On Jan 18, 6:54 am, Randy Poe <poespam-t...@yahoo.com> wrote:

On Jan 17, 8:14 pm, JSH <jst...@gmail.com> wrote:

Hint: If you find that your factoring of an RSA key is
projected to take 100 orders of magnitude more than
the age of the universe, then world civilization won't
end. We already know it's possible in principle to factor
these keys. Their safety lies in the fact that it takes
more time than anybody has to do so.

- Randy

Seemingly reasonable suggestions that ignore the realities of the
situation.

On Jan 18, 6:54 am, Randy Poe <poespam-t...@yahoo.com> wrote:

On Jan 17, 8:14 pm, JSH <jst...@gmail.com> wrote:

The alternative is a very public demonstration, like factoring RSA
public keys in posts on this newsgroup, but I project that afterwards
stock markets around the world would crumble, and what's happening now=
with them would look like the good 'ol days.

A convincing and perfectly safe demonstration would be
to factor RSA challenge numbers that have already been
factored, such as RSA-140.

Go for it.

Another convincing demonstration would be to factor
some small primes, say of 4, 5, ..., 9 digits, and give
us run times for your algorithm to show us how it scales.
Then we could project that up to hundreds of digits.

The advantage of using 9 and fewer digits is that you
don't need to use an arbitrary-precision library, you
can use your built-in integer types.

Hint: If you find that your factoring of an RSA key is
projected to take 100 orders of magnitude more than
the age of the universe, then world civilization won't
end. We already know it's possible in principle to factor
these keys. Their safety lies in the fact that it takes
more time than anybody has to do so.

=A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 - Randy

Seemingly reasonable suggestions that ignore the realities of the
situation.

For those who think that is a dodge, I've simplified a bit to get a
result that looks more directly related to factoring, where the point
I want to make is--these are fundamental relations.

Even without factoring as an issue because of encryption these would
be a big deal, but with it, they are a huge deal, but we have an
atypical reaction, as instead of buzzing about these results in a way
that everyone would notice, if there is any discussion going on in the
math community it's hidden away so that I don't see it.

Given an atypical response it is prudent to puzzle out the reasons for
that response before deciding on the best course, while also picking
the safest direction as determined by the simplest explanation--i.e.
using Occam's Razor.

Given a target composite T and integer factors f_1 and f_2, such that
f_1*f_2 =3D nT, and any prime p, the following relations must be true:

f_1 =3D ak mod p

and

f_2 =3D a^{-1}(1 + a^2)k mod p

where

k^2 =3D (a^2+1)^{-1}(nT) mod p.

Those represent the fundamental factoring relations that underpin ALL
composite factorizations.

IN other words, those are the rules. =A0No integer factorizations escape
them and what human beings have currently learned to date about
factoring is encompassed by them. =A0They're sort of like the F=3Dma of
integer factorizations.

Example: n=3D1, T=3D119, p=3D11, and a=3D2 gives

k^2 =3D (5)^{-1}(119) mod 11 =3D 9(9) mod 11 =3D 4 mod 11.

And k =3D 2 mod 11 is a solution, so f_1 =3D 4 mod 11, and

f_2 =3D 2^{-1}(1+4)(2) mod 11 =3D 5 mod 11.

In this case, subtracting 11 gives one prime factor as f_1 =3D -7 mod
11, and f_2 =3D -6 mod 11. =A0Subtracting 11 again from f_2 gives -17.

Faced with THE RULES of integer factorizations the mathematical
community is being rather silent, wouldn't you think?

So you may say, those must not be THE RULES, then, right? =A0But the
derivation of them is trivially easy requiring a bit of algebra, so
they are THE RULES, which forces you to find another explanation.

Like Sherlock Holmes said (probably paraphrasing a bit): Take away all
other explanations and the one that remains no matter how improbable,
must be the truth.

James Harris- Hide quoted text -

- Show quoted text -

On Jan 18, 6:54 am, Randy Poe <poespam-t...@yahoo.com> wrote:

On Jan 17, 8:14 pm, JSH <jst...@gmail.com> wrote:

The alternative is a very public demonstration, like factoring RSA
public keys in posts on this newsgroup, but I project that afterwards
stock markets around the world would crumble, and what's happening now
with them would look like the good 'ol days.

A convincing and perfectly safe demonstration would be
to factor RSA challenge numbers that have already been
factored, such as RSA-140.

Go for it.

Another convincing demonstration would be to factor
some small primes, say of 4, 5, ..., 9 digits, and give
us run times for your algorithm to show us how it scales.
Then we could project that up to hundreds of digits.

The advantage of using 9 and fewer digits is that you
don't need to use an arbitrary-precision library, you
can use your built-in integer types.

Hint: If you find that your factoring of an RSA key is
projected to take 100 orders of magnitude more than
the age of the universe, then world civilization won't
end. We already know it's possible in principle to factor
these keys. Their safety lies in the fact that it takes
more time than anybody has to do so.

- Randy

Seemingly reasonable suggestions that ignore the realities of the
situation.

For those who think that is a dodge,

I've simplified a bit to get a
result that looks more directly related to factoring, where the point
I want to make is--these are fundamental relations.

On Jan 18, 6:54 am, Randy Poe <poespam-t...@yahoo.com> wrote:

On Jan 17, 8:14 pm, JSH <jst...@gmail.com> wrote:

The alternative is a very public demonstration, like factoring RSA
public keys in posts on this newsgroup, but I project that afterwards
stock markets around the world would crumble, and what's happening now
with them would look like the good 'ol days.

A convincing and perfectly safe demonstration would be
to factor RSA challenge numbers that have already been
factored, such as RSA-140.

Go for it.

Another convincing demonstration would be to factor
some small primes, say of 4, 5, ..., 9 digits, and give
us run times for your algorithm to show us how it scales.
Then we could project that up to hundreds of digits.

The advantage of using 9 and fewer digits is that you
don't need to use an arbitrary-precision library, you
can use your built-in integer types.

Hint: If you find that your factoring of an RSA key is
projected to take 100 orders of magnitude more than
the age of the universe, then world civilization won't
end. We already know it's possible in principle to factor
these keys. Their safety lies in the fact that it takes
more time than anybody has to do so.

- Randy

Seemingly reasonable suggestions that ignore the realities of the
situation.

For those who think that is a dodge, I've simplified a bit to get a
result that looks more directly related to factoring, where the point
I want to make is--these are fundamental relations.

Even without factoring as an issue because of encryption these would
be a big deal, but with it, they are a huge deal, but we have an
atypical reaction, as instead of buzzing about these results in a way
that everyone would notice, if there is any discussion going on in the
math community it's hidden away so that I don't see it.

Given an atypical response it is prudent to puzzle out the reasons for
that response before deciding on the best course, while also picking
the safest direction as determined by the simplest explanation--i.e.
using Occam's Razor.

Given a target composite T and integer factors f_1 and f_2, such that
f_1*f_2 = nT, and any prime p, the following relations must be true:

f_1 = ak mod p

and

f_2 = a^{-1}(1 + a^2)k mod p

where

k^2 = (a^2+1)^{-1}(nT) mod p.

Those represent the fundamental factoring relations that underpin ALL
composite factorizations.