Final Exam Math 105: Topics in Mathematics Cryptology, the Science of Secret Writing Rhodes College Tuesday, 30 April :30 11:00 a.m.
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1 Final Exam Math 10: Topics in Mathematics Cryptology, the Science of Secret Writing Rhodes College Tuesday, 0 April 2002 :0 11:00 a.m. Instructions: Please be as neat as possible (use a pencil), and show your work. A correct answer with no work shown may not receive full credit. Start each numbered question on a new page. Unless otherwise indicated, use your calculator only to do basic arithmetic and function evaluation. Turn in your test sheets along with those containing your work. 1. (1 points) For each of the following indicate whether the statement is true or false. (a) If the greatest common divisor of integers a and b is 1, then either a or b is prime. (b) A scytale is a device that implements a type of polyalphabetic substitution. (c) A principle of cryptography is that the security of a ciphersystem resides in the key information. (d) In a substitution cipher, the letters in the plaintext message are rearranged. (e) If Alice and Bob have exchanged a great deal of Vigenère key data and they use this method, alway using a key as long as the plaintext and never re-using a key, then Eve, attempting a ciphertext-only attack and knowing that the underlying plaintext is English, will do no better than guessing at the plaintext. (f) If an English plaintext is enciphered using the Vigenère method with a long random keyword, then the index of coincidence of the resulting ciphertext will be close to 1/2. (g) A -bit linear shift register generates a sequence of 0 s and 1 s that never repeats. (h) In any sample of ordinary English text, the letter E will be the most frequent.
2 (i) If an efficient method for factoring very large numbers is found, then RSA would no longer be a secure means of encryption. (j) The number with base twenty-six representation EXAM is. (k) A principle of cryptography is that the security of a cipher system rests on keeping the method of encipherment secret. (l) (mod 09). (m) RSA has been proven unbreakable. (n) Public key cryptosystems tend to run on computers more slowly than secret-key systems. 2. ( points ) Decipher NOCJUT, which was enciphered with the affine encipherment function E(x) =(21x + 1)mod 2.. ( points) Decipher [ FCUWP, ] which was enciphered using the Hill cipher 11 1 matrix A = ( points ) Decipher TETST WTUEA HTEOU RTRRA OBTON U, which was enciphered with a keyword columnar transposition using the keyword URGENT.. ( points) Decipher FONIIOSJRJ which was enciphered using the Vigenère method with keyword NUMBER.. ( points) Decipher , which was enciphered using the binary Vigenère method with the key stream generated by the FSR b b + b 1 mod 2 with initial bit values b b b b 2 b 1 = (9 points ) (a) Find the greatest common divisor of 91 and 2. (b) The number 109 is a product of two distinct primes. At most how many trial divisions by primes will be required to find these primes (see the prime table)? (c) Use the corollary to Fermat s little theorem to help to compute 1 mod 9.. (10 points) Suppose that Alicia is implementing RSA with primes p =, q = 1, and public exponent e = 1.
3 (a) Explain what she does to set up for receiving encrypted messages and calculate all of the numbers that she will use with these choices of p, q, and e. (b) If Roberto wants to send Alicia the message x = 2encrypted using her public key, determine the ciphertext he produces. (c) Suppose Alicia receives the encrypted message y =. Write down the expression she will need to evaluate in order to decrypt. (Do not actually evaluate this expression.) 9. (9 points) Explain briefly but completely what is involved in a cryptanalysis of the Vigenere cipher. Mention the various techniques we have used. 10. (9 points) Shown here are letter distributions for three different ciphertexts, one from a monoalphabetic substitution, one from a polyalphabetic substitution, and one from a transposition. (i) (ii) (iii) A B C 9 D E F G H I J 1 A B C D E F G H I J A B C D E F G H I J M N 0 O 1 P Q R S T U V W X M N O P Q R S T U V W X M N O P Q R S T U V W X (a) Indicate which distribution corresponds to which cipher. Explain. (b) For the polyalphabetic one, compute the index of coincidence. (c) Assuming encipherment by the Vigenère method with a keyword, use the Friedman test to estimate the keyword length. 11. (9 points) Alice and Bob are going to use the Diffie-Hellman key agreement protocol. If the public prime modulus is p = and the public base is s =,Alice generates the number a = 10, and Bob generates b = 1. Determine the key on which they agree. x x 1 (mod 2)
4 A B C D E F G H I J M N O P Q R S T U V W X A B C D E F G H I J M N O P Q R S T U V W X A B C D E F G H I J M N O P Q R S T U V W X B C D E F G H I J M N O P Q R S T U V W X A C D E F G H I J M N O P Q R S T U V W X A B D E F G H I J M N O P Q R S T U V W X A B C E F G H I J M N O P Q R S T U V W X A B C D F G H I J M N O P Q R S T U V W X A B C D E G H I J M N O P Q R S T U V W X A B C D E F H I J M N O P Q R S T U V W X A B C D E F G I J M N O P Q R S T U V W X A B C D E F G H J M N O P Q R S T U V W X A B C D E F G H I M N O P Q R S T U V W X A B C D E F G H I J M N O P Q R S T U V W X A B C D E F G H I J M N O P Q R S T U V W X A B C D E F G H I J N O P Q R S T U V W X A B C D E F G H I J M O P Q R S T U V W X A B C D E F G H I J M N P Q R S T U V W X A B C D E F G H I J M N O Q R S T U V W X A B C D E F G H I J M N O P R S T U V W X A B C D E F G H I J M N O P Q S T U V W X A B C D E F G H I J M N O P Q R T U V W X A B C D E F G H I J M N O P Q R S U V W X A B C D E F G H I J M N O P Q R S T V W X A B C D E F G H I J M N O P Q R S T U W X A B C D E F G H I J M N O P Q R S T U V X A B C D E F G H I J M N O P Q R S T U V W A B C D E F G H I J M N O P Q R S T U V W X A B C D E F G H I J M N O P Q R S T U V W X
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