Saturday, June 24, 2006
SF: Progress, double factorization proven
For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.
And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.
Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!
They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.
This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.
Let
S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))
and
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
now multiply both out, which gives
S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and
T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and now subtract one from the other for one result, and add one to the other for another:
S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)
and
S+T = 2*k_1*k_3*x + 2*k_2*k_4*y
and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.
It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.
AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
of your taget composite.
And there is nothing in the algebra that says that if T is your public key, this method will not work!
But is there anything in there that says it will?
I don't know for sure, but I'm afraid it can be made to work.
And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.
And they are ignoring it now to my knowledge.
My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.
Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!
I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.
And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?
Think about it. What would you do if you were a math con and the story got out?
But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.
They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.
But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.
How did this happen?
Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?
If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?
Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.
To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.
The people who don't lie, like politicians who don't, don't get ahead.
But they picked a practical area with factoring, and here politics don't work.
Their social world ran up against a real world problem, and failed.
[A reply to someone who examined James' method and asked whether he made mistakes.]
I don't know. And I don't care.
You are a minor player in an area where if I'm right, far more intelligent people with a lot more expertise are already working on these equations, so you are probably way, way, way behind, and your input is irrelevant.
If they don't work, who cares? Seems to me that you seem to think they do work, but inefficiently, which means either you made a mistake, or better researchers CAN make these work, and, well, then it's the worst case scenario.
For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.
And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.
Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!
They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.
This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.
Let
S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))
and
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
now multiply both out, which gives
S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and
T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and now subtract one from the other for one result, and add one to the other for another:
S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)
and
S+T = 2*k_1*k_3*x + 2*k_2*k_4*y
and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.
It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.
AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
of your taget composite.
And there is nothing in the algebra that says that if T is your public key, this method will not work!
But is there anything in there that says it will?
I don't know for sure, but I'm afraid it can be made to work.
And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.
And they are ignoring it now to my knowledge.
My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.
Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!
I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.
And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?
Think about it. What would you do if you were a math con and the story got out?
But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.
They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.
But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.
How did this happen?
Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?
If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?
Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.
To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.
The people who don't lie, like politicians who don't, don't get ahead.
But they picked a practical area with factoring, and here politics don't work.
Their social world ran up against a real world problem, and failed.
[A reply to someone who said that the whole problem is about finding factors efficiently.]
If you look for integer k's, then there are a FINITE number of possibles.
There is no way to limit the possibilities with other methods.
So, like with an RSA key, someone with these ideas can absolutely guarantee factoring the key if they loop through all possible integer k's, but there is no such guarantee possible in a reasonable space, with any other known method.
Besides, you people talk as if finding new factoring methods was an everyday thing.
Ok, yes, for me, sure I can find new factoring methods easily, and up until now, none have been practical.
So maybe you've forgotten that in the real world, there are maybe 4 or 5 factoring methods that have been come up with by people besides me in the history of humanity.
[A reply to someone who wanted to know what qualifies James to describe the other poster as a minor player in a field where James is not respected at all.]
You really think this is about respect?
It's about the mathematics.
There is only ONE way those equations can't be important.
That one way is if you need to know the factorization of T ahead of time to get that finite set of integer values.
If you don't, then that's it. It's over. RSA is finished. The company will soon go bankrupt and the entire Internet will be changed.
Companies will die, while new ones may be born.
Billions of dollars and then trillions of dollars will shift around the world.
Something on this scale is so huge you cannot begin to comprehend it, and yes, you people are less than bit players on a stage where presidents and other heads of state may soon get phone calls telling them of the news.
All you people are good enough for is to show this to be wrong, but it's not, so you have no real role here.
I'm mostly just posting now to hear myself talk. None of you are actually important at this point.
You have no impact. No ability to greatly affect things one way or another.
You are just flotsam dragged along with the current in something that is huger than anything civilization has ever faced before.
The lives of millions in the balance? Yup. So yeah, you're a nobody on that scale.
None of you on sci.math actually matter any more.
Your role is done.
And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.
Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!
They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.
This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.
Let
S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))
and
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
now multiply both out, which gives
S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and
T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and now subtract one from the other for one result, and add one to the other for another:
S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)
and
S+T = 2*k_1*k_3*x + 2*k_2*k_4*y
and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.
It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.
AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
of your taget composite.
And there is nothing in the algebra that says that if T is your public key, this method will not work!
But is there anything in there that says it will?
I don't know for sure, but I'm afraid it can be made to work.
And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.
And they are ignoring it now to my knowledge.
My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.
Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!
I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.
And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?
Think about it. What would you do if you were a math con and the story got out?
But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.
They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.
But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.
How did this happen?
Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?
If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?
Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.
To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.
The people who don't lie, like politicians who don't, don't get ahead.
But they picked a practical area with factoring, and here politics don't work.
Their social world ran up against a real world problem, and failed.
[A reply to someone who examined James' method and asked whether he made mistakes.]
I don't know. And I don't care.
You are a minor player in an area where if I'm right, far more intelligent people with a lot more expertise are already working on these equations, so you are probably way, way, way behind, and your input is irrelevant.
If they don't work, who cares? Seems to me that you seem to think they do work, but inefficiently, which means either you made a mistake, or better researchers CAN make these work, and, well, then it's the worst case scenario.
For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.
And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.
Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!
They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.
This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.
Let
S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))
and
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
now multiply both out, which gives
S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and
T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y
and now subtract one from the other for one result, and add one to the other for another:
S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)
and
S+T = 2*k_1*k_3*x + 2*k_2*k_4*y
and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.
It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.
AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization
T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))
of your taget composite.
And there is nothing in the algebra that says that if T is your public key, this method will not work!
But is there anything in there that says it will?
I don't know for sure, but I'm afraid it can be made to work.
And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.
And they are ignoring it now to my knowledge.
My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.
Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!
I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.
And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?
Think about it. What would you do if you were a math con and the story got out?
But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.
They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.
But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.
How did this happen?
Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?
If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?
Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.
To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.
The people who don't lie, like politicians who don't, don't get ahead.
But they picked a practical area with factoring, and here politics don't work.
Their social world ran up against a real world problem, and failed.
[A reply to someone who said that the whole problem is about finding factors efficiently.]
If you look for integer k's, then there are a FINITE number of possibles.
There is no way to limit the possibilities with other methods.
So, like with an RSA key, someone with these ideas can absolutely guarantee factoring the key if they loop through all possible integer k's, but there is no such guarantee possible in a reasonable space, with any other known method.
Besides, you people talk as if finding new factoring methods was an everyday thing.
Ok, yes, for me, sure I can find new factoring methods easily, and up until now, none have been practical.
So maybe you've forgotten that in the real world, there are maybe 4 or 5 factoring methods that have been come up with by people besides me in the history of humanity.
[A reply to someone who wanted to know what qualifies James to describe the other poster as a minor player in a field where James is not respected at all.]
You really think this is about respect?
It's about the mathematics.
There is only ONE way those equations can't be important.
That one way is if you need to know the factorization of T ahead of time to get that finite set of integer values.
If you don't, then that's it. It's over. RSA is finished. The company will soon go bankrupt and the entire Internet will be changed.
Companies will die, while new ones may be born.
Billions of dollars and then trillions of dollars will shift around the world.
Something on this scale is so huge you cannot begin to comprehend it, and yes, you people are less than bit players on a stage where presidents and other heads of state may soon get phone calls telling them of the news.
All you people are good enough for is to show this to be wrong, but it's not, so you have no real role here.
I'm mostly just posting now to hear myself talk. None of you are actually important at this point.
You have no impact. No ability to greatly affect things one way or another.
You are just flotsam dragged along with the current in something that is huger than anything civilization has ever faced before.
The lives of millions in the balance? Yup. So yeah, you're a nobody on that scale.
None of you on sci.math actually matter any more.
Your role is done.
# posted by James Harris @ 18:35 
