hw9sol - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 9 Due at the beginning of class on Wednesday, November 28, 2007 Problem 1 Consider the following computational problem: Given N, a, b, x, y , where N 1 is prime, a, b, x, y are integers between 1 and N - 1, compute a x b y (mod N ). 1. Describe an algorithm that takes at most 4 k + C multiplications, where k = log N and C is a positive constant. 2. Design an algorithm that uses at most 2 k + C multiplications, for some constant C . Solution sketch: The naive algorithm computes a x (mod N ) then b y (mod N ) and then multiplies them modulo N . This takes roughly 4 k multiplications, using repeated exponentiation. To halve that number, let x k - 1 · · · x 1 x 0 and y k - 1 · · · y 1 y 0 be the binary representations of x and y . Set c = ab (mod N ) and z = 1. Then repeat for i = k - 1 down to 0: if ( x i , y i ) = (1 , 1) then set z to z 2 .c (mod N ); if ( x i , y i ) = (1 , 0) then set z to z 2 .a (mod N ); if ( x i , y i ) = (0 , 1) then set z to z 2 .b (mod N ); if ( x i , y i ) = (0 , 0) then set z to z 2 (mod N ). Problem 2 A common form of Fermat’s theorem is: a p = a (mod p ), for any prime p and integer a . Prove this by induction on a . (Hint: prove that ( a + b ) p = a p + b p modulo p ). Solution sketch: 1. ( p
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  • Fall '07
  • CharikarandChazelle
  • Exponentiation, Natural number, Prime number, Solution sketch

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