solution 1

# Introduction to Modern Cryptography: Principles and Protocols (Chapman & Hall/CRC Cryptography and Network Security Series)

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ICS 201: Cryptography and Communication Security 10/19/2007 Solutions to homework 1 Problem 2.2 If encryption is secure then by Definition 2.1 this condition implies that Pr [ M = m ] = Pr [ M = m ] for all m, m in the message space, which is obviously not true if M is sampled from any distribution over the message space. Problem 2.3 After throwing out the key k = 0 l the OTP encryption is no longer perfectly secure because then the key-space has one fewer element than the message space. A good example is that for any c ∈ { 0 , 1 } l the message m = c is impossible, because O l negationslash∈ K . But any other message is possible, which violates for example the condition in lemma 2.3, which says that encryption is perfectly secure only if for all c, m 0 , m 1 the probability that m 0 encrypts to c (over keys) is the same as the probability that m 1 encrypts to c (over keys). Problem 2.4(b) Since the substitution cipher has key-space K of size 26!, it can provide perfect secrecy only if the message space M has at most 26! elements too. Let M

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