Introduction to Cryptography Assignment

Introduction to Cryptography Assignment

In this assignment, you will analyze/investigate the Cryptographic problems related to number theory, key distribution and public-key encryption .

Please write down all details of your solutions as clearly as possible. Do not simply write the final answers.

Problem 1:Number Theory

(a). Answer the following:

(i) What is the size of group Z ∗ 11?

(ii) Is 3 a generator of Z ∗ 11? and

(iii) Is 2 a generator of Z ∗ 11?

Problem 2: Key Exchange

(a). Suppose p be a prime number and g be the generator of Z ∗ p . Consider a scenario where two users Alice and Bob want to generate a shared key secretly on an open channel using the Diffie-Hellman Key-Exchange Protocol. Let us assume that p = 11, g = 2 and say Alice and Bob choose 3 and 8, respectively, as their random input integers in the Diffie-Hellman Protocol. What would be the final key shared by Alice and Bob under this case? Please explain your answer.

Problem 3: Public-Key Encryption

(a). Describe why textbook RSA is not CPA-secure. A solution often used to address this issue is called Padded-RSA. Please explain how Padded-RSA addresses the issue of textbook RSA?

Problem 4

(a). Describe in detail a man-in-the-middle attack on the Diffie-Hellman key exchange protocol. Show how an adversary can act so that at the end of the protocol Alice has a key k1 (known to the adversary) and Bob has a key k2 (known to the adversary). In addition to the above, show how an adversary can act so that if Alice and Bob continue by carrying out an encrypted conversation (using their keys obtained from the protocol), then the adversary obtains all the plaintext messages sent and Alice and Bob don’t know that any attack was carried out.

(b). In the RSA scheme, the public key consists of n and e. Suppose the attacker is able to figure out the totient (Φ) of the modulus (n). Under this case, would the attacker be able to break the RSA scheme? Explain the consequences.

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