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MATH135 - Assignment8 Solutions

# MATH135 - Assignment8 Solutions - MATH 135 Assignment#8...

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MATH 135 Fall 2007 Assignment #8 Due: Thursday 22 November 2007, 8:20 a.m. Notes : In problems 3 and 4, please do your work by hand and show this work. (Of course, using MAPLE to check your answers is probably a good idea.) The last page of this document contains some handy MAPLE commands. The text file “Problem 6 Values” contains the values of the large numbers in Problem 6 in text format. There will tentatively be a TA in the Tutorial Centre to help with Maple issues from 1:30 to 2:30 on Wednesday 21 November. Hand-In Problems 1. Determine the complete solution to the congruence x 31 + 17 x 43 (mod 55). 2. Prove that n 21 n 3 (mod 21) for every integer n . 3. If p = 251, q = 281 and e = 73, find the associated RSA public and private keys. 4. (a) In an RSA scheme, the public key ( e, n ) = (23 , 19837) and the private key is ( d, n ) = (17819 , 19837). Using the Square and Multiply algorithm, encrypt the message M = 1975 using the appropriate key. (b) In an RSA scheme, the the public key is ( e, n ) = (801 , 1189), the private key is ( d, n ) = (481 , 1189). Note that 41 is a factor of 1189. Using the Chinese Remainder Theorem, decrypt the ciphertext C = 567 using the appropriate key. 5. Complete the following steps using MAPLE: Using the command restart , clear MAPLE’s memory. Using the command rand , create two random integers R1 and R2 each less than 10 50 . Using the command nextprime , find the next prime larger than each of R1 and R2 . (Call these p and q . Use the command isprime to verify that each is prime.) Calculate n and φ ( n ), calling them n and phi . Define e to be your UWID number. Using the command gcd , check if the gcd of e and phi is 1. If it is, go to the next step. If it is not, repeat this process to find the first value of e larger than your UWID that gives a gcd of 1 with phi . Using the command msolve , determine the value of d , part of the private key. (After using msolve , you will have to define d by copying and pasting your result.) Define M to be 2625242322212019181716151401020304050607080910111213 . Using the &^ command, encrypt M using e and n , to get C . Decrypt C using d and n . Print your output and hand it in with your Assignment.

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6. Complete the following steps using MAPLE: Using the command restart , clear MAPLE’s memory. Define n to be 28389368948326183561923542922514268376927952930167092496012482257483771333584 99454468046751209461074038548632181358620800764412077 (You may find it helpful to copy and paste the value of n available in the accompanying text file instead of trying to (correctly!) retype n .) Define e to be 123578176234876198276491234876148973 and d to be 14120514684168224392251360605222273900386543069562554506259906955949044365472 03725494350063143836462319733064004787101733886580301 Suppose your public key is ( e, n ) and your private key is ( d, n ) and that you receive the message C , which is 80128892902547886808096146110599124584248309969798722917949036154729920471947 1173555177025839977041228416176205654413823029580356 Decrypt the message using the appropriate key. Using the correspondence A 01 , B 02 , etc., write the message in English. (Note the order of letters in the English alphabet is ABCDEFGHIJKLMNOPQRSTUVWXYZ . You will probably find it easiest to do this last step by hand.) Print your output and hand it in with your Assignment.
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MATH135 - Assignment8 Solutions - MATH 135 Assignment#8...

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