Solovay–Strassen primality test
The Solovay -Strassen test ( according to Robert M. Solovay and Volker Strassen ) is a probabilistic primality test. The test checks for an odd number n, whether it is prime or composite. In the latter case, the test, however, is not a factor of the number n generally gives
The Solovay -Strassen test is how the Miller -Rabin test, a Monte Carlo algorithm. That is, it only provides a certain probability ( 50%) of a statement. By repeating this probability can also be enlarged. If the test ( repeated ) are not valid, so this can be as " n is probably prime " interpret.
Method
For an odd number n, which is to test for primality, a natural number a is chosen with 1
To evaluate the quality of the Solovay - Strassen test, must be distinguished whether n is prime or not.
To increase the power of the test for non- prime, the test with independently chosen random bases is repeated a sufficient number of times.
If the test is repeated k times, then the probability that all K iterations, the result is " no information " is (although N is not a prime number ), less than.
This is a pessimistic estimate - in most cases the quality will be much better.
The Solovay -Strassen test is efficient because the gcd, the powers and the Jacobi symbol can be computed efficiently.
The example of the composite number n = 91 ( a Fermat pseudoprime to - for example - the bases 17, 29 ) is shown a possible sequence of tests:
Is 91 a prime number?
Test 1: a = 29
Test 2: a = 17
Test 3: a = 23
Let n > 2 is an odd composite number.
Too prime number is called a false witness for the primality of respect to the Solovay - Strassen test, if
For so the bases are false witnesses.
Since the amount of false witnesses is a subgroup of the multiplicative group with order less than or is equal to (where denotes the Euler φ - function that the number of prime numbers less than indicates ) and is considered, are more than half of the available range of bases false witnesses.
Thus is achieved in runs a probability of an error (i.e., a compound number is not recognized as such ), which is smaller.
One is the Euler's theorem: For any prime p > 2
With this criterion, all numbers are screened out, which are neither prime nor Euler pseudo- prime to the base a.
The other property combines this with the Legendre symbol: For every prime number p> 2
Since one can not assume at the test numbers indicate that it is prime, one uses the Jacobi symbol.
This criterion also the Euler -Jacobi pseudoprimes fall out.
Assessment
Efficiency
Example
False witnesses