Complexity bounds on Semaev’s naive index calculus method for ECDLP

• Main Authors: Yokoyama Kazuhiro, Yasuda Masaya, Takahashi Yasushi, Kogure Jun
• Format: Article
• Language: English
• Published: De Gruyter 2020-10-01
• Series: Journal of Mathematical Cryptology
• Subjects: ecdlp; summation polynomials; index calculus methods; gröbner basis computation; fall degrees; primary 94a60; secondary 14g50
• Online Access: https://doi.org/10.1515/jmc-2019-0029
• ISSN: 1862-2976; 1862-2984

Since Semaev introduced summation polynomials in 2004, a number of studies have been devoted to improving the index calculus method for solving the elliptic curve discrete logarithm problem (ECDLP) with better complexity than generic methods such as Pollard’s rho method and the baby-step and giant-step method (BSGS). In this paper, we provide a deep analysis of Gröbner basis computation for solving polynomial systems appearing in the point decomposition problem (PDP) in Semaev’s naive index calculus method. Our analysis relies on linear algebra under simple statistical assumptions on summation polynomials. We show that the ideal derived from PDP has a special structure and Gröbner basis computation for the ideal is regarded as an extension of the extended Euclidean algorithm. This enables us to obtain a lower bound on the cost of Gröbner basis computation. With the lower bound, we prove that the naive index calculus method cannot be more efficient than generic methods.