Next suppose n is composite but is not a carmichael

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Next, suppose n is composite but is not a Carmichael number. Then the theorem follows from Theorem 11.1 and the fact that L 0 n L n . Finally, suppose that n is a Carmichael number. We claim that L 0 n P n , where P n is as defined in Theorem 11.3. To prove this, let α Z * n . Then if h 0 is as defined in Theorem 11.3, we have h 0 h and s i ( α ) = [1 mod n ] for h 0 i h . Now, if in addition, α L 0 n , then we have s h 0 - 1 ( α ) = [ ± 1 mod n ], which implies α P n . That proves the claim. Thus, we have shown that L 0 n is contained in a subgroup P n of Z * n , and by Theorem 11.3, P n ( Z * n . By the same argument as in the proof of Theorem 11.1, it follows that | L 0 n | ≤ ( n - 1) / 2. 2 67
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The above result is not the best possible. In particular, one can show without too much difficulty that | L 0 n | ≤ ( n - 1) / 4. We do not present this result here. Even this result is overly pessimistic from an “average case” point of view. It turns out that for “most” odd integers n of a given length, | L 0 n | is much smaller than this. The Miller-Rabin algorithm is widely used in practice. Of course, in a practical implementation, before applying this test, one would first perform a bit of trial division, testing if n is divisible by any primes up to some small bound B . 11.3 The Distribution of Primes In this section, we discuss some facts relating to the distribution of prime numbers, and algorithmic methods for generating prime numbers. For a real number x , the function π ( x ) is defined to be the number of primes up to x . Thus, π (1) = 0, π (2) = 1, π (7 . 5) = 4, and so on. The main theorem in the theory of the distribution of primes is the following. Theorem 11.5 (Prime Number Theorem) The number of primes up to x is asymptotic to x/ log x : π ( x ) x/ log x. A proof of the Prime Number Theorem is beyond the scope of these notes. However, one consequence of the Prime Number Theorem is that a random k -bit number (i.e., a number chosen at random from the interval { 2 k - 1 , . . . , 2 k - 1 } ) is prime with probability Θ(1 /k ). This fact suggests the following “generate and test” algorithm for generating a random k -bit prime: choose a k -bit number at random, test it for primality, and repeat until a number is found that passes the primality test. We leave it to the reader to verify the following assertions regarding this algorithm: The expected number of iterations of this algorithm is O ( k ). If we use a probabilistic primality test, such as the one in the previous section, that may erroneously report that composite number is prime with probability , the probability that this prime-generating algorithm erroneously outputs a composite number is O ( k ) (and not, in general, O ( )). If we use as our primality test the Miller-Rabin algorithm with a given error parameter t , then we have proven that 2 - t , and in fact (although we did not prove it here) 4 - t . However, as we already mentioned, these results are quite pessimistic, and in fact, the above prime-generating algorithm errs with much smaller probability, so that for t = 1 and sufficiently large k , the error probability is acceptably small for most practical purposes.
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