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(c) Consider the random plaintext on two letters, P = {A, B}. We have the proba- bility distribution on A, B, given by p(A) = p, and p(B) = q. We take K with o the identity map on P and T the permutation that interchange A with B. We put on K the uniform distribution, with p(o) = } = p(T). Finally, as a set С — Р. - [σ, ΤΗ 1. Show that if the probability on a sequence from P, is the-product of the probabilities for the n letters, then the probability that the cryptogram will be a specific sequence from Cn with m A's and n - т B's, is pm. q"-m + p"-m. q™ 2 2. Usir above, show that the entropy of Cn, is n H(C,) Σ • I'm · log(rm). -- m=0 m

(c) Consider the random plaintext on two letters, P = {A, B}. We have the proba- bility distribution on A, B, given by p(A) = p, and p(B) = q. We take K with o the identity map on P and T the permutation that interchange A with B. We put on K the uniform distribution, with p(o) = } = p(T). Finally, as a set С — Р. - [σ, ΤΗ 1. Show that if the probability on a sequence from P, is the-product of the probabilities for the n letters, then the probability that the cryptogram will be a specific sequence from Cn with m A's and n - т B's, is pm. q"-m + p"-m. q™ 2 2. Usir above, show that the entropy of Cn, is n H(C,) Σ • I'm · log(rm). -- m=0 m

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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(c) Consider the random plaintext on two letters, P = {A, B}. We have the proba- bility distribution on A, B, given by p(A) = p, and p(B) = q. We take K = {0,7} with o the identity map on P and 7 the permutation that interchange A with B. We put on K the uniform distribution, with p(σ) = ½ = p(7). Finally, as a set C = P. 1. Show that if the probability on a sequence from Pn is the product of the probabilities for the n letters, then the probability that the cryptogram will be a specific sequence from Cn with m A's and n - m B's, is I'm pm. qn-m + pn-m.qm 2 2. Using (1) above, show that the entropy of Cn, is N n H(Cn) == Σ · ™m · log(rm). m m=0 3. For p = 0.25. Find the first n > 0. such that En < 0.3.
Transcribed Image Text:(c) Consider the random plaintext on two letters, P = {A, B}. We have the proba- bility distribution on A, B, given by p(A) = p, and p(B) = q. We take K = {0,7} with o the identity map on P and 7 the permutation that interchange A with B. We put on K the uniform distribution, with p(σ) = ½ = p(7). Finally, as a set C = P. 1. Show that if the probability on a sequence from Pn is the product of the probabilities for the n letters, then the probability that the cryptogram will be a specific sequence from Cn with m A's and n - m B's, is I'm pm. qn-m + pn-m.qm 2 2. Using (1) above, show that the entropy of Cn, is N n H(Cn) == Σ · ™m · log(rm). m m=0 3. For p = 0.25. Find the first n > 0. such that En < 0.3.
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