Prime Numbers

3.5 Probable primes and witnesses 137
an odd composite number n, a witness is a base for which n is not a strong
pseudoprime.
A witness for n is thus the key to a short proof that n is composite.
Theorem 3.5.4 implies that at least 3/4 of all integers in [1,n − 1] are
witnesses for n, whenn is an odd composite number. Since one can perform a
strong pseudoprime test very rapidly, it is easy to decide whether a particular
number a is a witness for n. All said, it would seem that it is quite an easy task
to produce witnesses for odd composite numbers. Indeed, it is, if one uses a
probabilistic algorithm. The following is often referred to as “the Miller–Rabin
test,”, though as one can readily see, it is Algorithm 3.5.2 done with a random
choice of the base a. (The original test in [Miller 1976] was somewhat more
complicated and was a deterministic, ERH-based test. It was M. Rabin, see
[Rabin 1976, 1980], who suggested a probabilistic algorithm as below.)
Algorithm 3.5.6 (Random compositeness test). We are given an odd number
n>3. This probabilistic algorithm attempts to find a witness for n and thus
prove that n is composite. If a is a witness, (a, YES) is returned; otherwise, (a,
NO) is returned.
1. [Choose possible witness]
Choose random integer a ∈ [2,n− 2];
Via Algorithm 3.5.2 decide whether n is a strong probable prime base a;
2. [Declaration]
if(n is a strong probable prime base a) return (a, NO);
return (a, YES);
One can see from Theorem 3.5.4 that if n>9 is an odd composite, then the
probability that Algorithm 3.5.6 fails to produce a witness for n is < 1/4. No
one is stopping us from using Algorithm 3.5.6 repeatedly. The probability that
we fail to find a witness for an odd composite number n with k (independent)
iterations of Algorithm 3.5.6 is < 1/4 k . So clearly we can make this probability
vanishingly small by choosing k large.
Algorithm 3.5.6 is a very effective method for recognizing composite
numbers. But what does it do if we try it on an odd prime? Of course it
will fail to produce a witness, since Theorem 3.5.1 asserts that primes have
no witnesses.
Suppose n is a large odd number and we don’t know whether n is prime
or composite. Say we try 20 iterations of Algorithm 3.5.6 and fail each time
to produce a witness. What should be concluded? Actually, nothing at all
can be concluded concerning whether n is prime or composite. Of course,
it is reasonable to strongly conjecture that n is prime. The probability that
20 iterations of Algorithm 3.5.6 fail to produce a witness for a given odd
composite is less than 4 −20 , which is less than one chance in a trillion. So yes,
n is most likely prime. But it has not been proved prime and in fact might
not be.