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

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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 efficient 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 G i ’s are trivially stretching: | G i ( x ) | = 2 | x | for both
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