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Assignment 5
1. We know that ϕ(ab) = ϕ(a)ϕ(b) whenever gcd(a, b) = 1. Give
an example that shows that this identity is, in general, not true when
gcd(a, b) 6= 1.
2. Use the stream cipher discused in class (Section 2.6), with n = 7
and c0 = c1 = 1, c2 = c3 = 0, c4 = c5 = c6 = 1. Encrypt w =
1110011 1110001 1010001 using the key k = 1010011.
3. Find the sample space and probability distribution that model flipping
two coins. Describe the event “at least one coin comes up heads”
formally and compute its probability.
4. We throw two dice. Determine the probability that they both show
different numbers under the condition that the sum of both numbers is
5. (a) Determine the integer n such that the probability for two of n
people having the same birthday is at least 9/10.
(b) Suppose the 4-digit PINs are randomly distributed. How many people
must be in a room such that the probability that two of them have
the same PIN is at least 1/2? (Here “4 digits” means that the PIN
cannot start with a 0.)
Note: Pay special attention to the direction of the inequality in the
formula(s) used for this question.
Due: Thursday, February 25, 2021, 11:30 pm

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