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Mohamed G. Gouda                                                CS 337
Fall 2008                                                     Midterm 1
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Open Book                                                       80 minutes
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1. A set S of k-bit words is called 1-dependent iff xor(S) = 1, where xor(S)
is the k-bit word that results from applying the exclusive-or to the words in 
S, and 1 is the k-bit word whose bits are all 1's. Let T be the set of k-bit 
words that remains after removing one k-bit word x from a 1-dependent set S. 
Compute the removed word x as a function of the remaining set T.

Solution:
   x = inv(xor(T))
   where inv(y) is the k-bit word that results from inverting each bit
   in the k-bit word y.

   { By the definition of T and S}
   <=>   S = T U {x}
   { Apply xor to both sides }
   <=> xor(S) = xor(T) xor x
   { S is 1-dependent }
   <=> 1 = xor(T) xor x
   { x xor x = 0 }
   <=> x = 1 xor xor(T)
   { 1 xor x = inv(x) }
   x = inv(xor(T))

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2. A sender and a receiver know the following four 10-bit codewords.
	C0 = 0000000000
	C1 = 0000011111
	C2 = 1111100000
	C3 = 1111111111
The sender sends one of these codewords to the receiver. Zero or more bits in 
the sent codeword are inverted before the receiver receives a 10-bit word C. 
Then the receiver executes only one of the following three actions.
	(a) The receiver accepts word C as is. 
	(b) The receiver accepts one of the codewords C0, C1, C2, or C3 
	    instead of C.
	(c) The receiver accepts no word.
Under what condition, does the receiver execute each of these actions?

Solution:
   (a) c is one of the codewords c0, c1, c2, or c3.
   (b) The hamming distance between c and one of the codewords is at
   least 1 and at most 2 (since the smallest hamming distance between
   any two codewords is 5).
   (c) neither condition (a) nor condition (b) holds.

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3. Assume that RSA is used to generate a public key (e,n) to encode a message
M consisting of two symbols, where each symbol is a digit in the range 0..8.
What is the smallest possible value of n in this case? Explain your answer.

Solution:
The max value of M is 88. Thus, the possible values of n are
   89, 90, 91, ...
Only 91 can be factored into two prime numbers 7 and 13.
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4. Alice knows its own private key (da,na) and the public key (eb,nb) of Bob, 
and Bob knows its own private key (db,nb) and the public key (ea,na) of Alice.
To send a signed private message M to Bob, Alice can send either one of the 
following two messages.

	("Alice" | (M^da  mod na))^eb  mod nb

	("Alice" | (M^eb  mod nb))^da  mod na

Which of these two messages is better in this situation? Explain your answer.

Soluton:

 The first message is better because
   (1) If using the second message, an attacker in the middle can
       change the sigature of Alice since Alice's public key is known
       to everyone. Consequently, the attacker can sign the message
       using his/her own private key and thus Bob can not figure out
       who is the original sender since Bob may think the attacker is
       the original sender.
   (2) If using the second message, everybody can know the message is
       sent from Alice although nobody except Bob can know the content
       of the message.
   (3) The first message allows Bob to easily decrypt the message when
       there are many Alices.

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