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quiz6_solution - Quiz 6 solution Math 187 Prof Garsia Alex...

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Quiz 6 solution Math 187, Prof. Garsia Alex Brik UC San Diego May 21, 2010 1. a) H = 16 60 log 2 60 16 + 3 ° 4 60 log 2 60 4 + 8 60 log 2 60 8 + 24 60 log 2 60 24 = 2 : 2 Minimum number of binary registers 60 ° H = 60 ° 2 : 2 = 132 b) Hu/man tree: 1
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letter code length A 10 2 B 11110 5 C 11111 5 D 110 3 E 0 1 F 1110 4 bit= 2 ° 16 + 5 ° 4 + 5 ° 4 + 3 ° 8 + 24 + 4 ° 4 = 136 2. Heights h 1 , h 2 , ... , h n are a tree if S = n X i =1 1 2 h i ± 1 with equality i/ the tree is complete. Sequence 1: S = 1 4 + 1 4 + 1 8 + 1 8 + 1 16 + 1 16 + 1 32 = 29 32 incomplete tree Sequence 2: S = 1 2 + 1 8 + 1 8 + 1 8 + 1 16 + 1 32 + 1 32 = 1 complete tree. Sequence 3: S = 1 4 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 16 = 15 18 incomplete tree. 3. a). 2 i + 2 = 2 mod 4 i = 0 Plaintext: m 0 b) all keys are equally likely. Fix c s , m i s = 2 i + j mod 4 if i = 0 s = j - unique if i = 1 j = s ² 2 mod 4 unique Thus for every pair ( m; c ) there is a unique key k s.t. E k ( m ) = c . Thus by the theorem given in class the perfect secrecy is achieved. 2
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c) H ( K j C ) = H ( K ) ² H ( C ) + H ( M ) H ( K ) = 4 ° 1 4 ° 2 = 2 k 0 : 0 ! 0 1 ! 2 k 1 : 0 ! 1 1 ! 3 k 2 : 0 ! 2 1 ! 0 k 3 : 0 ! 3 1 ! 1 P ( C ) = 1 4 H ( C ) = 4 1 4 log 2 4 = 2 H ( M ) = 2 1 2 log 2 2 = 1 H ( K j C ) = 2 ² 2 + 1 = 1 d) P ( M = 0 ; C = 1) = P ( C = 1 j M = 0) P ( M = 0) = 3 8 ° 1 2 = 3 16 P ( M = 0 ; C = 0) = P ( C = 0 j M = 0) P ( M = 0) = 1 8 ° 1 2 = 1 16 P ( M = 0 ; C = 2) = P ( C = 2 j M = 0) P ( M = 0) = 1 8 ° 1 2 = 1 16 P ( M = 0 ; C = 3) = P ( C = 3 j M = 0) P ( M = 0) = 3 8 ° 1 2 = 3 16 P ( M = 1 ; C = 0) = P ( C = 0 j M = 1) P ( M = 1) = 1 16 P ( M = 1 ; C = 1) = 3 16 P ( M = 1 ; C = 2) = 1 16 P ( M = 1 ; C = 3) = 3 16 P ( C = 3) = P ( C = 1) = 1 2 ° 3 8 + 1 2 °
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