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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

3. Primality tests:

3. Primality tests: Search of prime numbers (2) (notice: trying to factor is much too slow) probabilistic; “Rabin-Miller” special case p � 3 mod 4 : a p prime � � � 0 mod p : � � 1 (mod p) 1 2 p not prime � for � of a´s : a � � 1 (mod p) 4 � test this for m different, independently chosen values of a, (doesn’t matter in general) error probability � a p�1 2 p�1 1 m 4 167

Z n : ring of residue classes mod n = {0, ... , n-1} • +, -, � fast Calculating with and without p,q (1) • exponentiation “fast” (square & multiply) 26 ( 11010) 7 2 example: ; from left s 7 � 7 1 7 10 7 110 7 1100 7 11010 m s m 7 11 7 1101 s • gcd (greatest common divisor) fast in Z (Euclidean Algorithm) s 168

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