 
Summary: CS5050 Homework 6 Chapter 10, 12
1. Consider a Monte Carlo algorithm A for a problem P whose expected running time is at most T (n)
on any instance of size n and that produces a correct solution with probability fl(n). Suppose further
that given a solution to P , we can verify its correctness in time t(n). Show how to obtain a Las
Vegas Algorithm that always gives a correct answer to P and runs in expected time at most (T(n) +
t(n))/fl(n).
2. Liz conjectures that for any n (2) , n1 is its own inverse mod n. Prove or disprove this conjecture.
3. Show the execution of ExtendedEuclidGCD(412,113) by constructing a table similar to Table 10.10
(page 465).
4. Compute successive powers for the elements of Z17 in a table similar to that of Table 10.5 (page
460).
5. Compute the multiplicative inverses of 113, 114, and 127 in Z299.
6. (Chapter 10) Alice and Bob enjoy weekly monopoly games every Friday night. When Bob
is sent to the wilds of western Slobovia while Alice stays in central city, they agree that
they'll just call each other up on the telephone and conduct their games that way. But they
run into some interesting problems: How exactly does draw a card in a phone call, so that
both know that the card draw was fair? This is what they came up with: Each card would
be given a number 0 through X1 (where X is the number of cards). Alice would pick a
number and tell Bob what it was. Bob would pick a number and tell Alice what it was. The
card chosen would be the sum of their numbers mod X. Will this work?
