In practice, a and k are both in the range p˜2150 · · ·2250, and computing T =a·
P and y0 = k ·P is done using the Double-and-Add algorithm as shown in Sect. 9.2. of understanding cryptography
1. Illustrate how the algorithm works for a = 19 and for a = 160. Do not perform elliptic curve operations, but keep P a variable.
2. How many (i) point additions and (ii) point doublings are required on average for one "multiplication"? Assume that all integers have n = [log2 p] bit.
3. Assume that all integers have n = 160 bit, i.e., p is a 160-bit prime. Assume one group operation (addition or doubling) requires 20 µsec. What is the time for one double-and-add operation?