Exercise 1.28. 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 , co
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