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Question:

For all of the following questions show your sage input/output.

  1. Compute the order of the curve defined by y^2 = x^3 + 7*x + 25 over the finite field with 47 elements
  2. On the curve defined by y^2 + x*y = x^3 + x over GF(2^8) compute the inverse of the point (1,1)
  3. On the curve defined by y^2 + y = x^3 + x^2 + x + 1 over the finite field with 701 elements, find a generator and show its order.
  4. On the curve defined by y^2 = x^3 + 4187*x + 3814 over finite field of size 6421 compute the sum of the points (3711,373) and (4376,2463).
  5. On the elliptic curve defined by y^2 = x^3 + 3361*x + 6370 over finite field of size 8461 compute 1001 times the point (1735, 3464).
  6. On the elliptic curve defined by y^2 = x^3 + 1800*x + 1357 over finite field of size 8191, let P1 = (1794, 1318) and P2 = (3514, 409), compute the sum of 13 times P1 plus 28 times P2.

 

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