Abstract
In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a numbern, the algorithm chooses “candidates”x 1,x 2, ...,x k uniformly and independently at random from ℤ n , and tests if any is a “witness” to the compositeness ofn. For either algorithm, the probabilty that it errs is at most 2−k.
In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen asx, x+1, ..., x+k−1, wherex is chosen uniformly at random from ℤ n . We prove that fork=[1/2log2 n], the error probability of the Miller-Rabin test is no more thann −1/2+o(1), which improves on the boundn −1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound ofn −1/2+o(1) if the number of distinct prime factors ofn iso(logn/loglogn).
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Peralta, R., Shoup, V. Primality testing with fewer random bits. Comput Complexity 3, 355–367 (1993). https://doi.org/10.1007/BF01275488
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DOI: https://doi.org/10.1007/BF01275488