$2\mathrm{.}72.$ A cryptographic hash takes a message as input and produces a fixed$-$length string as output, called the digital fingerprint. A brute force attack involves computing the hash for a large number of messages until a pair of distinct messages with the same hash is found. Find the number of attempts required so that the probability of obtaining a match is $\backslash (1/2\backslash ).$ How many attempts are required to find a matching pair if the digital fingerprint is $64$ bits long? $128$ bits long?