Consider a secure hash function H that produces a $60-$bit hash. $($a$)(5$ Points$)$ Suppose that H$(1)$ happens to hash to $0($i$.$e$.,60$ zero bits$).$ If you don't know anything further about H other than that fact and that it's a secure hash function, what is the probability that H$(2)$ also hashes to $0?($b$)(5$ Points$)$ What is the probability that H has at least one collision? Explain in ONE sentence. $($c$)(5$ Points$)$ Suppose that commodity hardware can compute a single computation of H in $10$ nanoseconds $(=10-8$ sec$).$ Within an order of magnitude, how many years will it take for an attacker using a single system to find an x such that H$($x$)=$ y for a specific y$?$ You can approximate one year as $3\mathrm{.}10$ sec$.($d$)(5$ Points$)$ Suppose now that a sustained form of Moore's Law means that after every year, H can be computed twice as quickly as for the previous year. $($For this problem, assume that this acceleration happens discretely year$-$by$-$year, rather than being spread across a given year, as is actually more realistic.$)$ Given this change, now about how many years will it take the attacker to find such an x$?$