# 椭圆曲线

$y^2=x^3+ax+b\,$

$y^2= P(x)\,$，其中P為任一沒有重根的三次或四次多項式，然後可得到一虧格1的無奇點平面曲線，其通常亦被稱為橢圓曲線。更一般化地，一虧格1的代數曲線，如兩個三維二次曲面相交，即稱為橢圓曲線。

## 群

$x_R = s^2 - x_P - x_Q\,$
$y_R = -y_P + s(x_P - x_R) \,$

$x_P = x_Q\,$

• $y_P = -y_Q\,$$P+Q = 0\,$
• $y_P = y_Q\,$$R = 2P\,$，其值為：
$s = \frac{3{x_P}^2 - p}{2y_P}\,$
$x_R = s^2 - 2x_P\,$
$y_R = -y_P + s(x_P - x_R)\,$

## 參考文獻

• I. Blake; G. Seroussi, N. Smart, N.J. Hitchin. Elliptic Curves in Cryptography. Cambridge Univ. Press. 2000. ISBN 0-521-65374-6.
• Richard Crandall; Carl Pomerance. Chapter 7: Elliptic Curve Arithmetic. Prime Numbers: A Computational Perspective 1st edition. Springer. 2001: 285–352. ISBN 0-387-94777-9.
• John Cremona. Alogorithms for Modular Elliptic Curves. Cambridge Univ. Press. 1992.
• Dale Husemöller. Elliptic Curves 2nd edition. Springer. 2004.
• Kenneth Ireland; Michael Rosen. Chapters 18 and 19. A Classical Introduction to Modern Number Theory 2nd edition. Springer. 1990.
• Anthony Knapp. Elliptic Curves. Math Notes 40, Princeton Univ. Press. 1992.
• Neal Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer. 1984.
• Neal Koblitz. Chapter 6. A Course in Number Theory and Cryptography 2nd edition. Springer. 1994. ISBN 0-387-94293-9.
• Serge Lang. Elliptic Curves: Diophantine Analysis. Springer. 1978.
• Joseph H. Silverman. The Arithmetic of Elliptic Curves. Springer. 1986.
• Joseph H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Springer. 1994.
• Joseph H. Silverman; John Tate. Rational Points on Elliptic Curves. Springer. 1992.
• Lawrence Washington. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC. 2003. ISBN 1-58488-365-0.