2. Overview of Elliptic Curves

一些概念：

• E, an elliptic curve defined over the field GF(q).
• |E| = (q +1) - t
• t, the trace of E over the field GF(q)
• E’, a quadratic twist（二次扭曲） of E

If R is a point on the curve that is also contained in 〈P〉, there is a unique integer k in the interval [0, l−1] so that R = kP, where l is the order of P in E. This number is called the discrete logarithm of R to the base P. The discrete logarithm problem is the problem of finding the discrete logarithm of R to the base P for any two points P and R on the curve, if such a number exists.

2.1 Non-binary Curves

2.1.1 Curves in Short-Weierstrass Form

维尔斯特拉斯形式曲线

Let GF(q) denote the finite field with q elements, where q is an odd prime power and is not
divisible by three.

Wa,b 代表这类曲线。
$$\begin{gathered} 4a^3+27b^2\ne 0\mathrm{mod}P \\ {y}^2=x^3+ax+b\mathrm{mod}P \\ a,b\in GF(P) \end{gathered}$$
A Seed value may be used to generate the parameters a and b as described in Appendix C.2.1.1

2.1.2 Montgomery Curves

蒙哥马利曲线

Ma,b 代表这类曲线
$$\begin{gathered} M_A{,}_B:B{v}^2=u(u^2+Au+1) \\ A\ne \pm 2 \\ B\ne 0 \end{gathered}$$

2.1.3 Twisted Edwards Curves

扭曲爱德华兹曲线

Ea,d 代表这类曲线
$$\begin{gathered} a{x}^2+y^2=1+d{x}^2{y}^2 \\ d\ne 0 \\ a\ne d \end{gathered}$$
and a is a square in GF(q) while d is not.

An Edwards curve is a twisted Edwards curve with a = 1

2.2 Binary Curves

2.2.1. Curves in Short-Weierstrass Form

Ba,b代表这类曲线

GF(q)代表有q个元素的有限域，q=2^m
$$y^2+xy=x^3+a{x}^2+b$$

3. Recommended Curves for U.S. Federal Government Use

3.1 Choice of Underlying Fields

3.1.2. Choice of Underlying Fields

Specified Curves	Allowed Usage
K-233, B-233 K-283, B-283 K-409, B-409 K-571, B-571	Deprecated
P-224 P-256 P-384 P-521	ECDSA, EC key establishment (see [SP_800-56A])
Edwards25519 Edwards448	EdDSA
Curve25519, W-25519 Curve448, E448, W-448	Alternative representations included for implementation flexibility. Not to be used for ECDSA or EdDSA directly

Security Strength

Security Strength	Recommended Curves
112	P-224, K-233, B-233
128	P-256, W-25519, Curve25519, Edwards25519, K-283, B-283
192	P-384, K-409, B-409
224	W-448, Curve448, Edwards448, E448
256	P-521, K-571, B-571

3.1.4 Choice of Curves

Two kinds of curves are given:

• Pseudorandom curves, coefficients = hash(seed). Appendix C.3
• Special curves, whose coefficients and underlying fields have been selected to
  optimize the efficiency of the elliptic curve operations.

3.2 Curves Over Prime Fields

这一部分介绍了各种素域上的曲线。

3.3 Curves Over Binary Fields

Deprecated