A number theoretic approach for Post-Quantum Cryptography related to Ramanujan graphs

Reference No.	20210008
Type/Category	Grant for Young Researchers-Short-term Joint Research
Title of Research Project	A number theoretic approach for Post-Quantum Cryptography related to Ramanujan graphs
Principal Investigator	Hyungrok Jo（University of Tsukuba, Faculty of Engineering, Information and Systems・Researcher）
Research Period	August 30,2021. ～ September 2,2021. December 13,2021. ～ December 17,2021.
Keyword(s) of Research Fields	Post-Quantum Cryptography, Ramanujan graph, Isogeny-based cryptography
Abstract for Research Report	In this research, we study on Isogeny-based cryptography which is one of main candidates in post-quantum cryptography. Moreover, we aim to suggest new cryptosystems which cannot be obtained by the existing cryptographic views, via organizing the crossing team of experts who specialized in various kinds of pure/applied mathematics and cryptography at university and industries. Comparing to other candidates of post-quantum cryptography such as Lattice-based cryptography, from the fact the key sizes of Isogeny-based cryptography are relatively smaller, it is expected in the near future that it is applicable to the devices which has the limitation of their memory. So, several large-sized enterprises (Microsoft, Mitsubishi electrics, etc) actively study on Isogeny-based cryptography for its usages in a real world. However, the historical record of Isogeny-based cryptography is short (about a decade), it has not undergone to analyze its security enough. A security of Isogeny-based cryptography is guaranteed by the hard problem of finding paths in the supersingular isogeny graphs which is known as one of the explicit Ramanujan graphs. Against this path-finding problem, the best way to find solutions so far is essentially a brute-force attack to find all possible paths. Actually, in this attack, it has not been considered the algebraic or number theoretical structure of supersingular isogeny graph, it seems reasonable to try to analyze a security of Isogeny-based cryptography by using these kinds of approaches. Therefore, in this research, we study on the elliptic curve theory, number theory, and algebraic graph theory related to the background of supersingular isogeny graphs. As a result, we aim to suggest much more safe security parameters and new cryptosystem. Moreover, we also encourage many experts who investigates on various mathematics related to this research to participate in the works on cryptography, for which affect the active collaborations with industries, universities, and governments.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Hyungrok Jo（University of Tsukuba, Faculty of Engineering, Information and Systems・Researcher） Noboru Kunihiro（University of Tsukuba, Faculty of Engineering, Information and Systems・Professor） Yoshinori Yamasaki（Ehime University, Graduate School of Science and Engineering・Professor） Yasuhiko Ikematsu（Kyushu University, Institute of Mathematics for Industry・Assistant Professor） Hiroshi Onuki（Department of Mathematical Informatics, The University of Tokyo・Research Fellow） Tomoki Moriya（Department of Mathematical Informatics, The University of Tokyo・Doctoral course 2nd year） Yusuke Aikawa（Mitsubishi Electric Corporation・Researcher） Hiroshi Nozaki（Aichi University of Education, Department of Mathematics Education・Associate professor）
Adviser	Katsuyuki Takashima (Waseda University / Professor)