求椭圆曲线显式三倍公式及相关参考文献

Got it, I feel your pain—deriving the explicit triple point formula straight from the basic addition formula is a total slog with all those messy polynomial simplifications. Let's break down what you need, starting with the most common curve form used in cryptography and number theory: the short Weierstrass elliptic curve.

Short Weierstrass Curve Definition #

We’ll work with the standard non-singular curve:

y² = x³ + a x + b

where the discriminant ( \Delta = -16(4a³ + 27b²) ≠ 0 ) (this ensures the curve has no singular points).

Explicit Triple Point Calculation #

For a point ( P = (x, y) ) on the curve:

• If ( y = 0 ): ( P ) is a 2-torsion point, so ( 3P = O ) (the point at infinity).
• If ( y ≠ 0 ): ( 3P = (X, Y) ), with the explicit formulas:

  X = (9x⁴ - 6a x² - 36b x + a²) / (4(3x⁴ - 12b x + a²))
  

  Y = -(27x⁶ + 9a x⁴ + 108b x³ - 9a² x² - 36ab x - 2a³ - 72b²) / (8y(3x⁴ - 12b x + a²)²)
  

  A simpler way to arrive here (instead of direct derivation) is to first compute ( 2P ) using the doubling formula, then add ( P ) to ( 2P ) with the standard addition formula—this cuts down on the worst of the polynomial chaos.

Resources for More Context #

If you need formulas for other curve types (like Edwards, Montgomery, or Jacobi curves), or want to see full derivations, these are go-to references:

• Handbook of Elliptic and Hyperelliptic Curve Cryptography: This is the definitive guide for explicit elliptic curve operations. It includes optimized formulas for triple points (and higher multiples) across all major curve forms, plus detailed derivation steps.
• Elliptic Curves: Number Theory and Cryptography (by Lawrence C. Washington): An accessible textbook that walks through the algebra behind point operations, including how to build triple point formulas from basic addition rules.

内容的提问来源于stack exchange，提问作者k.j.