Please solve 2,3,4,5,6 and 7 questions.

MATH010 Cryptography Practice Problems for Final Exam 1. Is the curve y2=x33x+2 over R singular or not? Why? 2. Consider the elliptic curve y2=x3+2x+2overF17. i) Show that the condition 4a3+27b2=0(modp) is fulfilled for the curve. ii) Perform the additions (2,7)(5,2) and (3,6)(3,6) in the group of the curve. iii) The elliptic curve has the group order 19. Verify Hasse's theorem for this curve. iv) Why are all points of this curve are primitive? 3. Consider the elliptic curve E:y2=x3+x+3 over the field F7. i) For which values of xF7,x3+x+3 is a perfect square in F7 ? ii) List E(F7). In other words, list the points on the curve E. (Do not forget the point at infinity). iii) Find the line through the two points (4,1) and (6,1). Find the third point of E which this line passes through (the existence of such a point is guaranteed by Bezout Theorem). Finally, find (4,1)(6,1). iv) Show that the elliptic curve group E(F7) is cyclic and find a generator. v) Find the Discrete Logarithm of (6,1) to the base (4,1). That is, find k such that (6,1)= k(4,1). vi) We know that possible orders of elements in the group E(F7) are the divisors of the group order 6 , namely 2 and 3 . Find a point on E(E7) of order 3 and a point of order 2 . 4. List the points on the elliptic curve y2=x3+2x+7overF11. Then, use double-and-add algorithm to compute the orders of the points. 5. Your task is to compute a session key in a Diffie Hellman key echange protocol based on elliptic curves. Your private key is a=6. You receive Bob's public key B=(5,9). The elliptic curve being used is defined by y2=x3+x+6(mod11). 6. What are the security criteria for a given hash function? 7. Let p be a prime and let be an integer with p. Let h(x)=(modp). Why h(x) is not a good cryptographic hash function