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12.14. EXERCISES 253
Bob is going to be sending you the name of a new business contact. But
so is Carley. If Bob’s RSA parameters are N = 7979 and e = 47 and
Carley’s are N = 3569 and e = 53, who sent the message and what does
it say
13. The statement of Euler’s Theorem begins “If p and q are two distinct
primes.” Why is the distinctness needed In other words, what happens
if p = q
14. By some amazing coincidence Alice and Bob independently chose the same
primes P and Q when making their RSA parameters. Fortunately they
chose different encryption and decryption keys e A and e B . Show that
given the ciphertexts C A = (M e A
)%N and C B = (M e B
)%N if e A and e B
are relatively prime then M can be recovered.
15. Show that P +Q = N −(P −1)(Q−1)+1 and P −Q = √ (P + Q) 2 − 4N.
Thus by adding or subtracting P and Q can be found directly.
This shows that knowledge of N and (P − 1)(Q − 1) suffices to break
RSA.
16. Shamir, Rivest and Adelman suggested that their RSA method could be
used to play poker over the phone [SRA]. Explain how using techniques
similar to signing a message, two players may use RSA to play poker.
Hint: Have one player begin by encrypting the fifty-two messages ace
of clubs, two of clubs, three of clubs, ..., king of spades, sending
them to the other player. There will need to be a lot of messages sent.