When encrypting, public key systems operate in a manner analogous to a block cipher in ECB mode. That is, the plaintext is chopped into blocks and each block is encrypted independently.

a. Why is ECB mode a bad idea when encrypting with a block cipher? Why is a chaining mode, such as CBC, a much better way to use a block cipher?

b. Why is it not necessary to perform any sort of chaining mode when using public key encryption?

c. Could your reasoning in part b be applied to block ciphers? Why or why not?

Suppose that Alice’s RSA public key is (N, e) = (33,3) and her private key is d = 7.

a. If Bob encrypts the message M = 19 using Alice’s public key, what is the ciphertext CI Show that Alice can decrypt C to obtain M.

b. Let S be the result when Alice digitally signs the message M = 25. What is 5? If Bob receives M and S, explain the process Bob will use to verify the signature and show that in this particular case, the signature verification succeeds.

Recall that with the RSA public key system it is possible to choose the same encryption exponent, e, for all users. For the sake of efficiency, sometimes a common value of e = 3 is used. Assume this is the case.

a. What is the cube root attack on RSA and when does it succeed?

b. Give two different ways of preventing the cube root attack. Both of your proposed fixes must still provide improved efficiency over the case where a common encryption exponent e = 3 is not used.

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