Problem 3 (RSA - Covid 19 Edition, sorry - I had to make at least one COVID reference this semester). Bob uses the RSA Public Key Cryptosystem to send a ciphertext to Alice using Alices's public information. He uses her public key N = 11567078666698476133 and e = 65537. If Eve intercepts Bob's message c = 7057468852010395394 what was Bob's message m? (Factor N to determine Alice's private information. Knowing Alice's primes p and q we can easily construct the decryption exponent d.) Note: ord("C) = 67, ord('0') = 79...., ord('9') = 57. 1 2 Remark 1. Suppose that Bob published two different encryption exponents ej and e2 for use with his public modulus N and that Alice encrypts a single plaintext m using both of Bob's exponents. If Eve intercepts the ciphertexts ci 3 m (mod N) and C2 = m2 (mod N) she can take a solution to the equation elu + 2. v = ged(e1,e2) and use it to compute 09. c = (m")" (m2)" = m%s'u+62v = msed(e1,99) (mod N). If it happens that ged(C1, C2) = 1, Eve has recovered the plaintext - without having to factor N! I.e., regardless of the size of N (strength of encryption) if you send the same message twice with different exponent it is bad news! 100