1/18 2/16 3/20 4/17 5/6 6/9 7/14 % Please do not write in the spaces above.

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1 1/18 2/16 3/20 4/17 5/6 6/9 7/14 % 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! J MATH Fall 2015 Dr. Morton Name: Exam III

2 1. (18 points) Short Answer: a. For a one-time pad, there are three rules that the key must satisfy. What are the three rules? The key must be: b. What is the difference between decryption and breaking a cipher? (In order to answer, explain what each one is.) Also make sure you tell me if it is Alice/Bob who are doing the work or Eve. (3-5 sentences) c. In Mr. Adamson s story, the furniture movements were given as E = R S T 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 Find G+A, showing your work., J = R S T S T R, d. Use the division algorithm with b= and m=2345 to find q and r. (No work is needed here just give the answer). Recall that we want b=mq+r. e. What was the name of one Japanese code during WWII? f. What does the name Oscar stand for? g. What does the name Mallory stand for? h. How do Oscar and Mallory differ from each other? (What does each one do? 2-3 sentences)

3 2. (16 points) True or False? (Circle the correct answer.) If false, tell me why it is false. Be as specific as possible. a) There is exactly one completely unbreakable code/cipher as of today: The One Time Pad. True False b) Zipping up a coat (to go outside) is an example of a one-way function. True False c) Steganography means to hide the existence of the message (for example, using a microdot or invisible ink). True False d) Alice s Adventures in Wonderland were all part of Alice s dream. True False e) All United States codes during WWII were broken at some point. True False f) Alan Turing, the father of modern computers, was heralded as a hero throughout England due to his work on breaking the Enigma machine. True False g) To break a random cipher, statistical analysis is used. True False h) Rejewski, Rozycki, and Zygalski spied for Poland during WWII. True False

4 3. (20 points) Recall that there is a standard way of associating the alphabet with integers < 26: 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 a. Encrypt me using the AtBash cipher. (No work is needed here.) b. Decrypt NHUUJ using the affine cipher x = (23y + 25)mod26, showing all work (be sure to be clear about what you are doing) and using the correct notation. c. Decrypt KIBRIL which was encrypted using a Vigenere cipher with keyword set, showing all work (be sure to be clear about what you are doing) and using the correct notation. d. Is Vigenere monoalphabetic or polyalphabetic? Illustrate using part c above (you only need give one illustration).

5 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 (17 points) Suppose we wanted to use the RSA algorithm with two primes, p=17 and q=29. Suppose that we also decide to choose e=5. It turns out that with these values, d=269. a. Find N and L, showing all work. b. List out the public key for this specific example (both the numerical values and the variable names). c. List out the private key for this specific example (both the numerical values and the variable names). d. Check that this value of d will work here to decrypt. (Give the formula you used.) e. Encrypt the plaintext letter s, showing all work. Give all formulas used. f. Give the formula (including all values) that you would use to decrypt 228, provided that you had a calculator large enough. Give all formulas used. (In other words, what would you plug into your calculator?) g. This example is not how we would realistically use RSA in today s world. Give one reason why it is not.

6 5.(6 points) We want to use a Playfair cipher: a. Fill in the 5x5 matrix below for the key star trek as the keyword: b. Separate the plaintext belittle as needed in order to encrypt. Then encrypt, being careful about notation. 6. (9 points) a. Suppose that you have an affine cipher of the form y = (ax + b)mod20. How many possible usable keys are there? Explain in detail how you arrived at your answer. b. Suppose that you have a random monoalphabetic cipher on a 55 letter alphabet. How many possible usable keys are there (this number is probably too big for your calculator, but tell me what you would plug in to your calculator).

7 7. (14 points) a. What would we see if a hypersphere came through our world? (As time progresses, what would we see, in detail?) b. Suppose we have a 290-dimensional simplex. How many 0-faces (i.e. vertices) are the in the 290-dimensional simplex? How many 1-faces (i.e. lines) are there in the 290-dimensional simplex? Show your work. How would you build a 291-dimensional simplex from this existing 290-dimensional simplex? c. A cube is going through Flatland, parallel to an edge (so not vertex first, and not side first). What would the Flatlanders perceive as the cube goes through their world (in detail)?