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Jan 29 notes

# A Concrete Introduction to Higher Algebra, 2nd Edition

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U.C. Berkeley — CS276: Cryptography Lecture Notes: 01/29/2002 Professors Luca Trevisan and David Wagner Scribe: Deepak Rajan Lecture Notes: 01/29/2002 1 Message Indistinguishability Semantic Security Last class, we showed that Semantic Security ( SS ) = Message Indistinguishability ( MI ). Next, we try to prove that Message Indistinguishability ( MI ) = Semantic Security ( SS ). In fact, we’ll prove that ( t, )- MI = ( t , 2 )- SS , where t = t poly ( n ). Before we do so, we need an equivalent definition for ( t, )- MI . Definition 1 For every m 0 , m 1 ∈ { 0 , 1 } n , for every algorithm A that runs in time t ( n ) O ( n ) , for every a ∈ { 0 , 1 } , | a | ≤ n Pr ( p k ,s k ) G ( n ) [ A ( E ( m 1 , p k ) , p k ) = a ] Pr ( p k ,s k ) G ( n ) [ A ( E ( m 0 , p k ) , p k ) = a ] 2 ( n ) ( ) (the distribution of outputs of A () is roughly the same given the encryption of m 0 or m 1 .) Proposition 1 ( t, ) - MI = ( ) . Proof: Assume that A, a such that Pr ( p k ,s k ) G ( n ) [ A ( E ( m 1 , p k ) , p k ) = a ] Pr ( p k ,s k ) G ( n ) [ A ( E ( m 0 , p k ) , p k ) = a ] > 2 ( n ) (i.e. ( ) does not hold) Define A ( c, p ) as follows A ( c, p ) = 1 if A ( c, p ) = a 0 otherwise Now, Pr ( p k ,s k ) G ( n ) [ A ( E ( m i , p k ) , p k ) = i ] = 1 2 Pr ( p k ,s k ) G ( n ) [ A ( E ( m 1 , p k ) , p k ) = 1] + 1 2 Pr ( p k ,s k ) G ( n ) [ A ( E ( m 0 , p k ) , p k ) = 0] = 1 2 Pr ( p k ,s k ) G ( n ) [ A ( E ( m 1 , p k ) , p k ) = a ] + 1 2 1 Pr ( p k ,s k ) G ( n ) [ A ( E ( m 0 , p k ) , p k ) = a ] = 1 2 + 1 2 Pr ( p k ,s k ) G ( n ) [ A ( E ( m 1 , p k ) , p k ) = a ] Pr ( p k ,s k ) G ( n ) [ A ( E ( m 0 , p k ) , p k ) = a ] > 1 2 + ( n ) = ( t, )- MI does not hold.

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