Cryptography(COT 5937) Lecture 12
6/26/01
Solutions to previous contest problems:
1)
φ
(18338400) =
φ
((2
5
)(3
4
)(5
2
)(283)) = (2
5
– 2
4
)(3
4
– 3
3
)(5
2
– 5
1
)(283 – 1)
= 4872960
2) n = (17)(19),
φ
(n) = (17 – 1)(19 – 1) = 288
e
≠
2 because gcd(288,2) > 1. Similarly, e
≠
3 and e
≠
4. But, e=5 is a viable solution.
Now, we need to solve 5d
≡
1 (mod 288) for d. Using the extended Euclidean
Algorithm or otherwise, we find that d=173 is the corresponding decryption key.
Probabilistic Primality Testing
First we need to introduce the idea of a probabilistic algorithm. What is most strange
about this kind of algorithm is that it does not necessarily produce the correct answer to a
question. However, within some tolerance (or probability) it will produce a correct
response. (ie. ½ of the answers will be correct.
..)
This does not seem very useful, but we can indeed repeatedly run these algorithms to
increase our likelyhood of correctness for certain types of problems. Probabilistic
algorithms are typically designed for problems that have no known polynomial time
solution. (If you can’t solve the problem exactly in a given amount of time, you might as
well try to approximate the solution.
..)
For our purposes, we will only define a Monte Carlo algorithm, which is one type of
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 Summer '09
 MonteCarlo algorithm, yesbiased MonteCarlo algorithm

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