Computer Security: Art and Science

32.4 Exercises

1:
Let X represent the roll of a red die, and let Y represent the sum of the values from rolling the red die and a blue die. Prove that p(X = 3| Y = 8) = 1/5.
2:
Prove that the maximum entropy for an unknown message chosen from the set of possible messages { "yes", "no" } occurs when the probability of each message is 1/2.
3:
Let X and Y be random variables that take values from finite sets. Prove that H(X, Y) H(X) + H(Y), with equality holding when X and Y are independent.
4:
Let X and Y be random variables that take values from finite sets. Prove that H(X, Y) = H(X | Y) + H(Y).
5:
Let M and C be random variables that take values from the set of possible plaintexts and the set of possible ciphertexts for some cryptosystem. Prove that the cryptosystem provides perfect secrecy if and only if p(M | C) = p(M).

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1:	Let X represent the roll of a red die, and let Y represent the sum of the values from rolling the red die and a blue die. Prove that p(X = 3\| Y = 8) = 1/5.
2:	Prove that the maximum entropy for an unknown message chosen from the set of possible messages { "yes", "no" } occurs when the probability of each message is 1/2.
3:	Let X and Y be random variables that take values from finite sets. Prove that H(X, Y) H(X) + H(Y), with equality holding when X and Y are independent.
4:	Let X and Y be random variables that take values from finite sets. Prove that H(X, Y) = H(X \| Y) + H(Y).
5:	Let M and C be random variables that take values from the set of possible plaintexts and the set of possible ciphertexts for some cryptosystem. Prove that the cryptosystem provides perfect secrecy if and only if p(M \| C) = p(M).