This section describes the improved version of the reduced point additive operation by applying the same modular arithmetic reduction on the parameter m as the reduced elliptic curve equation.

In previous tutorials, we learned how to apply the same modular arithmetic reduction as the reduced elliptic equation on coordinates (xR, yR) of the resulting point R of the point addition operation:

```Reduced addition operation based rule of chord operation:

For any two given points on the curve:
P = (xP, yP)
Q = (xQ, yQ)

R = P + Q is a third point on the curve:
R = (xR, yR)

Where:
xR = m2 - xP - xQ (mod p)     (11)
yR = m(xP - xR) - yP (mod p)  (12)

If P != Q, m is determined by:
yP - yQ
m = ---------                (10)
xP - xQ

If P = Q, m is determined by:
3(xP)2 + a
m = ---------                 (6)
2(yP)
```

Because of the associativity of modular arithmetic, we can actually apply the same modular arithmetic reduction on the parameter m, to bring it to integer.

```Improved reduced addition operation based rule of chord operation:

For any two given points on the curve:
P = (xP, yP)
Q = (xQ, yQ)

R = P + Q is a third point on the curve:
R = (xR, yR)

Where:
xR = m2 - xP - xQ (mod p)     (11)
yR = m(xP - xR) - yP (mod p)  (12)

If P != Q, m is determined by:
yP - yQ
m = --------- (mod p)        (16)
xP - xQ

If P = Q, m is determined by:
3(xP)2 + a
m = --------- (mod p)        (17)
2(yP)
```

If you are not comfortable with modular arithmetic division, we can rewrite those equations as in modular arithmetic multiplication:

```Improved reduced addition operation based rule of chord operation:

For any two given points on the curve:
P = (xP, yP)
Q = (xQ, yQ)

R = P + Q is a third point on the curve:
R = (xR, yR)

Where:
xR = m2 - xP - xQ (mod p)     (11)
yR = m(xP - xR) - yP (mod p)  (12)

If P != Q, m is determined by:
m(xP - xQ) = yP - yQ (mod p)  (18)

If P = Q, m is determined by:
2m(yP) = 3(xP)2 + a (mod p)   (19)
```

You prove that equations (11), (12), (18) and (19) will provide the same (xR, yR) as questions (11), (12), (10) and (6). But equations (11), (12), (10) and (6) are better, because only 1 possible modular multiplicative inverse operation is needed on m. And equations (11), (12), (18) and (19) requires 2 possible modular multiplicative inverse operations on xR and yR.

Last update: 2019.