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Old 2007-05-09, 16:24   #1
roger's Avatar
Oct 2006

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Default Probability of n-digit factor?

I am interested in what the probability of a n-digit factor is in a random x-digit composite.

The reason I ask is, I am trying to factor a C120 and a C160. (For both, I am using Alpertrons ECM applet.) For the C160, I don't have a single factor, and have sieved to (so far) curve 437 (greater than 25 digits, 55% likely greater than 30 digits). So far, the largest factor (not the result, but smaller prime factor) is a P43. P20's, P25's, and P30's are pretty common.

If it matters, it is a home prime sequence, but not one anyone is working on (a 18-digit starting number); and this is just a small hobby, nothing of mathematical importance

Thanks for anyones help!

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Old 2007-05-09, 16:45   #2
R.D. Silverman
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Nov 2003

746010 Posts

Originally Posted by roger View Post
I am interested in what the probability of a n-digit factor is in a random x-digit composite.
I answer this question (and others) in my joint paper with Sam Wagstaff Jr.
A Practical Analysis of the Elliptic Curve Factoring Algorthm.

As n -->oo, the probability that n has a factor between a and a^(1+e)
is e/(e+1). This applies uniformly for a^(1+e) < sqrt(n).

If the factor is greater than sqrt(n), then it must be the largest factor and
its distribution is given by Dickman's rho function. See Knuth, vol. II.
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Old 2007-05-09, 17:25   #3
akruppa's Avatar
Aug 2002

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Lets assume your "random" number N is chosen uniformly from large enough interval [1, B] so that we can estimate the probability that a prime p divides it as 1/p. Now you want the probability that there is at least one p in [10[I]n[/I]-1, 10[I]n[/I]-1] that divides N. This is messy to compute, lets look at the expected number of such primes instead, \sum_{p\in P, 10^{n-1} \leq p \leq 10^n-1} 1/p. If this expected value is much smaller than 1, it is quite close to the probability. Now \lim_{k \rightarrow \infty}\sum_{p\in P, p \leq k} 1/p - \log \log(k) = M for some constant M, so we can take the sum over the interval as roughly like log log (10[I]n[/I]) - log log (10[I]n[/I]-1) = log(n / (n-1)). If n/(n-1) is close to 1, you can approximate log(n / (n-1)) by 1/n, or better 1/(n-0.5).

However, this assumes that N was taken from all the integers not exceeding B. If you have done a PRP test already and it came out negative, you know that N is not any of the primes below B. A prime p divides a prime q iff p=q, so we should remove all the primes from the interval [1, B] except the ones in [10[I]n[/I]-1, 10^[I]n[/I]-1]. Since I assume 10[I]n[/I] << B, we simply remove all of the primes below B instead.

So the set of numbers you chose your random number from has cardinality about B-B/log(B), the set of non-prime numbers not exceeding B that have a prime factor in [10[I]n[/I]-1, 10[I]n[/I]-1] has size about B*(log(n /(n-1))) and if you number was chosen uniformly from all the non-primes ≤ B, the probability becomes

log(n / (n-1)) / (1 - 1/log(B))

Everything changes again if you did trial division/ECM/whatever on your number. You can use a Bayesian model to account for unsuccessful factoring attempts. Silverman and Wagstaff "Practical Analysis of the Elliptic Curve Factoring Algorithm" treats this in detail.


Last fiddled with by akruppa on 2007-05-09 at 23:09 Reason: small clarification
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Old 2007-05-09, 22:51   #4
roger's Avatar
Oct 2006

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Wow, information dump!

Thanks for the explanation, Akruppa; both of your posts pointed to Wagstaff/Silverman's paper, so I'll have a look.

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