Question

5. We must assume that keys are not secure forever, and will eventually be discovered; thus keys should be changed periodically. Assume Alioe 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 rn = 37). (c) Alice changes her encryption key to e = 31, Encrypt rn = 11 using the new exponent (d) Suppose d' satisfies ed Ξ 1 mod kp(N), where k is any integer. Show that d will function as a decryption exponent. (In partic- ular, show that redmod N for any z 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. に1491? (f) Find a decryption exponent for e 31, DO NOT attempt to find φ(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.

Answer 1

m RSA we found 匕 1-131 8t 2mod 3403 = (2765 x 1369 χ 37) mod 3402 6 ent

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We oot oudh we aue get n u ench yoked A) 1677 ve.

c me mod N 3 od 343 Sin bolw 2 - 147 encrypled moss