crypto6_new

crypto6_new - IS 5371 Cryptography CIS 5371 Cryptography 6....

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Unformatted text preview: IS 5371 Cryptography CIS 5371 Cryptography 6. An Introduction to Number Theory 1 Congruence and Residue classes Arithmetic modulo n, Z n Solving linear equations The Chinese Remainder Theorem Eulers phi function The theorems of Fermat and Euler Quadratic residues Legendre & Jacobi symbols 2 Arithmetic modulo n Examples Z ={0,1,2, , n-1}, n { , , , , }, Z p * = {1,2, , p-1}, for prime p , = {all integers < ith cd n = 1} Z n * = {all integers k, 0 < k < n, with gcd ( k,n ) = 1}. 3 Solving linear equations heorem Theorem For any integer n > 1 , ax b (mod n ) is solvable, if and only if, gcd( a,n ) | b . Examples 6 x 18 (mod 36) has 6 solutions: 3, 9, 15, 21, 27, 33. 2 x 5 (mod 10), has no solutions. 4 The Chinese Remainder Theorem . let and , 1 ) , gcd( with integers positive be , , Let 1 j i n m m m M m m m m ) (mod congruence Then the 1 1 2 1 r m c x ) (mod 2 2 m c x . solution a has ) (mod M r r Z z unique m c x 5 The Chinese Remainder Theorem...
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This note was uploaded on 12/11/2011 for the course CIS 5371 taught by Professor Mascagni during the Fall '11 term at FSU.

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crypto6_new - IS 5371 Cryptography CIS 5371 Cryptography 6....

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