Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 360 — #368
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Chapter 9
Number Theory
Problem 9.75.
Let
S k D 1 k C 2 k C C p k ;
where p is an odd prime and k is a positive multiple of p
b 2 . p::0 such that
S k a b .mod p/:
1. Find a 2 Œ0::p/ and
Problems for Section 9.11
Practice Problems
Problem 9.76.
Suppose a cracker knew how to factor the RSA modulus n into the product of
distinct primes p and q. Explain how the cracker could use the public key-pair
.e; n/ to find a private key-pair .d; n/ that would allow him to read any message
encrypted with the public key.
Problem 9.77.
Suppose the RSA modulus n D pq is the product of distinct 200 digit primes p and
q. A message m 2 Œ0::n/ is called dangerous if gcd.m; n/ D p, because such an m
can be used to factor n and so crack RSA. Circle the best estimate of the fraction
of messages in Œ0::n/ that are dangerous.
1
200
1
400
1 1 1 1
200 10 10 200 400 10 10 400
Problem 9.78.
Ben Bitdiddle decided to encrypt all his data using RSA. Unfortunately, he lost his
private key. He has been looking for it all night, and suddenly a genie emerges
from his lamp. He offers Ben a quantum computer that can perform exactly one
procedure on large numbers e; d; n. Which of the following procedures should Ben
choose to recover his data?
Find gcd.e; d/.
Find the prime factorization of n.