prelim208

# prelim208 - (e Assuming the string 010101 is received and...

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Mathematics 336 Prelim 2 April 4, 2008 4 problems, 25 points each No books, notes or electronic devices may be used. Your proofs may use anything that has been given in class or in the book, as long as you show clearly what you are using. You must EXPLAIN ALL ANSWERS! 1. Consider an RSA public key cryptosystem in which your public key is ( e 1 , n 1 ) = (27 , 55) and my public key is ( e 2 , n 2 ) = (7 , 22). (a) Your decoding key is ( d 1 , 55). Determine d 1 . (b) “Sign” the numerical message 02 03 06 so that I know it is from you. (c) Encrypt the result of part (b) for transmission to me. (There is no need to carry out the arithmetic in this part; just indicate what has to be done.) 2. Let H = 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 be a parity check matrix for a binary code. (a) Determine a generator matrix G for this code. (b) How many codewords are there in this code? (c) Can it correct all single errors? Why or why not? (d) Is this a Hamming code? Why or why not?
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Unformatted text preview: (e) Assuming the string 010101 is received, and at most one error occurred (if any), determine the codeword that was sent. 3. Suppose that G is a ﬁnite group of order < 100 and that H and K are subgroups of G of orders 18 and 30, respectively. (a) What is the order of G ? Why? (b) How many left cosets of H are there? Explain. 4. (a) Let f 1 and f 2 be elements of F [ x ], where F is some ﬁeld. Suppose g = gcd( f 1 ,f 2 ) in F [ x ]. Let V := { α ∈ F | f 1 ( α ) = f 2 ( α ) = 0 } be the set of common roots of f 1 and f 2 in F , and U := { α ∈ F | g ( α ) = 0 } be the set of roots of g in F . Show V = U . (b) Find gcd( x 4 + 3 x 3 + 2 x + 1 ,x 3 + x 2 + x + 1). (c) Use parts (a) and (b) to ﬁnd all common rational roots of the polynomials x 4 + x 3 + x + 1 and x 3 + 2 x 2 + 2 x + 1 . 1...
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