generating functions

DMTH237 Assignment 5 DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE Please sign the declaration below, and staple this sheet to the front of your solutions. Your assignment must be submitted at the Science Centre, E7A Level 1. Your assignment must be STAPLED, please do not put it in a plastic sleeve. PLAGIARISM Plagiarism involves using the work of another person and presenting it as one’s own. For this assignment, the following acts constitute plagiarism: a) Copying or summarizing another person’s work. b) Where there was collaborative preparatory work, submitting substantially the same final version of any material as another student. Encouraging or assisting another person to commit plagiarism is a form of improper collusion and may attract the same penalties. STATEMENT TO BE SIGNED BY STUDENT 1. I have read the definition of plagiarism that appears above. 2. In my assignment I have carefully acknowledged the source of any material which is not my own work. 3. I am aware that the penalties for plagiarism can be very severe. 4. If I have discussed the assignment with another student, I have written the solutions independently. SIGNATURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Use generating functions to solve these recurrences. (a) an − 5 an−1 + 6 an−2 = 5 × 3 n with a0 = 1 and a1 = 2 . (b) an − 2 an−1 = 3 n − 4 with a0 = 2. 2. Find the chromatic polynomial of each of these graphs. • • • •• • • • • • • • • • • • • • • • • • • (a) (b) (c) First downloaded: 1/6/2014 at 5:4::1 1 3. You can do the following sub-questions without a calculator. (a) Find the inverse of 135 in Z223. (b) Use Fermat’s theorem to find the remainder when 2971 is divided by 31. [Hint: rewrite 35 as a negative number modulo 37.] (c) Use Euler’s theorem to find the remainder when 114805 is divided by 48. (d) Find the remainder when 40! is divided by 43. State any theorems that you use. 4. In an RSA Public Key system, Eve’s modulus is 862093, which is the product of two primes, and her encoding key is 137. Bob sends Eve a message, represented by the number 25. (a) What is the encoded message? (b) Eve’s decoding key is 288833. Use this information to find the two prime factors of 862093. (c) Hence find ϕ(862093). (d) This particular RSA system has not been set up securely, so that Alice has the same modulus as Eve. If Alice’s public encoding key is 17, use Eve’s information to find Alice’s secret decoding key. (e) Eve intercepts a message being sent to Alice, as the number 242435. What number represents the original content of this message? 5. [Optional] Your task, should you accept it, is to decode the personalised message represented by the following set of numbers, resulting from an RSA encoding as described below. You will know that you have successfully decoded the message, since your own student ID appears at some place within the message surrounded by a pair of colons (:), starting at character k say. Your answer should consist of the text of the message, along with the number k at which character the colon preceding the student ID appears, as well as a description of how you obtained it. You are free to use any software that you like in attempting to decode your message. Your personalised encoded message is as follows. 84224, 43981, 204780, 272905, 314496, 25172, 109841, 29543, 187027, 229841, 245073, 278439, 243226 Messages are built using the 40 characters comprising digits, capital letters, space character and some punctuation as shown in the following ordered list — the space character is at position 11 (shown in quotes), with ‘A’ at position 12. 0 1 2 3 4 5 6 7 8 9 ‘ ’ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z , : . Let ind(c) denote the position of character c within the above list — starting at 1, so ind(0) = 1. Three characters are combined to form a single number using base 41; e.g. ‘ABC’ corresponds to 412 × ind(A) + 41 × ind(B) + ind(C) . Or is it ind(A) + 41 × ind(B) + 412 × ind(C) ? (Nevermind, only one of these works correctly with your message!) The 39 characters in the message (of which 10 are devoted to the student ID) are split into groups of three successive characters, with each group converted to a number as above. These 13 numbers are then each encoded using RSA with a modulus of m = p q = 359951 (with p and q being primes) and encoding exponent of 11, to produce your given encoded message. What is the corresponding decoding exponent? You must work this out, given that {1019, 285875} is a valid encoding–decoding pair for the given modulus. Use this information, or other means, to determine φ(m) and the primes p and q. Then determine the required decoding exponent, and decode each of the 13 numbers to recover the numbers for 3-letter groups in the original message. Use base 41 to convert those numbers back into characters. Finding your student ID confirms having done everything correctly. Successful completion of this optional task is worth a bonus of 1 mark for this assignment. 2

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