1) Your RSA public key is n=115. Bob sends you an encrypted letter of

the alphabet, which he encrypted by cubing it modulo n.

The transmission you received from Bob is 03. Use your decryption key

to figure out what letter of the alphabet Bob sent you.

2) Alice's RSA public key is n=1050589. Bob wants to send her a message consisting of one letter

of the alphabet together with some nonsensical padding. (The padding could be anywhere in the message.)

Bob encrypts his message by cubing it modulo n. By eavesdropping, you discover that

the transmission Alice receives from Bob is 306720. Using the knowledge that Alice carelessly

chose her two primes too close together, crack her RSA code, and then figure out what letter of the alphabet

Bob sent to her in his message. (Crack the code means find her decryption key.)

Hint: To simplify the computations, I chose 306720 so that its order mod n is only 10.

You may use a calculator for this problem (e.g., the Google calculator).

3) Let a and b be (congruent to) squares mod p, and let c and d be (congruent to) nonsquares mod p,

where p is an odd prime not dividing abcd. (For example, if p=7, then a could be 23 and b could be 11

and c could be 12 and d could be 13.) In general, which of the following are always squares mod p,

and which of the following are always nonsquares mod p?

(i) ab (ii) ac (iii) cd

Justify.