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 modulon,'' that is, a number not equal to 1 orn − 1whose square is equal to 1 modulon. It is possible to prove that if such a nontrivial square root of 1 exists, thennis not prime. It is also possible to prove that ifnis an odd number that is not prime, then, for at least half the numbers a < n , computinga^(n − 1)in this way will reveal a nontrivial square root of 1 modulon. (This is why the Miller-Rabin test cannot be fooled.) Modify theexpmodprocedure to signal if it discovers a nontrivial square root of 1, and use this to implement the Miller-Rabin test with a proceduremiller-rabin-testanalogous tofermat-test. Check your procedure by testing various known primes and non-primes. Hint: One convenient way to makeexpmod` signal is to have it return 0.