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Question 1 (8 marks)

This question tests your knowledge of what constitutes a group. Let n be a composite number. Let X = {1, 2, . . . , n − 1}.
(a) Let x ∈ X. Show that if there are integers a, b such that ax + bn = 1, then gcd(x, n) = 1. (2 marks)
(b) Let x ∈ X. Show that if gcd(x, n) 6= 1 then ax ≡ 1 (mod n) has no solution for any integer a. (2 marks)
(c) Show that there is at least one element x ∈ X, such that gcd(x, n) 6= 1. (2 marks)
(d) Let ∗ be multiplication modulo n. That is, for x, y ∈ X, x ∗ y = xy (mod n). Prove that (X, ∗) is not a group. (2 marks)

Question 2 (4 marks)

An encryption system is considered malleable if given a ciphertext of some unknown plaintext, it is possible to obtain a valid ciphertext of a related plaintext without even knowing the contents of the plaintext. This is problematic depending on the application. For instance, in bidding for a contract, a company might outbid its competitor by simply multiplying it’s rival company’s encrypted bid by 0.9, without even knowing the
bid [DDN03]. The following questions relates to the malleability of the Elgamal cryptosystem taught in the lecture.

(a) Suppose we are given the ciphertext c = (c1, c2) of some unknown message m, where c1 ≡ gk (mod p)for some unknown random integer k ∈ Zp−1 and c2 ≡ m · hk(mod p), where h is the public key of some unknown private key x, in the Elgamal cryptosystem. Let m0 be a message that you know. Can you obtain a valid ciphertext of the message m · m0 without knowing m? (2 marks)

(b) Let p = 739. Given the following ciphertext of some message m1 encrypted using randomized Elgamal, what is the ciphertext of m1 · m2, where m2 ≡ 2 (mod p)? Show the steps in PARI/GP. (1 mark) (c1, c2) = (246, 609)Note that the integers are reduced modulo p.

(c) Continuing from part(c), suppose the private key is x = 419. What is the message m1? (1 mark)

Question 3

(b) Show that if X is not uniformly distributed then F∞(X)

(c) Argue that min-entropy is the highest if X is uniformly distributed. What is the min-entropy (E∞(X)) in this case

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