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sample_quiz6_solution

# sample_quiz6_solution - Math 187 Prof Garsia typed by Alex...

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Math 187 Prof. Garsia typed by Alex Brik 05-18-2010 SAMPLE QUIZ 6 solution. 1. Solution : m1: HELP m2: COME m3: FOOD m4: FIRE m5: LOST a) need at least 5 keys and 5 ciphertexts. Keys k1, k2, k3, k4, k5 M : m1 m2 m3 m4 m5 k1 1 2 3 4 5 k2 2 3 4 5 1 k3 3 4 5 1 2 k4 4 5 1 2 3 k5 5 1 2 3 4 The entry ( i; j ) of the matrix M provides the encoding of m j with the key k i . Let c 1 = m 1 , c 2 = m 2 , ..., c 5 = m 5 . Thus E k i ( m j ) = c M ( i;j ) Note that all keys are equally likely. For each pair ( m i ; c j ) there is a unique key k s s.t. E k s ( m i ) = c j . b) P ( c 1 ) = P ( c 2 ) = P ( c 3 ) = P ( c 4 ) = P ( c 5 ) = 1 5 c) We will be chosing keys from teh collection of all 4 letter sequences. Hence every key is equally likely. For every message m and a four letter ciphertext c there is a unique key k s.t. E k ( m ) = c Thus the perfect secrecy is achieved. 2. 1

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a. k 1 (0 ; 0) ! (0 ; 0) (0 ; 1) ! (0 ; 1) (1 ; 0) ! (1 ; 0) (1 ; 1) ! (1 ; 1) k 2 (0 ; 0) ! (0 ; 0) (0 ; 1) ! (1 ; 0) (1 ; 0) ! (0 ; 1) (1 ; 1) ! (1 ; 1) k 3 (0 ; 0) ! (0 ; 0) (0 ; 1) ! (0 ; 1) (1 ; 0) ! (1 ; 1) (1 ; 1) ! (1 ; 0) k 4 (0 ; 0) ! (0 ; 0) (0 ; 1) ! (1 ; 0) (1 ; 0) ! (1 ; 1) (1 ; 1) ! (0 ; 1) H = 4 ° 1 4 ° 2 = 2 b.
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