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# tut09sols - 2 The University of Sydney MATH2068 Number...

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The University of Sydney MATH2068 Number Theory and Cryptography (http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/) Semester 2, 2008 Lecturer: R. Howlett Tutorial 9 1. The Secret Society of SilverSmiths encrypts messages as follows. They represent the letters A to Z by the numbers 1 to 26 and a space by the number 0. Thus each message is identified with a sequence of residues mod 27. They add a space at the end if necessary to ensure that each message consists of an even number of residues. So each plaintext has the form a 1 a 2 a 3 a 4 . . . a 2 k - 1 a 2 k for some positive integer k , with 0 a i < 27 for all i ∈ { 1 , 2 , . . . , 2 k } . The encryption key is a sequence ( k 1 , k 2 , k 3 , k 4 , k 5 , k 6 ) consisting of six residues mod 27. The message sender computes t 2 i - 1 t 2 i = k 1 k 2 k 3 k 4 a 2 i - 1 a 2 i + k 5 k 6 for each i from 1 to k , and then obtains the ciphertext as the alphabetical equivalent of b 1 b 2 b 3 b 4 . . . b 2 k - 1 b 2 k , where b j is the residue of t j modulo 27. If the encryption key is (0 , 5 , 11 , 1 , 4 , 13), what is the ciphertext for the message FRIDAY? Solution. Numerically, FRIDAY is 6,18,9,4,1,25. Now 0 5 11 1 6 18 + 4 13 = 13 16 0 5 11 1 9 4 + 4 13 = 24 8 0 5 11 1 1 25 + 4 13 = 21 22 and so the ciphertext is MPXHUV. 2. As a member of the SSSS you receive the ciphertext message YIXIUV, but, search though you might, you cannot find the piece of paper on which you wrote the decryption instructions. But you can remember the encryption instructions and key (from Exercise 1). Find the plaintext.

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tut09sols - 2 The University of Sydney MATH2068 Number...

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