1. When elliptic curves are used for cryptography, why are elliptic curves over finite fields better than elliptic curves over the real numbers?

2. An elliptic curve y2 = x3 + ax + b (mod 29) includes points P = (7,15) and = (16,13).

(a) Determine with justification the equation of the curve. (b) Determine with justification all values oft for which there is no point (x, y) on the curve.

3. Sometimes students wonder why the geometric construction P + Q requires the reflection step. Suppose instead that we used a simpler no-reflection definition to add elliptic curve points, letting R = P+ e2 where P,Q, R are collinear points on an elliptic curve (i.e. removing the reflection step from the definition of addition).

(a) Show that with a no-reflection definition of addition, we could get 2P = 0 for every choice of P.

(b) What advantage does the actual definition of addition (that is with the reflection step) have over the no-reflection definition of R = P + Q? 

Attachment:- CNS_Question.pdf