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PrimTest-notes - Handout #18 ma187s: Cryptography April 25,...

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Handout #18 ma187s: Cryptography April 25, 2006 PRIMALITY TESTING 1 Quadratic Residues In solving congruence equations of higher degree the following result of Lagrange is basic Theorem 1.1 For a prime p the equation P ( x )= a 0 + a 1 x + a 2 x 2 + ··· + a n x n = 0 (mod p ) (1.1) has at most n solutions. Proof. The result is clear if n = 1. Thus we shall proceed by induction and assume that our assertion is true for polynomial equations of degree n - 1 or less. This given, note that for any b whatever we can write P ( x ) - P ( b a 1 ( x - b )+ a 2 ( x 2 - b 2 a 3 ( x 3 - b 3 + a n ( x n - b n ) Factoring out x - b we get P ( x ) - P ( b )=( x - b ) Q ( x ) (1.2) where Q ( x a 1 + a 2 ( x + b a 3 ( x 2 + xb + b 2 + a n ( x n - 1 + x n - 2 b + + b n - 1 ) Assume then that a and b are both solutions of (1.1). Setting x = a in (1.2) gives ( a - b ) Q ( a P ( a ) - P ( b )=0 - 0 = 0 (mod p ) (1.3) Now if a is not equal to b (mod p ), that is a - b is not divisible by p , then (1.3) implies that Q ( a (mod p ). In other words, except for at most one solution (say b ) of the equation P ( x ) = 0, all the others are solutions of Q ( x ) = 0. However, Q ( x ) is a polynomial of degree n - 1 at most and by the induction hypothesis the equation Q ( x ) = 0 can have no more that n - 1 solutions. Thus (1.1) itself can have no more the n distinct solutions altogether. This completes the induction argument and the proof of the theorem. Remark. It is customary to call the solutions of a polynomial equation P ( x ) = 0 the roots of the polynomial P ( x ). Thus Lagrange’s theorem may be rephrased by saying that any polynomial of degree n has no more than n roots mod p . Note that these congruence equations may have no solution at all. For instance we can find no x such that x 2 = 2 (mod 5) nor can we solve x 2 = 8 (mod 11) (1.4) This can be easily checked for (mod 5) we have 1
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Handout #18 ma187s: Cryptography April 25, 2006 1 2 =1,2 2 =4,3 2 =4,4 2 =1 a similar reasoning gives that (1.4) has no solution. Taking this into account we shall say that an integer a is a quadratic residue mod p if and only if the equation x 2 - a = 0 (mod p ) has a solution x .
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PrimTest-notes - Handout #18 ma187s: Cryptography April 25,...

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