代数的トーラスにおける離散対数問題の困難性に関する考察

摘要

It is expected that we can reduce the sizes of ciphertext and key and construct compact public-key cryptosystems by using algebraic tori. However, it is often said that there remains room for verifying whether the discrete logarithm assumption in the algebraic tori is as sure as those in the finite fields and the elliptic curves. In fact, it was pointed that there are group isomorphisms between tori and weak curves. In this article, we analyze the feasibility of such approach for the discrete logarithm problem in the algabraic tori, and we indicate that, it is infeasible to solve the discrete logarithm problem.%代数的トーラスを用いて暗号文や鍵サイズを圧縮し，コンパクトな暗号が構成できる．しかし，トーラスにおける離散対数問題が従来の有限体や楕円曲線における離散対数問題と同様に安全であるか議論の余地があった．実際，トーラスから弱い楕円曲線への準同型写像によって離散対数問題（CDH問題やDDH問題を含む）が容易に解けるかも知れないという指摘があった．今回，その指摘に関して考察を行い，指摘されている問題を回避できることを示す．