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Unformatted text preview: CSci 5471: Modern Cryptography, Spring 2010 Homework #2 Yongdae Kim Due: Thursday, Feb. 16, 9:00 AM Show all steps. Please ask any questions, if the problems are not clear. Total 120 points. 1. Number Theory! (30 points) (a) (7 pts) Show that ax + by = c has integer solution ( x,y ) if gcd( a,b ) | c . (b) (8 pts) Show that for n > 1 , the number 1 + 1 / 2 + 1 / 3 + ... + 1 /n is not an integer. (c) (8 pts) Show that if a n- 1 is a prime, then a = 2 and n is a prime. ( n > 1 ) (Hint) You can use one of the last homework problems as a fact. (d) (7 points) If n > 1 is an integer and is not a prime, then show that there is a prime such that p | n and p n. 2. Affine Cipher (10 pts) : As we discussed in the class, Caeser cipher is not secure. To improve the Caesar cipher, Yongdae proposed a new encryption algorithm called affine cipher defined as below: c = E a,b ( m ) am + b (mod 26) . (a) To be decryptible, what kind of condition a must satisfy? (b) Write decryption algorithm D a,b ( c ) and show why the ciphertext c will be decrypted to m . (c) Suppose you are given two (plaintext, ciphertext) pairs (i.e. ( m 1 ,c 1 ) and ( m 2 ,c 2 ) ). With these two pairs, show that how one can find two keys a,b efficiently. 3. Modes of Operation (20 points) : When there is a lot of data to encrypt, using DES ECB mode is insecure because of the block permutation attack. In other words, blocks of encrypted data can be swapped without the receiver (decryptor) noticing. Suggest a secure way to address these issues. The rules of the game are:receiver (decryptor) noticing....
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