homework 5

homework 5 - ICS 180: Introduction to Cryptography...

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ICS 180: Introduction to Cryptography 5/27/2004 Homework 5 Due Thursday , 6/03/2004 1 Constructing a PRG from a PRF This question is designed so that you see a relation between a PRF and a PRG. You have seen in class that with some work one can build a PRF out of any PRG. But PRF does seem like a more powerful construct, so the other direction, construction of a PRG from a PRF should be easy. But how shall this be done exactly? Let { f s | s ∈ { 0 , 1 } τ } τ =1 , 2 ,... be a PRF family, where for each τ and each s ∈ { 0 , 1 } τ , function f s maps domain { 0 , 1 } τ onto the same range { 0 , 1 } τ . (Using the notation from the lecture and the notes, we’d say that l ( τ ) = L ( τ ) = τ .) Consider the following attempts to construct a PRG from this PRF family. For each of the attempts, either prove that the PRG is secure or prove that it is not, by showing an e±cient algorithm that distinguishes its outputs from random strings: 1. G 1 ( x ) = [ f x (0 τ ) | f x (1 τ )] for x ∈ { 0 , 1 } τ 2. G 2 ( x ) = [ f 0 τ ( x ) | f 1 τ ( x )] for x ∈ { 0 , 1 } τ Note that both constructions, on purpose, are done in a way so that the
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homework 5 - ICS 180: Introduction to Cryptography...

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