There are several kinds of defining equation for elliptic curves, but this section deals with

Weierstrass equations

. For the prime finite field

GF(p), p>3

, the Weierstrass equation is

E : y

2

= x

3

+ a*x + b

, where

a

and

b

are integers modulo

p

. Number of points on the elliptic curve

E

is denoted by

#E

.

For purpose of cryptography some additional parameters are presented:

The set of

p, a, b, n

and

G

parameters are Elliptic Curve (EC) domain parameter. This section deals with three NIST recommended Elliptic Curves those domain parameters are known and published in [SEC2] (Standards for Efficient Cryptography Group, "Recommended Elliptic Curve Domain Parameters", SEC 2, September 2000).

Private key is a positive integer

u

in the range

[1, n-1]

. Public key

V

, which is the point on elliptic curve

E

, where

V = [u]*G

. In cryptography, there are two types of key pairs: regular (or longterm) and ephemeral (or nonce - number that can only be used once). From the math point of view, they are similar.

Input:

Output: The signature, which is a pair of integers

(r, s)

, where

r

and

s

belongs the range

[1. r-1]

.

Operation:

Input:

Output: The derived shared secret value

z

, which is the

GF(p)

field element

Operation: