What traits make a number a good choice for NFSNET?

[wblipp]

What traits make a number a good choice for NFSNET? Is the size of the base number (227 digits for 10^227-1) important? Is ths size of the composite cofactor (212 digits for 10^227-1) important? How big is "too big to be interesting?" How much previous work should have been done by other methods?

[xilman]

What traits make a number a good choice for NFSNET? Is the size of the base number (227 digits for 10^227-1) important? Is ths size of the composite cofactor (212 digits for 10^227-1) important? How big is "too big to be interesting?" How much previous work should have been done by other methods?

Several characteristics are taken into account. Perhaps the most important is that it should not be too hard (or we'd never get it finished with the resources we have) or too easy (or we'd spend more time on administration than computation. For SNFS, the range is about 180 to 240 digits. For GNFS the range is perhaps 135 digits to 165 digits.

For the time being we want to do numbers which are of some mathematical interest, which means in practice they come from a number of well know tables of factorizations associated with various projects. Almost all so far have come from the Cunningham project.

We don't want to spend a great deal of effort finding a factor which could have been found much more easily with the Elliptic Curve Method, so we generally require that a NFSNET candidate have had enough ECM work done on it that it's unlikely to have a factor below 50 digits or so.


Paul

[wblipp]

Originally Posted by xilman

For SNFS, the range is about 180 to 240 digits. For GNFS the range is perhaps 135 digits to 165 digits.

So looking at the 195 digit cofactor from 2^1188+1, it's not a good candidate.

1. The 195 digit cofactor is bigger than the 165 digit limit for GNFS.

2. The 358 digit 2^1188+1 is bigger than the 240 digit limit for SNFS.

3. The primitive polynomial is only 217 digits, but

(x^33+1)*(x+1)/((x^11+1)*(x^3+1))

isn't special enough for SNFS to take advantage of.

Is that all correct?

[xilman]

Originally Posted by wblipp

So looking at the 195 digit cofactor from 2^1188+1, it's not a good candidate.

1. The 195 digit cofactor is bigger than the 165 digit limit for GNFS.

2. The 358 digit 2^1188+1 is bigger than the 240 digit limit for SNFS.

3. The primitive polynomial is only 217 digits, but

(x^33+1)*(x+1)/((x^11+1)*(x^3+1))

isn't special enough for SNFS to take advantage of.

Is that all correct?

Sounds like a good summary to me.

If you can find a polynomial of degree at most 7 with coefficients all of which are smaller than, say, 9 digits and a corresponding root modulo N (where N is the 217-digit number) then we could run SNFS on it. I haven't found any such polynomial.


Paul

[wblipp]

Originally Posted by xilman

If you can find a polynomial of degree at most 7 with coefficients all of which are smaller than, say, 9 digits and a corresponding root modulo N (where N is the 217-digit number) then we could run SNFS on it. I haven't found any such polynomial.

Is it sufficient for the root to be modulo the C195 cofactor, or does the root really need to be modulo the C217 primitive factor?

[xilman]

Originally Posted by wblipp

Is it sufficient for the root to be modulo the C195 cofactor, or does the root really need to be modulo the C217 primitive factor?

Modulo the C195 is fine. Oh, I forgot: the polynomial has to have degree at least 4.

Paul

[wblipp]

Originally Posted by xilman

For SNFS, the range is about 180 to 240 digits. For GNFS the range is perhaps 135 digits to 165 digits.
...
If you can find a polynomial of degree at most 7 (and at least 4) with coefficients all of which are smaller than, say, 9 digits and a corresponding root modulo N (where N is the 217-digit number (or the 195-digit number)) then we could run SNFS on it.

For cases where we can find such a polynomial, is the feasible range then same 180-240 digits quoted above? Or would such a "found" polynomial be less effective, so that the practical range is something between these ranges?

William

[xilman]

Originally Posted by wblipp

For cases where we can find such a polynomial, is the feasible range then same 180-240 digits quoted above? Or would such a "found" polynomial be less effective, so that the practical range is something between these ranges?

William

The range 180-240 remains the same.

Paul

[nfortino]

Could someone point me to a good paper on the SNFS? I was able to find one on the GNFS, but I haven’t found one detailing the differences.

[TauCeti]

Originally Posted by nfortino

Could someone point me to a good paper on the SNFS? I was able to find one on the GNFS, but I haven’t found one detailing the differences.

An excellent and freely available 142-page paper (diploma thesis) with an explanation of the general ideas behind NFS and a GNFS/SNFS comparison can be found here. But AFAIK it's only available in german language.

There are also large sections about GNFS and SNFS in the book 'Prime Numbers" from Crandall and Pomerance.

So far i have not looked into 'The Development of the Number Field Sieve' from Lenstra but maybe i should if i ever want to succeed understanding the SNFS ;)

[junky]

Originally Posted by nfortino

Could someone point me to a good paper on the SNFS? I was able to find one on the GNFS, but I haven’t found one detailing the differences.

really good question, if ya find something before me, feel free to post it, so I and a lot a ppl would have the possibility to take a look on it.

im sure we could find such papers somewhere in US' universities. Maybe Jeff would knows :)

later