2. Find all solutions to [39][44] + [14] [$13] = [2} in 2.91. Express your answer as [57} for some integer n: with c g: m c: 91. 3. Prove that there exist 135 consecutive positive integers chug, .. . ,e135 such that for each integer i with 1 5 i E 135, there is a perfect square k,- > 1 such that k1- [ (1g. Hint: Set up a system of linear congruences and use the Generalized Chinese Remainder Theorem. 4. (a) Suppose that in setting up BSA, Alice chooses p = T, q = 13, and e = 25. i. What is Alice's public ke ii. 1What is Alice's private key? (b) Bob has set up an RSA scheme. He provides Alice with the public key (13,323). Alice wishes to send Bob the message M = 10. Determine the ciphertext corresponding to M. 5. Alice, Bob, and Charlie set up REA schemes. It happens that the their moduli n3, rib, and ac are coprime to one another, and each uses a = 3 to generate a public key. A single encrypted message M is sent to Alice, Bob, and Charlie using each of their schemes. An eavesdropper has intercepted the ciphertexts Ca, 01,, and Cc, and has access to the three public keys. (a) Let N = nanbnc. Prove that there exists a unique integer m E {(1, 1, 2, . .. ,N ._ 1} such that m E Ca (mod nu), m E 05 {mod at), and m E Cc {mod nc). (b) Preve that m = M3 is the unique integer from (a). (c) Explain how the eavesdropper can use (a) and (b) to decrypt the original message M