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Unformatted text preview: Introduction Public Key Cryptography Discrete Mathematics Andrei Bulatov Discrete Mathematics Public Key Cryptography 35-2 Previous Lecture Linear congruences Chinese Remainder Theorem Discrete Mathematics Public Key Cryptography 35-3 Fermats Theorem Fermats Great (Last) Theorem . For any n > 2, the equation does not have integer solutions x,y,z > 0 It had remained unproven for 358 years (posed in 1637, proved in 1995) Andrew Wiles proved it in 1995 n n n z y x = + Discrete Mathematics Public Key Cryptography 35-4 Fermats Little Theorem Fermats Little Theorem. If p is prime and a is an integer not divisible by p, then Clearly, it suffices to consider only residues modulo p. p) (mod 1 a 1 p - 5 Z 1 2 3 4 1 2 3 4 1 2 3 4 2 4 1 3 3 1 4 2 4 3 2 1 Discrete Mathematics Public Key Cryptography 35-5 Fermats Little Theorem (cntd) Fermats Little Theorem was improved by Euler Fermats Little Theorem improved For any integers m and a such that they are relatively prime where (m) denotes the Euler totient function, the number of numbers 0 < k < m relatively prime with m Example: m) (mod 1 a (m) 8 Z Discrete Mathematics Public Key Cryptography 35-6...
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- Spring '10