Need some help with my Cryptography review questions, I've having a tough time with these two. Please write out the proofs completely! No code is required.

2. Prove the following generalization of the results in this chapter: Fix a value x{0,1}. Then when taking q uniform samples from {0,1}, the probability that there exist two distinct samples whose xOR is x is BirthdayProb(q,2). 3. Mining Bitcoin requires evaluating the SHA-256 cryptographic hash function as many times as possible, as fast as possible. When mining Bitcoin today, the only economically rational choice is to use special-purpose hardware that does nothing except evaluate SHA256, but is millions (maybe billions) of times more energy efficient than a general-purpose CPU evaluating SHA-256. (a) The relevant specs for Bitcoin mining hardware are wattage and giga-hashes (or terahashes) per second, which can be converted into raw energy required per hash. Search online and find the most energy efficient mining hardware you can (e.g., least joules per hash). (b) Find the cheapest real-world electricity rates you can, anywhere in the world. Use these to estimate the monetary cost of computing 240,250,,2120 SHA-256 hashes