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Research and Improvement of the key algorithms elliptic curve cryptosystems

Author: GaoFuQiang
Tutor: JiZuoQin
School: Changchun University of
Course: Applied Computer Technology
Keywords: Elliptic Curve Schoof algorithm SEA algorithm Ekies prime BSGS
CLC: TN918.1
Type: Master's thesis
Year: 2010
Downloads: 72
Quote: 0
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Abstract


In 1985,Miller proposed the use of elliptic curves in public-key cryptography, and so did koblitz in 1987.The rational points of elliptic curve forms an additive group, the discrete logarithm of this group is called elliptic curve discrete logarithm problem(ECDLP). As yet, there is no efficient method to solve ECDLP. The security of elliptic curve cryptosystem is based on this mathematic problem. The elliptic curve cryptosystem provides the highest strength of any cryptography ever known. Because it has shorter key size, less memory and less bandwidth. So it has wide application prospect in information security fields.The article first carries on the development status of elliptic curve cryptography, then analyzes the elliptic curve cryptosystem related theory, include elliptic curve basic concept, elliptic curve cryptosystem mathematic theory and so on. Based on above theory, make a in-depth analysis of the Schoof algorithm,Elkies and Atkin’s improvements on Schoof algorithm.There is also an introduction to improvements on SEA algorithm, such as: Isogeny cycle, Atkin Index, Chinese&match etc.In using the elliptic curve cryptography, it is very important to choose a secure elliptic curve. As yet, we have two methods to choose the secure elliptic curve in practical use. The first one is Complex multiplication; the second one is counting the number of rational points of elliptic curves generated randomly. At present, the most import way is choosing a elliptic curve generated randomly first, then counting the number of rational points of this elliptic curve,until find the elliptic curve which is suitable for the security consideration. Hence, solving the point counting problem plays an important role in the design of elliptic curve cryptography. Schoof-Elkies-Atkin (SEA) is the a mature method to solve the points counting problem. In this article, we proposed two improvements on the implementation of SEA algorithm. one is based on Elkies prime, and another is based on baby-step-giant-step strategy. It improves the original SEA implementation for elliptic curves defined over big prime field.

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