5. We must assume that keys are not secure forever, and will eventually be discovered; thus keys should be changed periodically. Assume Alice sets up a RSA cryptosystem and announces N = 3403, e = 11. (a) Encrypt m 37 using Alice's system (b) At some point, Eve discovers Alice's decryption exponent is d 1491. Verify this (by decrypting the encrypted value of m-37) (c) Alice changes her encryption key to e = 31, Encrypt m = 11 using the new exponent (d) Suppose d' satisfies ed' 1 mod kp(N), where k is any integer. Show that d will function as a decryption exponent. (In partic- ular, show that mod N for any r relatively prime to N. This means that to break the system, Eve doesn't have to find Alice's decryption exponent d, but can find any d that "works") (e) How can Eve use this to find the decryption exponent for e = 31, based on her knwoledge of the encryption/decryption pair e = 11, d = 1491? (f) Find a decryption exponent for e = 31 . DO NOT attempt to find p(N). (g) Verify that the decryption exponent works. (You can do this by taking any message m and encrypting it, using e 31, Then show that the decryption exponent works to recover the original message (h) Eve intercepts Bob's message 2292. Decrypt this message