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

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

(a) Determine with justification the equation of the curve.

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

problem . Sometimes students wonder why the geomet1c construction P + Q requires the refection step.

Suppose instead that we used a simpler no refection definition to add elliptic curve points, letting R = P +

Q where P, Q, Rare collinear points on an elliptic curve (i. e. removing the refection 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 01 R = P + Q?

problem: For this question, you may work by hand or use the applet Elliptic Curves Applet: over Zp in the Content for Module 5. Computations are over the elliptic curve y2 = x3 +l1x +6 over Z23. To support your answer, you can quote calculations without great detail. For example, you could say that 2(2, 6) = (19,17), without detailing the calculations of m, x, y.

Tip: Organize your work to avoid unnecessary repetition.I

Given a positive integer k, define a set of Pfints S(k) on the elliptic curve as follows:

PE S(k) IF AND ONLY IF [(2k)P =/0 AND (2k-1)p ~ 0].

(a) Determine with justification all points in S(l).

(b) Determine with justification all points in S(2).

(c) Determine with justification the largest value of k for which S(k) is not empty, and the corresponding points in S(k).

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