Miller-Rabin test

One variant of the Fermat test that cannot be fooled is called the Miller-Rabin test (Miller 1976; Rabin 1980). This starts from an alternate form of Fermat's Little Theorem, which states that if n is a prime number and a is any positive integer less than n , then a raised to the ( n − 1 )st power is congruent to 1 modulo n . To test the primality of a number n by the Miller-Rabin test, we pick a random number a < n and raise a to the ( n − 1 )st power modulo n using the expmod procedure. However, whenever we perform the squaring step in expmod , we check to see if we have discovered a ``nontrivial square root of 1 modulo n ,'' that is, a number not equal to 1 or n − 1 whose square is equal to 1 modulo n . It is possible to prove that if such a nontrivial square root of 1 exists, then n is not prime. It is also possible to prove that if n is an odd number that is not prime, then, for at least half the numbers a < n , computing a^(n − 1) in this way will reveal a nontrivial square root of 1 modulo n . (This is why the Miller-Rabin test cannot be fooled.) Modify the expmod procedure to signal if it discovers a nontrivial square root of 1, and use this to implement the Miller-Rabin test with a procedure miller-rabin-test analogous to fermat-test . Check your procedure by testing various known primes and non-primes. Hint: One convenient way to make expmod signal is to have it return 0.

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