homework 1 - or not and prove your answer In each case say...

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ICS 180: Introduction to Cryptography 4/8/2004 Homework 1 Due Thursday, 4/15/2004, at the beginning of class! 1 Substitution cipher [15 points] Have a look at the substitution cipher in Lecture Notes 1 (section 3.2) and recall the definition of perfect secrecy. Prove that the substitution cipher is perfectly secure for the special case of = 1, and that it it is not perfectly secure if 2. 2 OTP cipher variations [30 points] We showed that One-Time Pad encryption satisfies perfect secrecy if M = K = { 0 , 1 } , for any . Consider some variations of the OTP cipher, where the messages and/or keys are binary strings as before but with some strings missing. Consider set S of three 2-bit strings, S = { 00 , 01 , 10 } . Consider the following three variations on the OTP cipher. In all these variations the key generation algorithm chooses k ∈ K uniformly, and encryption and decryption work as in OTP, i.e. Enc ( k, m ) = k m and Dec ( k, c ) = k c . For each of the OTP variations below, say whether the resulting cipher is perfectly secure
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Unformatted text preview: or not, and prove your answer . In each case, say what the space C oF the ciphertexts is, and note the sizes oF the message space M and key space K . Do these sizes correlate somehow with whether or not the cipher is secure? Can you explain why? 1. Let M = S ℓ and K = { , 1 } 2 ℓ , i.e. both the message and the key are (2 ℓ )-long bit strings 1 However, not every (2 ℓ )-bit string can be a valid message. ±or example, For ℓ = 3, we could have m = [00 , 01 , 00] = 000100 but m = [11 , 00 , 11] = 110011 is not in M because 11 n∈ S . 2. Let M = { , 1 } 2 ℓ and K = S ℓ 3. Let M = K = S ℓ . [[Hint: This one is actually not perfectly secure. ..]] 1 We use notation A n to denote a set of n-long sequences [ A 1 , A 2 , ..., A n ] where each A i is an element of A . Using this notation, { , 1 } n denotes a set of all n-long binary strings. H1-1...
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