橢圓曲線密碼系統製作與效能分析

Abstract

有限體上的橢圓曲線理論背景是建立於求解離散對數的難題上，近年來，關於此方面的理論已經發展得非常快速，人們已經將此種技術引進公鑰密碼系統（如RSA、ElGamal等）中，希望藉著橢圓曲線技術在相同的安全度下縮短系統所需的密鑰長度，但其付出的代價則是較複雜的運算，其中絕大部分都在做有限體上的算術運算。因此，如何加快有限體上的算術運算速度就成為一個重要的研究課題。目前提出來最有效率的方法有兩種，分別是基於正規基底下的運算模式以及架構於最佳擴展體下的運算模式。 在本論文中，我們將這兩種方法分別實作在不同的機器平台下，分析比較其執行效能，並整理歸納出不同的環境平台下選擇運算模式的標準以及最佳的參數，希望能對實作橢圓曲線密碼系統有相當的貢獻。

In recent years, several studies have been conducted on software implementation of elliptic curve cryptosystem (ECC). The security of ECC is based on the intractability of the discrete logarithm problem (DLP) on elliptic curve (EC). It requires much shorter key lengths as compared to the existing scheme while still retaining the same security level. However, the complicated computation over the finite field required in ECC is quite a challenging task. Consequently, it is critical for us to figure out how to accelerate the computation speed. Recently, two efficient schemes, normal basis and optimal extended field, have been proposed for ECCs to solve the aforementioned problem. In this thesis, we implemented these two schemes on different platforms, firstly. Then we compared the arithmetic performance over finite field. According to the experimental results, we not only draw conclusion about whether both schemes are suitable for each platform, but also gain the knowledge about how to tune the parameters to optimize each configuration.