Recall the (Randomised) Elgamal Cryptosystem - IT Assignment Help

Assignment Task

Question 1 Recall the (randomised) Elgamal cryptosystem. Let p be a prime. Let G be the group Z∗ p = f1; 2; : : : ; p - 1g under multiplication modulo p. Let g 2 G be a generator of the group. Let (x; h) denote Bob’s (private key, public key) pair, where x is a random element of Zp-1 = f0; 1; : : : ; p - 1g, and h ≡ gx (mod p).

(a) Suppose Alice sends a message m 2 G with ciphertext c = (c1; c2) to Bob, encrypted using a random k 2 Z p-1. Suppose Eve, the eavesdropper, comes to know both the message m, and its ciphertext

- Explain what can Eve do if Alice were to encrypt m0 2 G using the same k, resulting in ciphertext c0 = (c0 1; c0 2).
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.

(b) 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 2 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?

Question 2
Let X be a random variable taking on n values with probabilities p1; p2; : : : ; pn. Recall from Lecture 9 that the min-entropy of X, denoted E1(X), is given by
E1(X) = min i 1 pi:
If X is uniformly distributed then p1 = p2 = · · · = pn = n1 .
(a) Show that if X is not uniformly distributed then necessarily one of the pi’s is > 1=n.
(b) Show that if X is not uniformly distributed then E1(X) < n.
(c) Argue that min-entropy is the highest if X is uniformly distributed. What is the minimum entropy in this case?

Question 3
Let H be a hash function that has the properties of hiding and collision-resistance. Recall that collision
resistance property means that it is infeasible to find two inputs x; y with x 6= y such that H(x) = H(y).
Consider the commitment scheme from Lecture 9. To commit to a message m, Alice samples a random nonce r, and computes C = H(rjjm);
and publishes C as her commitment. A commitment scheme is binding if it is infeasible to find two pairs (m; r) and (m0; r0) with m 6= m0 such that H(rjjm) = H(r0jjm0):
Show that if you can find a collision in H, this does not necessarily break the binding property of the commitment scheme.

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