Assignment 3 (Due on April 13, 2012)
Undergrad students will need to specify which 6 out
of 8 problems that she/he is submitting.

Problem 1: Find multiplicative inverses (if it exists):
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a) 177 mod 1077:
                
b) 311 mod 2431:



Problem 2: Compute 53^165 mod 401 using the square and multiply algorithm
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You are required to show all steps.


Problem 3: pg 182 no. 3
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Problem 4: pg 183	 no. 8
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Problem 5: Show that 129 is not a prime without factoring: (Apply testing
algorithm) --------------------------------------------------------- 
(without any division)



Problem 6: In RSA scheme, suppose Trudy knows the public key of Alan. 
Suppose further that Alan is stupid enough to publicly annouce the the
difference of
the 2 primes p-q == S.  How can Trudy break this scheme?

Problem 7: Suppose Michael has an RSA cryptosystem with modulus n and
encryption exponent b1, and Stanley has an RSA cryptosystem with the
same modulus n with encryption exponent b2.  Suppose also
that GCD(b1,b2)=1.  Now, consider the situation that arises if Diana
encrypts the same plaintext x to send to both Michael and Stanley.
Suppose Alan sees both ciphertext that Diana sends to Michael and
Stanley.  How can Alan obtain the message x?

Problem 8: In class, we discuss the problem of using e=3.  Suppose the
public key of Bob, Bart and Bert are (e,n)= (3,161), (3,209),
(3,221).  Alan sends the same message to Bob, Bart, and Bert, encrypted
with their respective public key.  Alan sends Bob, Bart, and Bert, 6 mod
161, 113 mod 209 and 177 mod 221 respectively.  Using Chinese reminder
theorem as illustrated in class (without factoring, or guessing), find the
message that Alan sends to them.