Do all ten problems.

1. Give the asymptotic time complexity of the following, using $\Theta$-notation.

   for (int i = 1; i * i <= n; i++) {
       for (int j = 1; j <= n; j += j) {
           System.out.println("*");
       }
   }
   

2. Suppose $f(n)=\sqrt{n^3}$ and $g(n)=5^{\log_3{n}}$. Is $f$ in $O(g)$, $\Omega(g)$, or $\Theta(g)$? Justify your answer (rigorously), appealing to the definitions of $O$, $\Theta$, or $\Omega$ as needed.
3. The following items are inserted, in this order, into a binary max heap: 1 5 4 17 22 3 19. Then we remove an element. What does the heap now look like? Draw both the tree form and the array form.
4. Here’s a famous function:

   def c(n):
       return 0 if n==1 else c(n/2 if n%2==0 else 3*n+1)+1
   

   Write this function tail-recursively.
5. An algorithm with complexity function $T(n) = 2^n$ can process a 30-element list on our PC in 256 seconds.
   1. How many days would it take to process a 40-element list?
   2. How many years would it take to process a 50-element list?
   3. If we ran the algorithm on a machine that was one billion times faster than our PC, how large of a list could we process in one day? (Note 1 day = 86400 seconds)
   4. If we needed a computer that could process 80 elements in a week, how much faster than our original PC does this computer need to be? (1 week = 604800 seconds)
6. An algorithm with complexity function $T(n) = 8 \log_2 n$ can process a 512-element list on our PC in 144 seconds.
   1. How long would it take to process a 2048-element list?
   2. How long would it take to process a 1048576-element list?
   3. If we ran the algorithm on a machine that was one thousand times faster than our PC, how large of a list could we process in one day? (Note 1 day = 86400 seconds)
   4. If we needed a computer that could process 4096 elements in a minute, how much faster than our original PC does this computer need to be?
7. In the reading for the course, you’ve seen that Fermat’s Little Theorem and corollaries give rise to probabilistic primality tests; for example we know that if, for all $a$ not divisible by $p$, $a^{p-1} \equiv 1 \pmod p$ then $p$ might be prime. But it turns out that there is actually a perfectly deterministic primality test: Wilson’s Theorem says that $(p-1)! \equiv -1 \pmod p$ if and only if $p$ is prime. Write a function called is_prime or isPrime that returns a boolean value indicating whether its argument is prime or not. Then answer whether there are any disadvantages to using this function. Okay there is a huge one. What is it? Answer using technical language and back it up with real numbers.
8. Given $p=876872342387$, $q=276872342389$, and $e=5$, what are the RSA public and private keys?
9. How many possible “keys” are there in a block transposition cipher that transposes individual bytes within blocks of 16 bytes?
10. In the mod-17 world, what is the discrete logarithm, base 3, of 13?