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1. Let P be a prime that has 1024 bits and let a be a primitive root of p. Let h(x) - a ( mod p). We analyze if h is a good hash function. a) Is h(x) preimage resistant? Say YES or NO and justify your claim. b) Is h(x) weakly collision resistant? Say YES or NO and justify your claim. 2. In a family of five, what is the probability that no two people are born the same month? Explain how you have computed the probability. 3. Bob is using the El Gamal signature scheme. His public key is (p, a, B) = (97,23,15) and his secret key is a= 67. a) Calculate Bob's signature for message m-17 with ephemeral random k-31. b) You receive allegedly from Bob the signed messages (ml,rl, s1) = (22,37,33) and (m2,r2,s2)=(82,13,65). Verify if these messages both originate from Bob.

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According the definition, the function h(x) is preimage resistant if given an output y it is infeasible to find the input, namely x such that αx= y. On the other hand, together with the fact that α is primitive root mod p, we obtain the definition of the discrete logarithm problem (which is known as NP-Hard problem for a prime p chosen with length 1024 bits). Hence the chosen hash function is preimage resistant....

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