What traits make a number a good choice for NFSNET?

[R.D. Silverman]

"Quote:
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 :)"

If you will send me a private note, with your email address, I will send
a paper (Postscript) describing details of SNFS.

[junky]

hey bob, im still waiting for your papers :)

thanks.

[wpolly]

What about the C228 cofactor of 12^256+1?

[R.D. Silverman]

Originally Posted by wpolly

What about the C228 cofactor of 12^256+1?

It is too big. The C277 for 12,256+ is too big for SNFS and the C228 is
too big for GNFS.

[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.

Does the root need to be small, too? If yes, then it's probably rare for such a polynomial and root to exist because there are only 10^81 combinations of coefficients and roots versus 10^217 possibe modulo values. If the root can be as large as N, then there are probably many such polynomials although it may be difficult to find one of them.

[wblipp]

Originally Posted by wblipp

Does the root need to be small, too?

A bit more musing shows me the root can probably be large. It not, then the highest possible value for the polynomial is about 10⁷². so the only way to make the modular value zero would be to make the polynomial value zero, which would mean that (x-r) is a factor of the polynomial. Since it's trivial to create such polynomials as (x-r) times anything, it's unlikely such polynomials are of any use.

[xilman]

Originally Posted by wblipp

A bit more musing shows me the root can probably be large. It not, then the highest possible value for the polynomial is about 10⁷². so the only way to make the modular value zero would be to make the polynomial value zero, which would mean that (x-r) is a factor of the polynomial. Since it's trivial to create such polynomials as (x-r) times anything, it's unlikely such polynomials are of any use.

The root can be any size you like. What's important is the size of the coefficients of the polynomials.

Actually, that's an oversimplification, but it's a good first cut. If you want more details, dig out Brian Murphy's thesis.

An example may help. NFSNET is currently sieving the 201-digit composite cofactor of 10^223+1. We are using the polynomials x^6+10 and 1 - (10^37)*x which share a root 10^(-37) modulo 10^223+1. The latter value is a 201-digit number.


Paul

[R.D. Silverman]

Originally Posted by xilman

The root can be any size you like. What's important is the size of the coefficients of the polynomials.

Actually, that's an oversimplification, but it's a good first cut. If you want more details, dig out Brian Murphy's thesis.

An example may help. NFSNET is currently sieving the 201-digit composite cofactor of 10^223+1. We are using the polynomials x^6+10 and 1 - (10^37)*x which share a root 10^(-37) modulo 10^223+1. The latter value is a 201-digit number.


Paul

Actually Paul, the size of the root does matter. Let's use your example
and look at a 'typical' lattice point. (say (3x10^6, 3x10^6)) (choose
another if you like)

We have two polynomials. The corresponding norms at lattice point (b,a)
are a + 10^37 b and a^6 + 10b^6. For our typical point, the norms are
about 3x10^43 and 7x10^39. Note that the linear norm is larger. The
product is about 2x10^82

If we were to use a quintic, the linear norm becomes about 3 x 10^51
while the algebraic norm shrinks to 2 x 10^33. The product is now about
6 x 10^84, i.e. larger.

A septic would yield norms of about 3x10^38 and 2 x 10^46.

Having equal norms would be optimal.

The size of the root affects the norm of the linear polynomial. There is a
ying-yang effect. Reducing one norm increases the other and vice versa.
We want the product to be as small as possible averaged over the sieve
region.

See my recent paper for a more detailed analysis.

[xilman]

Originally Posted by Bob Silverman

Actually Paul, the size of the root does matter. Let's use your example
and look at a 'typical' lattice point. (say (3x10^6, 3x10^6)) (choose
another if you like)

We have two polynomials. The corresponding norms at lattice point (b,a)
are a + 10^37 b and a^6 + 10b^6. For our typical point, the norms are
about 3x10^43 and 7x10^39. Note that the linear norm is larger. The
product is about 2x10^82

If we were to use a quintic, the linear norm becomes about 3 x 10^51
while the algebraic norm shrinks to 2 x 10^33. The product is now about
6 x 10^84, i.e. larger.

A septic would yield norms of about 3x10^38 and 2 x 10^46.

Having equal norms would be optimal.

The size of the root affects the norm of the linear polynomial. There is a
ying-yang effect. Reducing one norm increases the other and vice versa.
We want the product to be as small as possible averaged over the sieve
region.

See my recent paper for a more detailed analysis.

Bob, I agree with you (with the reservations noted in my original about the analysis only being a first cut) if the linear polynomial is of the form x-m where m is the root.

Note there is nothing in the NFS which requires a polynomial to be linear, though a linear polynomial is very frequently used because of the difficulty in finding good polynomials of higher degree which share a common root with other polynomial(s) in use.

Neither is there any requirement that a linear polynomial by x-m. I gave an explicit example where the root m is very large but the poynomial norms are quite small. That polynomial was x/m - 1.

I further note that Kleinjung's method of finding quintics for GNFS finds linear polynomials of the form ax+b where neither a nor b are equal to the value of the common root.

I stand by my claim that (subject to the agreed disclaimer) that the size of the coefficients of the polynomials are much more important than the size of the root. By taking x-m as your linear polynomial you are implicitly agreeing with me as in this particular case the root is a coefficient of a polynomial.

I do, of course, agree that ideally the degrees and coefficients of the polynomials should be chosen that the norms are as close to each other as possible and as small as possible (again, subject to considerations which can be found in Murphy's thesis).


Paul

[jasong]

Somewhat off-topic: Would this Brian Murphy you keep mentioning happen to be a dark-haired fat dude with a winning smile? I knew a Brian Murphy that went to college here in Conway, AR, but I don't know his major. Could he be the same guy?