I need help with number 5 on my number theory hw.

13" = 18 (mod 31) 5. (20 points) Alice and Bob are using Diffie-Hellman key exchange to agree on a private shift cipher key. Alice chooses the prime number p = 29 and the modular number g = 7 and sends these to Bob. Bob chooses the exponent b = 5, and Alice chooses the exponent a = 11. (a) What is the next modular number that Alice sends to Bob? (b) What is the next modular number the Bob sends to Alice? (c) What is the private shift cipher key that they compute? (d) Use the shift cipher key obtained in part (c) to encrypt the following message from Bob to Alice: HIALICE 6. (20 points) Alice publishes the following RSA public key: n = 21, and e = 5. (a) Encrypt the message m = 11 for Bob to send to Alice. (b) Bob sends Alice the encrypted message c= 18. Help Eve break the cipher and decrypt this message (your answer will be a number modulo 21). (c) Comment on why this particular choice of n and e was not a good implementation of RSA. 7. (20 points) Solve the following quadratic congruence using the "completing the square" method: 9x +66x +49 = 0 (mod 79) Hint: 18-1 = 22 (mod 79). 8. (20 points) Evaluate the following Legendre symbols using the specified method: (a) Use Euler's criterion to evaluate (b) Use Eisenstein's criterion to evaluate (c) Use quadratic reciprocity to evaluate 13" = 18 (mod 31) 5. (20 points) Alice and Bob are using Diffie-Hellman key exchange to agree on a private shift cipher key. Alice chooses the prime number p = 29 and the modular number g = 7 and sends these to Bob. Bob chooses the exponent b = 5, and Alice chooses the exponent a = 11. (a) What is the next modular number that Alice sends to Bob? (b) What is the next modular number the Bob sends to Alice? (c) What is the private shift cipher key that they compute? (d) Use the shift cipher key obtained in part (c) to encrypt the following message from Bob to Alice: HIALICE 6. (20 points) Alice publishes the following RSA public key: n = 21, and e = 5. (a) Encrypt the message m = 11 for Bob to send to Alice. (b) Bob sends Alice the encrypted message c= 18. Help Eve break the cipher and decrypt this message (your answer will be a number modulo 21). (c) Comment on why this particular choice of n and e was not a good implementation of RSA. 7. (20 points) Solve the following quadratic congruence using the "completing the square" method: 9x +66x +49 = 0 (mod 79) Hint: 18-1 = 22 (mod 79). 8. (20 points) Evaluate the following Legendre symbols using the specified method: (a) Use Euler's criterion to evaluate (b) Use Eisenstein's criterion to evaluate (c) Use quadratic reciprocity to evaluate