Speeding up SEA Algorithm for Elliptic Curves

• Main Authors: Yung-Hsiang Liu, 劉用翔
• Other Authors: Rong-Jaye Chen
• Format: Others
• Language: en_US
• Published: 2008
• Online Access: http://ndltd.ncl.edu.tw/handle/33206618553618099196

碩士 === 國立交通大學 === 資訊科學與工程研究所 === 96 === In 1985, Miller proposed the use of elliptic curves in public-key cryptosystem, and so did Koblitz in 1987. The rational points of an elliptic curve forms an additive group. The discrete logarithm problem of this group is called elliptic curve discrete logarithm problem(ECDLP). There is no method to solve ECDLP efficiently. The security of elliptic curve cryptosystem(ECC) is based on ECDLP. Therefore, The key of ECC can be shorter than that of RSA in order to reach the same secure strength. In using the elliptic curve cryptosystem, it is important to select a secure elliptic curve. There are three methods to select secure elliptic curves. The suggested method is counting the number of rational points of elliptic curves generated randomly. Therefore, we can determine whether a randomly generated elliptic curve is suitable for the security consideration. Hence, solving the point counting problem plays a crucial role in the design of elliptic curve cryptosystems. Schoof-Elkies-Atkin(SEA) algorithm is an important method to solve the point counting problem. In this thesis, we propose strategies of Atkin primes, Elkies primes, and Baby-step-giant-step. It improves the original SEA algorithm a lot for elliptic curves defined over big prime fields.