1/ 17 2/20 3/19 4/12 5/14 6/13 7/10 Total /105. Please do not write in the spaces above.

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1 1/ 17 2/20 3/19 4/12 5/14 6/13 7/10 Total /105 % Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before attempting any problems. You must show all work, or risk losing credit. Be sure to answer all questions asked. To receive full credit on problems, they must not only be mathematically correct, but they must also be solved using the correct notation and terminology. Good luck! MATH Version I Spring 2018 Dr. Morton Name: Exam IV My dear students,remember this: I believe in you. Believe in yourself. You CAN do this. -Dr. M

2 1. (17 points) Groups: a. In Mr. Adamson s story, the furniture movements were given as R S T E = R S T, A = R S T R T S, B = R S T T S R, C = R S T S R T, G = R S T T R S, J = R S T S T R, Start here: Find B+J, showing your work. After first movement: After second movement: Thus B+J=. b. For each of the following sentences, classify the sentence as a Statement or a Non-statement. If it is a statement, then classify it as True or False. Sentence Is this a statement? (Circle one): Yes No If it is a statement, is it (circle one): q: The Lord of the Rings was an amazing movie. Statement? Yes No r: 6+5=2 Statement? Yes No s: Alice met a March hare during her adventures in Wonderland. Statement? Yes No t: Why are you so angry? Statement? Yes No u: Paul Erdos actively researched mathematics even Statement? Yes No in his old age v: Paul Erdos never married but did have two children. Statement? Yes No c. Negate the following statements: All frogs are hogs. Negation: No students sleep. Negation: 6+5=2. Negation:

3 2. (20 points) The following is a table for the set {A, B, C, D, E, F} with operation denoted. a. Does this table exhibit the property of closure? (Circle one: ) Yes No A B C D E F A D C B A F E B F D E B C A C E A F C B D D A B C D E F E C F A E D B F B E D F A C Explain, carefully (I must be convinced that you understand what closure means here). b. Does this table exhibit the property of identity? (Circle one: ) Yes No If yes, give the identity: c. Does this table exhibit the property of inverses? (Circle one: ) Yes No Fill in the table below, writing an X if no inverse for that element is found: Element A B C D E F Inverse d. Does this table exhibit the property of associativity? (Circle one: ) Yes No First, how many total things would we need to test to be sure that this was associative? Find the following, showing all work: o A (B F)= o (A B ) F= Did the previous example work out the way it needed to? How do you know? e. Is this a group? (Circle one): Yes No f. Does this table exhibit the Abelian property? (Circle one: ) Yes No Explain if yes, and if no, give an illustrative example. g. Is it possible that the set above is isomorphic to either of the GROUPS below? Explain for each. Note: You do not have to prove that either of these sets are groups I am telling you that they are. * M N M M N N N M M N O P Q R M M N O P Q R N N O P Q R M O O P Q R M N P P Q R M N O Q Q R M N O P R R M N O P Q

4 3. (19 points) We want to use RSA to send messages. Alice chooses p=29, q=23, e=9 and d=137. a. Find the values of N and L, giving the formulas you used, and showing all work. b. List the public key for this example (give the variable names and the values). c. List the private key for this example (give the variable names and the values). d. Why is this value of e allowed here? To answer, give the condition that e must satisfy and show that this value of e does satisfy the conditions here. e. Why is this value of d allowed here? To answer, give the condition that d must satisfy and show that this value of d does satisfy the conditions here. f. Give the (general) formula for encryption in RSA. g. The message Bob wants to send to Alice is simply the message f. Bob knows that the numerical value of f is 5. Help Bob encrypt the message 5 and find the ciphertext, showing all work. h. Give the (general) formula for decryption in RSA. i. Bob encrypts a new message, and thus Alice receives the ciphertext message 660. Help Alice decrypt the message, by telling her what to enter into her calculator. Show all work, but note that these numbers are too big to find the actual answer. j. This scenario is not quite realistic. How, in reality, would Alice s key differ if she truly wanted to use RSA securely? (1-2 sentences)

5 4. (12 points) The set is {1,2,3,4,5,6,7} and the operation is multiplication mod 8. a. First, fill in the rest of the table (I have filled in some of the table for you): * mod b. Does this table have the property of closure? (Circle one): Yes No Explain how you know, being specific. c. Does this table have the property of an Identity? (Circle one): Yes No If yes, what is it? If no, explain. d. Does this table have the property of Inverses? (Circle one): Yes No Fill in the table below to answer, writing X for any element that has no inverse: Element Inverse

6 5. (14 points) True or False? (Circle the correct answer.) If false, tell me why it is false. Be as specific as possible. a. Walking across the street is an example of a one-way function. b. Paul Erdos is a disproof of G.H. Hardy s theory that mathematics is a young man s game c. The following table of the set consisting of elements {Q,R,S} with operation & displays the property of closure. & Q R S Q Q R S R R S M S S M N d. In the following table of the set consisting of elements {Q,R,S} with operation & the element R s inverse is S. & Q R S Q Q R S R R S M S S M N e. The end of Alice s Adventures in Wonderland has Alice realizing it was all a dream. f. The odd integers (recall: integers are whole numbers) have an identity under the operation of addition: It is the number zero.

7 6. (13 points) Consider the following table of the symmetries of a regular hexagon. a. Fill in the unfilled spaces in the and K columns ONLY of the table. + E R F G H I J K E R 1 R 5 R1 R2 R3 R4 5 R 2 R 3 F G H I R 4 R 5 J K b. What is the identity element here? c. What is K s inverse? d. Is this Abelian? Yes No How do you know? Note you have done enough work just in what you filled into the table above to answer.

8 7. (10 points) a. Give a real-world (non-mathematical) example of a one-way function. b. First, here are some prime factorizations that may or may not come in handy: You may use these without confirming this work, and you do not need moset of them: 9688 = 2 ) = 2 / = = 2 ) Now to the actual problem: Suppose we want to use RSA. Alice chooses p=31, q=347. Then she chooses e=12. Is this value of e allowed? Why or why not? Be specific in your answer. Suppose we still want to use RSA, and Alice again chooses p=31, q=347. Then she chooses e=49 and d=636. Is this value of d allowed? Why or why not? Be specific in your answer. c. What is 1+2? (Not a trick question; just a bonus point for you after a hard week!)