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Unformatted text preview: CSci 5471: Modern Cryptography, Fall 2010 Homework #3 Yongdae Kim – Due: Mar. 2 9:00 AM – Show all steps. Please ask any questions, if the problems are not clear. – Send me your solution by e-mail. – Total 100 + 40 points. 1. Number Theory! (20 points) In the following problems, a,b,c,m,x,y are all integers and n is a positive integer. (a) (3 pts) Prove or disprove: if ax ≡ ay (mod n ) , then x ≡ y (mod n ) . (b) (3 pts) Prove or disprove: if a | c and b | c , then ab | c . (c) (3 pts) Prove or disprove: If d = gcd( a,b ) , then gcd( ma,mb ) = md . (d) (3 pts) Prove or disprove: a | b implies a ≤ b . (e) (3 pts) Let a,b be distinct integers. Show that there exists an infinite number of integers x satisfying gcd( a + x,b + x ) = 1 . (f) (5 pts) Show that n 2 + 23 is divisible by 24 for infinitely many n . 2. DES (10 points) : Let DES K ( m ) represent the encryption of plaintext m with key K using the DES cryptosystem. Suppose y = DES K ( m ) and y = DES c ( K ) ( c ( m )) , where c ( · ) denotes the bitwise complements of its argument. Prove that y = c ( y ) 3. Doubling the domain (30 points) DES is based on the Feistel network, and it can be considered as a method for doubling the input/output size of a random permutation. Consider the following, which is another such a method:doubling the input/output size of a random permutation....
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