1007/978-3-642-41383-4_8
  • ISBN: 9783642413827, 9783642413834
  • Peer Reviewed
• [Journal Article] Extended Algorithm for Solving Underdefinedned Multivariate Quadratic Equations (2013)
  • Author(s): Hiroyuki Miura, Yasufumi Hashimoto, Tsuyoshi Takagi
  • Journal Title: Fifth International Conference on Post-Quantum Cryptography, PQCrypto 2013 Volume: LNCS 7932 Pages: 118-135
  • DOI: 10.1007/978-3-642-38616-9_8
  • ISBN: 9783642386152, 9783642386169
  • Peer Reviewed
• [Journal Article] Attacks on Multi-Prime RSA with Small Prime Difference (2013)
  • Author(s): Hui Zhang, Tsuyoshi Takagi
  • Journal Title: 18th Australasian Conference on Information Security and Privacy, ACISP 2013 Volume: LNCS 7959 Pages: 41-56
  • DOI: 10.1007/978-3-642-39059-3_4
  • ISBN: 9783642390586, 9783642390593
  • Peer Reviewed
• [Presentation] Recent Developments of Post-Quantum Cryptography (2016)
  • Author(s): Tsuyoshi Takagi
  • Organizer: Future Cryptographic Technology Forum: Cryptographic Technologies in the Era of Quantum Computation
  • Place of Presentation: Seoul National University
  • Year and Date: 2016-01-14
  • Int'l Joint Research / Invited
• [Presentation] Security Analysis of Quantum-Safe Cryptography (2015)
  • Author(s): Tsuyoshi Takagi
  • Organizer: 3rd ESTI Workshop on Quantum-safe Cryptography
  • Place of Presentation: Seoul
  • Year and Date: 2015-10-07
  • Int'l Joint Research / Invited
• [Presentation] MQ Challenge: Hardness Evaluation of Solving Multivariate Quadratic Problems (2015)
  • Author(s): Takanori Yasuda, Xavier Dahan, Yun-Ju Huang, Tsuyoshi Takagi and Kouichi Sakurai
  • Organizer: NIST Workshop on Cybersecurity in a Post-Quantum World
  • Place of Presentation: Washington D.C
  • Year and Date: 2015-04-03
  • Int'l Joint Research
• [Presentation] 多変数多項式暗号の安全性評価 (2015)
  • Author(s): 高木 剛
  • Organizer: 第6回暗号フロンティア研究会
  • Place of Presentation: 北陸先端科学技術大学院大学
  • Year and Date: 2015-03-18
  • Invited
• [Presentation] Matsumoto-Imai中間写像のTame分解に関する考察 (2015)
  • Author(s): 矢城信吾、高木剛
  • Organizer: 2015年暗号と情報セキュリティシンポジウム, SCIS2015
  • Place of Presentation: リーガロイヤルホテル小倉
  • Year and Date: 2015-01-20 – 2015-01-23
• [Presentation] MQ challenge: hardness evaluation of solving multivariate quadratic problems (2015)
  • Author(s): Tsuyoshi Takagi
  • Organizer: DIMACS Workshop on The Mathematics of Post-Quantum Cryptography
  • Place of Presentation: Rutgers University
  • Year and Date: 2015-01-12 – 2015-01-16
  • Invited
• [Remarks] 九州大学マス・フォア・インダストリ研究所 高木研究室
  • URL: http://imi.kyushu-u.ac.jp/~takagi/
• [Remarks] 九州大学 マス・フォア・インダストリ研究所 高木研究室
  • URL: http://imi.kyushu-u.ac.jp/~takagi/