3.2 Arithmetic in an Elliptic Curve Group over Fp

Note that these rules are exactly the same as those for elliptic curve groups over real numbers, with the exception that computations are performed modulo p.

There are several major differences between elliptic curve groups over Fp and over real numbers. Elliptic curve groups over Fp have a finite number of points, which is a desirable property for cryptographic purposes. Since these curves consist of a few discrete points, it is not clear how to "connect the dots" to make their graph look like a curve. It is not clear how geometric relationships can be applied. As a result, the geometry used in elliptic curve groups over real numbers cannot be used for elliptic curve groups over Fp. However, the algebraic rules for the arithmetic can be adapted for elliptic curves over Fp. Unlike elliptic curves over real numbers, computations over the field of Fp involve no round off error - an essential property required for a cryptosystem.

### 3.2.1 Adding distinct points P and Q

The negative of the point P = (xP, yP) is the point -P = (xP, -yP mod p). If P and Q are distinct points such that P is not -Q, then

P + Q = R where

s = (yP - yQ) / (xP - xQ) mod p

xR = s2 - xP - xQ mod p and yR = -yP + s(xP - xR) mod p

Note that s is the slope of the line through P and Q.

### 3.2.2 Doubling the point P

Provided that yP is not 0,

2P = R where

s = (3xP2 + a) / (2yP ) mod p

xR = s2 - 2xP mod p and yR = -yP + s(xP - xR) mod p

Recall that a is one of the parameters chosen with the elliptic curve and that s is the slope of the line through P and Q.

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