In class we discussed a cipher based on remainder arithmetic. Here's how it worked.

Given a plaintext message, say the word "fab", first convert the letters into numbers using the following table:
```a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p  q  r  s  t  u  v  w  x  y  z
2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
```
The word fab turns into the numbers 7, 2, 3. Then use the following encryption formula encrypt the numbers:

y = xe mod p

For this formula p should be a prime number bigger than the biggest number in our table above. In class we picked p = 29. The encryption exponent e must be a number between 1 and p-1 such that gcd(e, p-1) = 1. In class we picked e = 3.

To encrypt the numbers 7, 2, 3, we take the number for each letter and plug it in for x to get y.

Now, f is 7 so we compute 73 mod 29. Since 73 = 343 and dividing 343 by 29 leaves a remainder of 24, we see that 7 encrypts to 24.

Likewise, a is 2 so we compute 23 mod 29. Since 23 = 8 and dividing 8 by 29 leaves a remainder of 8, we see that 2 encrypts to 8.

Finally, b is 3 so we compute 33 mod 29. Since 33 = 27 and dividing 27 by 29 leaves a remainder of 27, we see that 3 encrypts to 27.

Thus encrypting "fab" gives a ciphertext of 24, 8, 27.

Problem [1]: Encrypt the word "Hello" using the procedure described above (i.e., using the cipher described above).

You can use this web page to do the arithmetic for you.