Page 1 of 2 SIT192 Summative Assessment Task 1 Trimester 2, 2017 Due Friday, 12pm, 25th August 2017 Special Instructions Part 1 - Weekly Practicals { 5% of total mark { (1% each week) The weekly practicals are about hands-on experience with using MAPLE for most of the contents we will teach in this unit. A set of problems/Maple practical exercises will be given each week. Attempt the problems/exercises, take a screen shot of your Maple worksheet as you work through them, and submit a PDF on CloudDeakin Assignment Dropbox under Weekly Practicals by Sunday 11:59PM. You can also ask the practical instructor any questions you may have with the contents of this unit, including the practical questions/exercises. CAMPUS students should attend all practicals and CLOUD students, the Bb Collaborate Sessions.SIT281 Summative Assessment Task 1 2017 Trimester 2 Page 2 of 2 Part 2 { Problem Solving Questions { (15% of total mark for the unit) Q.1) Affine Cipher (a) You are given a plaintext: \exam", encrypt it using the affine function 5x + 2 (mod26). (b) Work out the decryption formula and use it to decrypt the ciphertext obtained from Part (a). (c) Explain why the affine function 13x + 2 (mod 26) cannot be used for encryption. Q.2) Hill Cipher (a) You have found out that the exam is easy, and wanted to tell your friends. Encrypt the plaintext \easy" using the matrix M =  2 3 1 3 . (b) What is the decryption matrix M −1? Q.3) LFSR Consider the linear recursion relation xn+2 = c0xn+1 +c1xn +c2 ( mod 2) that generates the sequence: 0; 1; 1; 0; 1; : : : (a) Find the value of the coefficients c0; c1, and c2. (b) Find the next 3 elements of the sequence, and determine the period of the recursion relation. Q.4) Extended Euclidean Algorithm Apply the Extended Euclidean Algorithm to: (a) find the value of gcd(242; 129); and (b) solve 242x + 129y = gcd(242; 129) for integers x and y. Q.5) Chinese Remainder Theorem and Euler’s Theorem (a) Given that x = 5 (mod 13) and x ≡ 7 (mod 11), solve for x by hand using the Chinese Remainder Theorem. (b) Now use Maple to solve for x, if x = 1 (mod 19) and x ≡ 17 (mod 31) (c) The Euler’s Theorem states that: If gcd(a; n) = 1, then aφ(n) ≡ 1 mod n. Use it to find, by hand, the value of: 11722 mod 225: Q.6) Modulus Arithmetic (i) Express 113 in binary, using the method for base conversion you have learned in SIT192. (ii) Hence express 3113 as a product of powers of 3 from the set (31, 32, 34, 38, 316, 332, 364). (iii) Determine each of 31, 32, 34, 38, 316, 332, 364( mod 217), using only the +, −, ×, ÷ functions on a calculator. Show all intermediate steps. (iv) Hence determine 3113( mod 217), again using only the +, −, ×, ÷ functions on a calculator. Show all intermediate steps. (v) Confirm your final answer using Wolfram Alpha. Show your query given to Wolfram Alpha.