CSCI 4116 Cryptography

MATH/CSCI 4116
Cryptography
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 =
3. 1110001 1010001 using the key k = 1010011.
4. 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.
5. We throw two dice. Determine the probability that they both show
   different numbers under the condition that the sum of both numbers is
   even.
6. (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