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# 34 - Chinese Remainder Introduction Theorem Discrete...

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Introduction Chinese Remainder Theorem Discrete Mathematics Andrei Bulatov

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Discrete Mathematics – Chinese Remainder Theorem 34-2 Previous Lecture Residues and arithmetic operations Caesar cipher Pseudorandom generators Divisors of zero Inverse
Discrete Mathematics – Chinese Remainder Theorem 34-3 Linear Congruences A congruence of the form ax b (mod m) where m is a positive integer, a and b are integers, and x is a variable, is called a linear congruence . We will solve linear congruences If a is relatively prime with m, then it has the inverse . Then ax b (mod m) x b (mod m) Find the inverse of 3 modulo 7 Solve the linear congruence 3x 4 (mod m) 1 a - 1 a - 1 a - 1 a -

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Discrete Mathematics – Chinese Remainder Theorem 34-4 The Chinese Remainder Theorem A linear congruence is similar to a single linear equation. What about systems of equations (Sun Tzu’s puzzle, 400 – 460 BC): “There are certain things whose number is unknown. When divided by 3, the remainder is 2;
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34 - Chinese Remainder Introduction Theorem Discrete...

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