If Trudy can factor the modulus N, then she can break the RSA public key cryptosystem. The complexity class for the factorization problem is not known. Suppose that someone proves that integer factorization is a “really hard problem,” in the sense that it belongs to a class of (apparently) intractable problems.

What would be the practical importance of such a discovery?

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Suppose that Bob uses the following variant of RSA. He first chooses N, then he finds two encryption exponents e_{o} and e1 and the corresponding decryption exponents do and d_{1}. He asks Alice to encrypt her message M to him by first computing C_{o} = M^{e}° mod N, then encrypting C_{o} to obtain the ciphertext, C_{1} = CQ1 mod N. Alice then sends C_{1} to Bob.

Does this double encryption increase the security as compared to a single RSA encryption? Why or why not?

Suppose that Alice signs the message M = “I love you” and then encrypts it with Bob’s public key before sending it to Bob. As discussed in the text, Bob can decrypt this to obtain the signed message and then encrypt the signed message with, say, Charlie’s public key and forward the resulting ciphertext to Charlie. Could Alice prevent this “attack” by using symmetric key cryptography?

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