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1 1/16 2/17 3/17 4/7 5/10 6/14 7/19 % Please do not write in the spaces above. Directions: You have 75 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 Fall 2014 Dr. Morton Name: Exam III
2 1. (16 points) Short Answer: a. There is exactly one completely unbreakable code/cipher as of today. What is the name of that code/cipher? b. For the code in part a above, there are three rules that the key must satisfy. What are the three rules? c. Give a real world (i.e. non-mathematical) example of a one-way function. d. What was the only code/cipher that was unbroken by the end of WWII? Which country used it? e. In which war was Enigma (mostly) used? By whom? Which country was originally responsible for breaking Enigma? f. What is the difference between decryption and breaking a cipher? (In order to answer, explain what each one is.)
3 2. (17 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. Decrypt QVI using the affine cipher x = (15y +1)mod26, showing all work (be sure to be clear about what you are doing) and using the correct notation. b. Decrypt OEZRAD which was encrypted using a Vigenere cipher with keyword wars, showing all work (be sure to be clear about what you are doing) and using the correct notation. c. Is Vigenere monoalphabetic or polyalphabetic? Illustrate using part b above (you only need give one illustration).
4 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 (teeny tiny) primes, p=7 and q=17. Suppose that we also decide to choose e=5. It turns out that with these values, d=77. 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), giving any important formulas that you might need. 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 q, showing all work. Give all formulas used. f. Give the formula (including all values) that you would use to decrypt 40, provided that you had a calculator large enough. Give all formulas used.
5 4.(7 points) We want to use a Playfair cipher: a. Fill in the 5x5 matrix below for the key snacks as the keyword: b. Split the plaintext little up as needed in order to encrypt. Then encrypt, being careful about notation. 5. (10 points) a. Suppose that you have an affine cipher of the form y = (ax + b)mod18. 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 37 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).
6 6. (14 points) a. Do the even integers have the following group properties under the operation of multiplication? Does it have closure? Explain. Does it have an identity? Explain. b. In the article, Group Theory and the Postulational Method, we know that E = R S T A = R S T B = R S T C = R S T G = R S T J = R S T R S T R T S T S R S R T T R S S T R Find G+C, showing all work. Make sure you tell me what your answer is. c. ( points) The following is a group (you don t have to check this). We will call this group 1. % A B C D E F A C D E F A B B D E F A B C a. Find the identity for group 1. C E F A B C D D F A B C D E b. What is B s inverse here? E A B C D E F F B C D E F A c. What is F s inverse here? d. Here is the table for group 1 again. Fill in the missing entry (bottom right) so that this table no longer has the property of closure. % A B C D E F A C D E F A B B D E F A B C C E F A B C D D F A B C D E E A B C D E F F B C D E F
7 7. (19 points) a. Fill in the remainder of the following table for multiplication mod 7 (no work is needed): * mod b. Using the table that you filled in in part a above, answer the following questions: What is the set here? What is the operation here? Does this table have the property of closure?? (circle one): YES NO How do you know (be precise)? Does this table have an identity? (circle one): YES NO If so, what is it? Fill in the inverses, if they exist, of each element. If no inverse exists for a certain element, write ni : Element Inverse Does this table have the associative property? (circle one): YES NO How many things would we have to check to be completely sure? Check the following: Is (3*5)*2=3*(5*2)? Show all work, step by step. Is this table a group? (circle one): YES NO Is this table Abelian? (circle one): YES NO How do you know?
1/18 2/16 3/20 4/17 5/6 6/9 7/14 % Please do not write in the spaces above.
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
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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1/19 2/25 3/8 4/23 5/24 6/11 Total/110 % Please do not write in the spaces above.
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