# assign4 - Algorithms in the Real World (15-853), Fall 09...

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Algorithms in the Real World (15-853), Fall 09 Assignment #1 Due: Nov. 3, 2009 Problem 1: Number Theory basics (10pt) A. For what values of n is φ ( n ) odd? B. Show that if d | m (i.e. d divides m ), then φ ( d ) | φ ( m ) . Problem 2: Difﬁe-Hellman (10pt) Extend the Difﬁe-Hellman scheme to enable three parties to share a single secret. Problem 3: RSA (10pt each) The following two questions exhibit what are called protocol failures . They shows how one can break a cryptosystem if the cryptosystem is used carelessly. You can solve either one of the two questions or both. A. Joe Hacker decides that he wants to have two public-private key pairs to be used with RSA— he feels that two is more prestigious than one. In his inﬁnite wisdom, he decides to use a common value for n = pq and selects two separate encryption exponents e 1 and e 2 , giving two distinct decryption keys d 1 and d 2 . He makes e 1 , e 2 and n public. You can assume that e 1 and e 2 are relatively prime. Assume Alice and Bob send the same secret plaintext message

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## This note was uploaded on 01/26/2010 for the course COMPUTER S 15-853 taught by Professor Guyblelloch during the Fall '09 term at Carnegie Mellon.

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assign4 - Algorithms in the Real World (15-853), Fall 09...

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