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A Monte Carlo method for factorization. (English) Zbl 0312.10006
The following simple method will find small prime factors $$p$$ of a number $$n$$ in $$O(p^{1/2})$$ arithmetical operations as opposed to the $$O(p)$$ operations required by trial division. Let $$x_0=2$$, $$x_{i+1}\equiv x_i^2-1\pmod n$$ (other similar sequences may be used). Generate in turn the pairs $$(x_j,x_{2j})$$, accumulating the product (mod $$n$$) of the numbers $$x_{2j}.x_j$$. At intervals, compute the g.c.d. of this product and $$n$$, which may give a factor, not necessarily prime. Probabilistic arguments predict that on average one should take about $$1.03\,p^{1/2}$$ steps to find a prime $$p$$; this is confirmed by experiment. The method has been further discussed by R. K. Guy [Proc. 5th Manitoba Conf. numer. Math., Winnipeg 1975, 49–89 (1976; Zbl 0338.10001)].

##### MSC:
 11Y05 Factorization 11Y11 Primality 65C05 Monte Carlo methods
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##### References:
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