$($a$)$ Suppose you write a computer program to decrypt an encrypted password using "brute force." That is$,$ your program tries every possible encryption key until it finds the key that successfully decrypts the password. Suppose that your program can test one billion keys per second. On average, how long would the program need if the key is known to be $30$ bits long?

What about $200-$bit keys, or $1000-$bit keys? Compare your answers with comprehensible units, such as days, years, or the age of the universe. In general, each time we add a single bit to the length of the key, what happens to the expected running time of your program?