4. Given an elliptic curve E y2 = x3 + 4x + 20 over Z29 and the base point P = (8, 10). The order of this curve is known to be #E = 37. Furthermore, an additional point Q = 15P = (14, 23) on the curve is also given. Determine the result of the following point scalar multiplications by using as few group operations as possible, i.e., make smart use of the known point Q. Specify how you simplified the calculation each time. Hint: in addition to using Q, use the fact that it is easy to compute P.

1) 16P

2) 38P

3) 53P

4) 14P + 4Q

5) 23P + 11Q

You should be able to perform the scalar multiplications with considerably fewer steps than a straightforward application of the double-and-add algorithm would allow.