Factoring an integer using NFS has 3 main steps:
1. Select Polynomial
2. Collect Relations via Sieving (NFS@Home is dedicated to this step)
3. Combine Relations
1. Polynomial Selection
Step 1 of NFS involves choosing a polynomial-pair (customarily shortened to 'a polynomial') to use in the other NFS phases. The polynomial is completely specific to the number you need factored, and there is an effectively infinite supply of polynomials that will work. The quality of the polynomial you select has a dramatic effect on the sieving time; a *good* polynomial can make the sieving proceed two or three times faster compared to an average polynomial. So you really want a *good* polynomial, and for large problems should be prepared to spend a fair amount of time looking for one.
Just how long is too long, and exactly how you should look for good polynomials, is currently an active research area. The approximate consensus is that you should spend maybe 3-5% of the anticipated sieving time looking for a good polynomial.
We measure the goodness of a polynomial primarily by its Murphy E score; this is the probability, averaged across all the possible relations we could encounter during the sieving, that an 'average' relation will be useful for us. This is usually a very small number, and the E score to expect goes down as the number to be factored becomes larger. A larger E score is better.
Besides the E score, the other customary measure of polynomial goodness is the 'alpha score', an approximate measure of how much of an average relation is easily 'divided out' by dividing by small primes. The E score computation requires that we know the approximate alpha value, but alpha is also of independent interest. Good alpha values are negative, and negative alpha with large aboslute value is better. Both E and alpha were first formalized in Murphy's wonderful dissertation on NFS polynomial selection.
With that in mind, here's an example polynomial for a 100-digit input of no great significance:
skew 7872388.07, size 9.334881e-014, alpha -5.410475, combined = 1.161232e-008
As mentioned, this 'polynomial' is actually a pair of polynomials, the Rational polynomial R1 * x + R0 and the 4-th degree Algebraic polynomial
A4 * x^4 + A3 * x^3 + A2 * x^2 + A1 * x + A0
The algebraic polynomial is of degree 4, 5, or 6 depending on the size of the input. The 'combined' score is the Murphy E value for this polynomial, and is pretty good in this case. The other thing to note about this polynomial-pair is that the leading algebraic coefficient is very small, and each other coefficient looks like it's a fixed factor larger than the next higher- degree coefficient. That's because the algebraic polynomial expects the sieving region to be 'skewed' by a factor equal to the reported skew above.
The polynomial selection determined that the 'average size' of relations drawn from the sieving region is smallest when the region is 'short and wide' by a factor given by the skew. The big advantage to skewing the polynomial is that it allows the low-order algebraic coefficients to be large, which in turn allows choosing them to optimize the alpha value. The modern algorithms for selecting NFS polynomials are optimized to work when the skew is very large.
NFS polynomial selection is divided into two stages. Stage 1 chooses the leading algebraic coefficient and tries to find the two rational polynomial coefficients so that the top three algebraic coefficients are small. Because stage 1 doesn't try to find the entire algebraic polynomial, it can use powerful sieving techniques to speed up this portion of the search. When stage 1 finds a 'hit', composed of the rational and the leading algebraic polynomial coefficient, Stage 2 then finds the complete polynomial pair and tries to optimize both the alpha and E values. A single stage 1 hit can generate many complete polynomials in stage 2. You can think of stage 1 as a very compute-intensive net that tries to drag in something good, and stage 2 as a shorter but still compute-intensive process that tries to polish things.
2. Sieving for Relations
The sieving step is not the theoretically most complex part of the algorithm of factorization, but it is the most time consuming part because it iterates over a large domain with some expensive calculations like division and modulo, although some of these can be avoided by using logarithms.
In general optimization of the sieving step will give the biggest reduction in actual running time of the algorithm. It is easy to use a large amount of memory in this step, and one should be aware of this and try to reuse arrays and use the smallest possible data types. The factor bases can for record factorizations contain millions of elements, so one should try to obtain the best on-disk/in-memory tradeoff.
The purpose of the sieving step is to find usable relations, i.e. elements (a, b) with the following properties
• gcd(a, b) = 1
• a + bm is smooth over the rational factor base
• b^deg(f)*f(a/b) is smooth over the algebraic factor base
Finding elements with these properties can be done by various sieving methods like the classical line sieving or the faster lattice sieving, the latter being used at NFS@Home.
The lattice sieving was proposed by John Pollard in "Lattice sieving, Lecture Notes in Mathematics 1554 (1991), 43–49.". The factor bases are split into smaller sets and then the elements which are divisible by a large prime q are sieved. The sizes of the factor bases have to be determined empirically, and are dependent on the precision of the sieving code, if all smooth elements are found or if one skips some by using special-q methods.
One advantage the lattice siever has is the following. The yield rate for the line siever decreases over time because the norms get bigger as the sieve region moves away from the origin. The lattice siever brings the sieve region "back to the origin" when special-q's are changed. This might be its biggest advantage (if there is one).
3. Combine Relations
The last phase of NFS factorization is a group of tasks collectively referred to as 'NFS postprocessing'. You need the factor base file described in the sieving section (only the polynomial is needed, not the actual factor base entries), and all of the relations from the sieving. If you have performed sieving in multiple steps or on multiple machines, all of the relations that have been produced need to be combined into a single giant file. And by giant I mean *really big*; the largest NFS jobs that I know about currently have involved relation files up to 100GB in size.
Even a fairly small 100-digit factorization generates perhaps 500MB of disk files, so you are well advised to allow plenty of space for relations. Don't like having to deal with piling together thousands of files into one? Sorry, but disk space is cheap now.
With the factor base and relation data file available, is is time to perform NFS postprocessing.However, for larger jobs or for any job where data has to be moved from machine to machine, it is probably necessary to divide the postprocessing into its three fundamental tasks. These are described below:
The first phase of NFS postprocessing is the filtering step. This analyzes the input relation file, sets up the rest of the filtering to ignore relations that will not be useful (usually 90% of them or more), and produces a 'cycle file' that describes the huge matrix to be used in the next postprocessing stage.
To do that, every relation is assigned a unique number, corresponding to its line number in the relation file. Relations are numbered starting from zero, and part of the filtering also adds 'free relations' to the dataset. Free relations are so-called because it does not require any sieving to find them; these are a unique feature of the number field sieve, although there will never be very many of them. Filtering is a very complex process. If you do not have enough relations for filtering to succeed, no output is produced other than complaints to that effect. If there are 'enough' relations for filtering to succeed, the result is a 'cycle file'.
How many relations is 'enough'? This is unfortunately another hard question, and answering it requires either compiling large tables of factorizations of similar size numbers, running the filtering over and over again, or performing simulations after a little test-sieving. There's no harm in finding more relations than you strictly need for filtering to work at all, although if you mess up and find twice as many relations as you need then getting the filtering to work can also be difficult. In general the filtering works better if you give it somewhat more relations than it stricly needs, maybe 10% more. As more and more relations are added, the size of the generated matrix becomes smaller and smaller, partly because the filtering can throw away more and more relations to keep only the 'best' ones.
NFS Linear Algebra
The linear algebra step constructs the matrix that was generated from the filtering, and finds a group of vectors that lie in the nullspace of that matrix. Finding nullspace vectors for a really big matrix is an enormous amount of work. To do the job, Msieve uses the block Lanczos algorithm with a large number of performance optimizations. Even with fast code like that, solving the matrix can take anywhere from a few minutes (factoring a 100-digit input leads to a matrix of size ~200000) to several months (using the special number field sieve on 280-digit numbers from the Cunningham Project usually leads to matrices of size ~18 million). Even worse, the answer has to be *exactly* correct all the way through; there's no throwing away intermediate results that are bad, like the other NFS phases can do. So solving a big matrix is a severe test of both your computer and your patience.
Multithreaded Linear Algebra
The linear algebra is fully multithread aware. Note that the solver is primarily memory bound, and using as many threads as you have cores on your multicore processor will probably not give the best performance. The best number of threads to use depends on the underlying machine; more recent processors have much more powerful memory controllers and can continue speeding up as more and more threads are used. A good rule of thumb to start off is to try two threads for each physical package on your motherboard; even if it's not the fastest choice, just two or four threads gets the vast majority of the potential speedup for the vast majority of machines.
Finally, note that the matrix solver is a 'tightly parallel' computation, which means if you give it four threads then the machine those four threads run on must be mostly idle otherwise. The linear algebra will soak up most of the memory bandwidth your machine has, so if you divert any of it away to something else then the completion time for the linear algebra will suffer.
As for memory use, solving the matrix for a 512-bit input is going to require around 2GB of memory in the solver, and a fast modern processor running the solver with four threads will need about 36 hours. A slow, less modern processor that is busy with other stuff could take up to a week!
NFS Square Root
With the solution file from the linear algebra in hand, the last phase of NFS postprocessing is the square root.
'an NFS square root' is actually two square roots, an easy one over the integers and a very complex one over the algebraic number field described by the NFS polynomial we selected. Traditionally, the best algorithm for the algebraic part of the NFS square root is the one described by Montgomery and Nguyen, but that takes some quite sophisticated algebraic number theory smarts.
Every solution generated by the linear algebra is called 'a dependency', because it is a linearly dependent vector for the matrix we built.
The square root in Msieve proceeds one dependency at a time; it requires all the relations from the data file, the cycle file from the filtering, and the dependency file from the linear algebra. Technically the square root can be speed up if you process multiple dependencies in parallel, but doing one at a time makes it possible to adapt the precision of the numbers used to save a great deal of memory toward the end of each dependency.
Each dependency has a 50% chance of finding a nontrivial factorization of the input.
msieve is the client used for post-processing phase