橢圓曲線是個古老的數學問題，在 1985 年首次分別被 Victor Miller 和 Neal Koblitz 兩位學者應用於公開金鑰密碼系統上，由於有限體上橢圓曲線所成的群的特性，使得解橢圓曲線上的離散對數問題較困難，因此橢圓曲線密碼系統可以使用比 RSA 較短的金鑰長度，也因為這優點使得橢圓曲線密碼系統廣泛被使用。

使用橢圓曲線於密碼系統上的首要工作就是建構一個橢圓曲線，而其中最關鍵的步驟在於計算有限體上橢圓曲線的點個數，也就是計算橢圓曲線的 group order，在 1985 年 Schoof 提出了一個 deterministic polynomial time 演算法，這方法成為計算橢圓曲線上的點個數的核心計算方法。本論文依照 IEEE P1363 的方法與 schoof 的演算法實作了『產生假隨機橢圓曲線』。

Elliptic curve cryptosystems were proposed in 1985 by Victor Miller and by Neal Koblitz independently. Since elliptic curve discrete logarithm problem is harder to solve than discrete logarithm problem in finite fields. If is believed that the key length of elliptic curve cryptosystems can be shorter then that of RSA with the same security strength.

The most important work of using elliptic curve cryptosystem is constructing a group from a proper elliptic curve. The major work of constructing an elliptic curve is counting points on elliptic curves over finite fields. In 1985, Schoof published a deterministic polynomial time algorithm for computing the number of points on the elliptic curves over finite fields. We consult IEEE P1363 to implement pseudo random elliptic curve.

1 簡介
2 橢圓曲線密碼學
2.1 橢圓曲線基本性質
2.2 有限體上的橢圓曲線
2.3 橢圓曲線數位簽章演算法 ECDSA
2.4 如何產生適合密碼系統的橢圓曲線
3 計算橢圓曲線在有限體上的點個數
3.1 Baby Step, Giant Step
3.2 Schoof’s Algorithm
3.3 Schoof-Elkins-Atkin Algorithm
4 實驗步驟與結果
4.1 產生一個假隨機橢圓曲線
4.2 檢驗 nearly prime
4.3 產生一個指定 order 的點
4.4 實驗結果
5 結論

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