Sum of 2 squares implies efficient factorization












2












$begingroup$


I'm concerning myself with factoring semi-primes and believe that if given a large semi-prime ($N$) one finds a non-trivial sum of squares representation:



$$ x^2 + y^2 = N$$



Then one can efficiently retrieve a factorization of $N$. My intuition for this belief is that I think the "factoring" problem over the gaussian integers is difficult and note that such a representation can be in linear time converted into a factorization $(x + yi) ( x- yi)$ when considering gaussian integers.



But it's not clear to me how to leverage this information for regular integers.



More Notes:



So I realized after @Jack D'Aruzio's comment that this information could be exploited by computing $i$ mod $N$



That is finding some constant $c$ such that $c^2 = -1 mod N$. Then we have that



$$ (x + cy)(x - cy) equiv 0 mod N$$



And therefore at least one of $(x + cy)$ and $(x - cy)$ must be a zero divisor and can be used to recover factors of (assuming we don't have the trivial, non-trivial pair) $N$.



Of course there is no guarantee that we can find such a $c$, and even if its guaranteed to exist such a $c$ might be computationally as expensive to find as factoring itself










share|cite|improve this question











$endgroup$












  • $begingroup$
    You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:12










  • $begingroup$
    The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:18








  • 2




    $begingroup$
    meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
    $endgroup$
    – Will Jagy
    Jan 2 at 22:58










  • $begingroup$
    you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
    $endgroup$
    – user25406
    Jan 9 at 0:22
















2












$begingroup$


I'm concerning myself with factoring semi-primes and believe that if given a large semi-prime ($N$) one finds a non-trivial sum of squares representation:



$$ x^2 + y^2 = N$$



Then one can efficiently retrieve a factorization of $N$. My intuition for this belief is that I think the "factoring" problem over the gaussian integers is difficult and note that such a representation can be in linear time converted into a factorization $(x + yi) ( x- yi)$ when considering gaussian integers.



But it's not clear to me how to leverage this information for regular integers.



More Notes:



So I realized after @Jack D'Aruzio's comment that this information could be exploited by computing $i$ mod $N$



That is finding some constant $c$ such that $c^2 = -1 mod N$. Then we have that



$$ (x + cy)(x - cy) equiv 0 mod N$$



And therefore at least one of $(x + cy)$ and $(x - cy)$ must be a zero divisor and can be used to recover factors of (assuming we don't have the trivial, non-trivial pair) $N$.



Of course there is no guarantee that we can find such a $c$, and even if its guaranteed to exist such a $c$ might be computationally as expensive to find as factoring itself










share|cite|improve this question











$endgroup$












  • $begingroup$
    You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:12










  • $begingroup$
    The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:18








  • 2




    $begingroup$
    meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
    $endgroup$
    – Will Jagy
    Jan 2 at 22:58










  • $begingroup$
    you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
    $endgroup$
    – user25406
    Jan 9 at 0:22














2












2








2





$begingroup$


I'm concerning myself with factoring semi-primes and believe that if given a large semi-prime ($N$) one finds a non-trivial sum of squares representation:



$$ x^2 + y^2 = N$$



Then one can efficiently retrieve a factorization of $N$. My intuition for this belief is that I think the "factoring" problem over the gaussian integers is difficult and note that such a representation can be in linear time converted into a factorization $(x + yi) ( x- yi)$ when considering gaussian integers.



But it's not clear to me how to leverage this information for regular integers.



More Notes:



So I realized after @Jack D'Aruzio's comment that this information could be exploited by computing $i$ mod $N$



That is finding some constant $c$ such that $c^2 = -1 mod N$. Then we have that



$$ (x + cy)(x - cy) equiv 0 mod N$$



And therefore at least one of $(x + cy)$ and $(x - cy)$ must be a zero divisor and can be used to recover factors of (assuming we don't have the trivial, non-trivial pair) $N$.



Of course there is no guarantee that we can find such a $c$, and even if its guaranteed to exist such a $c$ might be computationally as expensive to find as factoring itself










share|cite|improve this question











$endgroup$




I'm concerning myself with factoring semi-primes and believe that if given a large semi-prime ($N$) one finds a non-trivial sum of squares representation:



$$ x^2 + y^2 = N$$



Then one can efficiently retrieve a factorization of $N$. My intuition for this belief is that I think the "factoring" problem over the gaussian integers is difficult and note that such a representation can be in linear time converted into a factorization $(x + yi) ( x- yi)$ when considering gaussian integers.



But it's not clear to me how to leverage this information for regular integers.



More Notes:



So I realized after @Jack D'Aruzio's comment that this information could be exploited by computing $i$ mod $N$



That is finding some constant $c$ such that $c^2 = -1 mod N$. Then we have that



$$ (x + cy)(x - cy) equiv 0 mod N$$



And therefore at least one of $(x + cy)$ and $(x - cy)$ must be a zero divisor and can be used to recover factors of (assuming we don't have the trivial, non-trivial pair) $N$.



Of course there is no guarantee that we can find such a $c$, and even if its guaranteed to exist such a $c$ might be computationally as expensive to find as factoring itself







number-theory elementary-number-theory algebraic-number-theory factoring cryptography






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 3 at 16:24







frogeyedpeas

















asked Jan 2 at 21:47









frogeyedpeasfrogeyedpeas

7,44071949




7,44071949












  • $begingroup$
    You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:12










  • $begingroup$
    The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:18








  • 2




    $begingroup$
    meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
    $endgroup$
    – Will Jagy
    Jan 2 at 22:58










  • $begingroup$
    you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
    $endgroup$
    – user25406
    Jan 9 at 0:22


















  • $begingroup$
    You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:12










  • $begingroup$
    The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
    $endgroup$
    – Jack D'Aurizio
    Jan 2 at 22:18








  • 2




    $begingroup$
    meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
    $endgroup$
    – Will Jagy
    Jan 2 at 22:58










  • $begingroup$
    you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
    $endgroup$
    – user25406
    Jan 9 at 0:22
















$begingroup$
You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
$endgroup$
– Jack D'Aurizio
Jan 2 at 22:12




$begingroup$
You are just re-discovering the key ingredients of the quadratic sieve and of the general field sieve.
$endgroup$
– Jack D'Aurizio
Jan 2 at 22:12












$begingroup$
The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
$endgroup$
– Jack D'Aurizio
Jan 2 at 22:18






$begingroup$
The point is that the factorization problem in $mathbb{Z}$ and $mathbb{Z}[i]$ is just as difficult, but if $N$ is represented by some binary quadratic form (with a negative discriminant) (and a small class number), $x$ and $y$ have to fulfill many arithmetic constraints, so they can be sifted by exploiting the informations provided by Legendre/Jacobi/Kronecker symbols.
$endgroup$
– Jack D'Aurizio
Jan 2 at 22:18






2




2




$begingroup$
meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
$endgroup$
– Will Jagy
Jan 2 at 22:58




$begingroup$
meanwhile, if you get two different expressions as sum of two squares, you get a factoring. zakuski.utsa.edu/~jagy/Brillhart_Euler_factoring_2009.pdf
$endgroup$
– Will Jagy
Jan 2 at 22:58












$begingroup$
you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
$endgroup$
– user25406
Jan 9 at 0:22




$begingroup$
you may want to look at this method using the sum of 2 squares to factor numbers. It is different from Euler's method. math.stackexchange.com/questions/3066531/…
$endgroup$
– user25406
Jan 9 at 0:22










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060010%2fsum-of-2-squares-implies-efficient-factorization%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060010%2fsum-of-2-squares-implies-efficient-factorization%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

SQL update select statement

WPF add header to Image with URL pettitions [duplicate]