On computational hard problems and cryptographic applications based on tropical elliptic curve theory

Reference No.	20200030
Type/Category	Grant for Young Researchers- Short-term Visiting Researcher
Title of Research Project	On computational hard problems and cryptographic applications based on tropical elliptic curve theory
Principal Investigator	Hyungrok Jo（University of Tsukuba ・ Faculty of Engineering, Information and Systems・Researcher）
Research Period	March 23,2021. ～ March 26,2021.
Keyword(s) of Research Fields	Cryptography, Elliptic Curve Cryptography (ECC), Tropical geometry
Abstract for Research Report	This research mainly studies on computational hard problems of tropical elliptic curves, which have not been used for cryptographic applications in order to propose a new generation of cryptographic schemes. Since there is a view of “tropicalized” elliptic curves over a valuation ring in tropical geometry, it is a deep relationship with algebraic geometry. Especially, there are some variational approaches corresponding to the classical theory of elliptic curves over a projective plane, in recent, Riemann-Roch theorem is established in a way of tropical elliptic curves. While, in an arithmetic of elliptic curves, Elliptic Curve Cryptography (ECC) is considered from a discrete logarithm problem based on the group structure of elliptic curves, which put to practical use especially for supporting a security in the era of IoT (Internet of Things). In fact, even though there are some cryptographic approaches of tropical algebra or tropical matrix algebra (which is considered as semirings) which is essentially a Diffie-Hellman-like key establishment mechanism based on the elementary tropicalized algebraic structures so far, it is not known that the study on a discrete logarithm problem over tropical elliptic curves. From these reasons, this research on the tropicalization of Elliptic Curve Discrete Logarithm Problem (ECDLP) is challenging and also has a potentiality which can be considered as a seed to propose a new hard computational problem for cryptographic applications. In addition, we investigate on the possibilities of usages for cryptographic applications, which come from the tropicalization of the existing hard computational problems over elliptic curves such as an isogeny path problem. We study on the possibilities of tropical elliptic curves which have not been suggested as cryptographic applications, so we expect to broaden a way of cryptography.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Hyungrok Jo（University of Tsukuba ・ Faculty of Engineering, Information and Systems・Researcher）
Adviser	Takuro Abe (IMI, Kyushu University / Professor)