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A Monte Carlo Factoring Algorithm With Linear Storage

C. P. Schnorr and H. W. Lenstra, Jr.
Mathematics of Computation
Vol. 43, No. 167 (Jul., 1984), pp. 289-311
DOI: 10.2307/2007414
Stable URL: http://www.jstor.org/stable/2007414
Page Count: 23
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A Monte Carlo Factoring Algorithm With Linear Storage
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Abstract

We present an algorithm which will factor an integer $n$ quite efficiently if the class number $h(-n)$ is free of large prime divisors. The running time $T(n)$ (number of compositions in the class group) satisfies $\operatorname{prob}\lbrack T(m) \leqslant n^{1/2r} \rbrack \gtrsim (r - 2)^{-(r - 2)}$ for random $m \in [n/2, n]$ and $r \geqslant 2$. So far it is unpredictable which numbers will be factored fast. Running the algorithm on all discriminants $-ns$ with $s \leqslant r^r$ and $r = \sqrt{\ln n/\ln \ln n}$, every composite integer $n$ will be factored in $o(\exp \sqrt{\ln n \ln \ln n})$ bit operations. The method requires an amount of storage space which is proportional to the length of the input $n$. In our analysis we assume a lower bound on the frequency of class numbers $h(-m), m \leqslant n$, which are free of large prime divisors.

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