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prelim2

# prelim2 - 33 34 35 36(a Find the decoding exponent d(b...

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Mathematics 3360 Prelim 2 April 15, 2010 No books, notes, or electronic devices may be used. You may use anything that has been given in class or in the book as long as you show clearly what you are using. SHOW ALL OF YOUR WORK AND JUSTIFY YOUR ANSWERS. 1 (5 points) . If a binary linear code has exactly 4 codewords, what is the dimension of the code? Explain. 2 (20 points) . Find the multiplicative inverse of x 3 + x +1 in F 2 [ x ] mod x 4 + x 3 +1. [Please work slowly and carefully and try to avoid careless mistakes.] 3 (15 points) . Let G be the multiplicative group U 17 of units in Z 17 , and let H be the cyclic subgroup of G generated by [2] 17 . Write down the cosets of H in G . 4 (20 points) . Consider the ternary linear code with generator matrix G = 1 0 0 1 2 0 1 0 1 0 0 0 1 2 1 . (a) Find the minimal distance of the code. (b) Is 12121 a codeword? Justify your answer. 5 (20 points) . Consider an RSA cryptosystem with modulus m = 51, encoding exponent e = 11, and our usual translation table: A B C D E F G H · · · R S T U V W X Y Z 11 12 13 14 15 16 17 18 · · · 28 29 30 31 32 33 34 35 36 (a) Find the decoding exponent

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Unformatted text preview: 33 34 35 36 (a) Find the decoding exponent d . (b) Decode the 3-letter message 46, 23, 29. You may use the following table giving a 1 , a 2 , a 4 , and a 8 mod 51 for a = 46 , 23 , 29. 1 2 4 8 46 46 25 13 16 23 23 19 4 16 29 29 25 13 16 You must show all your work to receive credit. 6 (20 points) . How many solutions does the equation x 2 = 1 have in the ring Z 136 ? Justify your answer. [You do not have to ﬁnd the solutions.] This is the end of the exam. If you have the time and inclination, there is an extra credit problem on the back. It will have only a marginal eﬀect on your ﬁnal grade, so don’t work on it unless you want to. The purpose of the following problem is to explain the “coincidence” that all entries in the last column of the exponentiation table are 16 in the RSA problem above. 7 (e.c.) . For any integer a relatively prime to 51, show that a 8 ≡ 1 or 16 (mod 51)....
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prelim2 - 33 34 35 36(a Find the decoding exponent d(b...

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