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 27The numbers were then encrypted using the following encryption formula, with p = 29 and with an encryption exponent of e = 5:

y = x

The resulting ciphertext turned out to be:

8, 6, 23, 25, 4, 19, 3, 2, 5.

Problem [1]: Find the decryption exponent d.

Problem [2]: Decipher the secret message.

Recall that d is a positive number less than p-1 such that ed = 1 mod p-1. This is then used as the exponent in the decryption formula x^{d} mod p. So to decipher the secret message you must first find a positive number d such that de leaves remainder 1 when divided by p-1. (You can find d by trial and error when the numbers are small. In general, you can also use Euclid's algorithm.) Once you know d, to decrypt the first letter in the secret message, which is 8, you just compute 8^{d} mod 29, and then use the table to see which letter this is.