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Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method.

The success of Fermat's method depends on finding integers $x$ and $y$ such that $x^{2} - y^{2} = N$ , where $N$ is the integer to be factored. An improvement (noticed by Kraitchik) is to look for integers $x$ and $y$ such that $x^{2} \equiv y^{2} (\mod N)$ . Finding a suitable pair $(x, y)$ does not guarantee a factorization of $N$ , but it implies that $N$ is a factor of $x^{2} - y^{2} = (x - y) (x + y)$ , and there is a good chance that the prime divisors of $N$ are distributed between these two factors, so that calculation of the greatest common divisor of $N$ and $x - y$ will give a non-trivial factor of $N$ .

A practical algorithm for finding pairs $(x, y)$ which satisfy $x^{2} \equiv y^{2} (\mod N)$ was developed by Shanks, who named it Square Forms Factorization or SQUFOF. The algorithm can be expressed in terms of continued fractions, or in terms of quadratic forms. Although there are now much more efficient factorization methods available, SQUFOF has the advantage that it is small enough to be implemented on a programmable calculator.

Algorithm

Input: $N$ , the integer to be factored, which must be neither a prime number nor a perfect square, and a small multiplier $k$ .

Output: a non-trivial factor of $N$ .

The algorithm:

Initialize $P_{0} = ⌊ \sqrt{k N} ⌋, Q_{0} = 1, Q_{1} = k N - P_{0}^{2} .$

Repeat

$b_{i} = ⌊ \frac{⌊ \sqrt{k N} ⌋ + P_{i - 1}}{Q_{i}} ⌋, P_{i} = b_{i} Q_{i} - P_{i - 1}, Q_{i + 1} = Q_{i - 1} + b_{i} (P_{i - 1} - P_{i})$

until $Q_{i}$ is a perfect square.

Initialize $b_{0} = ⌊ \frac{⌊ \sqrt{k N} ⌋ - P_{i - 1}}{\sqrt{Q_{i}}} ⌋, P_{0} = b_{0} \sqrt{Q_{i}} + P_{i - 1}, Q_{0} = \sqrt{Q_{i}}, Q_{1} = \frac{k N - P_{0}^{2}}{Q_{0}}$

Repeat

$b_{i} = ⌊ \frac{⌊ \sqrt{k N} ⌋ + P_{i - 1}}{Q_{i}} ⌋, P_{i} = b_{i} Q_{i} - P_{i - 1}, Q_{i + 1} = Q_{i - 1} + b_{i} (P_{i - 1} - P_{i})$

until $P_{i + 1} = P_{i} .$

Then if $f = \gcd (N, P_{i})$ is not equal to $1$ and not equal to $N$ , then $f$ is a non-trivial factor of $N$ . Otherwise try another value of $k$ .

Shanks' method has time complexity $O (\sqrt[4]{N})$ .

Stephen S. McMasters (see link in External Link section) wrote a more detailed discussion of the mathematics of Shanks' method, together with a proof of its correctness.

Example

N = 11111, k = 1

P₀ = 105 Q₀ = 1 Q₁ = 86

P₁ = 67 Q₁ = 86 Q₂ = 77

P₂ = 87 Q₂ = 77 Q₃ = 46

P₃ = 97 Q₃ = 46 Q₄ = 37

P₄ = 88 Q₄ = 37 Q₅ = 91

P₅ = 94 Q₅ = 91 Q₆ = 25

Here Q₆ is a perfect square

P₀ = 104 Q₀ = 5 Q₁ = 59

P₁ = 73 Q₁ = 59 Q₂ = 98

P₂ = 25 Q₂ = 98 Q₃ = 107

P₃ = 82 Q₃ = 107 Q₄ = 41

P₄ = 82

Here P₃ = P₄

gcd(11111, 82) = 41, which is a factor of 11111.

References

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External links

Template:Number theoretic algorithms

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+'''Shanks' square forms factorization''' is a method for [[integer factorization]] devised by [[Daniel Shanks]] as an improvement on [[Fermat's factorization method]].
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+The success of Fermat's method depends on finding integers <math>x</math> and <math>y</math> such that <math>x^2-y^2=N</math>, where <math>N</math> is the integer to be factored. An improvement (noticed by [[Maurice Kraitchik|Kraitchik]]) is to look for integers <math>x</math> and <math>y</math> such that <math>x^2\equiv y^2\pmod{N}</math>. Finding a suitable pair <math>(x, y)</math> does not guarantee a factorization of <math>N</math>, but it implies that <math>N</math> is a factor of <math>x^2-y^2=(x-y)(x+y)</math>, and there is a good chance that the [[prime divisor]]s of <math>N</math> are distributed between these two factors, so that calculation of the [[greatest common divisor]] of <math>N</math> and <math>x-y</math> will give a non-trivial factor of <math>N</math>.
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+A practical algorithm for finding pairs <math>(x,y)</math> which satisfy <math>x^2\equiv y^2\pmod{N}</math> was developed by Shanks, who named it Square Forms Factorization or SQUFOF. The algorithm can be expressed in terms of continued fractions, or in terms of quadratic forms. Although there are now much more efficient factorization methods available, SQUFOF has the advantage that it is small enough to be implemented on a programmable calculator.
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+==Algorithm==
+'''Input''': <math>N</math>, the integer to be factored, which must be neither a [[prime number]] nor a [[Square number|perfect square]], and a small multiplier <math>k</math>.
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+'''Output''': a non-trivial factor of <math>N</math>.
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-== the National Women's Council or Ireland Air Max Online ==
+'''The algorithm:'''
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+Initialize <math>P_0=\lfloor\sqrt{kN}\rfloor,Q_0=1,Q_1=kN-P_0^2.</math>
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- </ul>
-== 19 New Balance 993 Sale ==
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+<math>b_i=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor+P_{i-1}}{Q_i}\right\rfloor,P_i=b_iQ_i-P_{i-1},Q_{i+1}=Q_{i-1}+b_i(P_{i-1}-P_i)</math>
-   <li>[http://enseignement-lsf.com/spip.php?article64#forum24485330 http://enseignement-lsf.com/spip.php?article64#forum24485330]</li>
+until <math>Q_i</math> is a perfect square.
-   <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/]</li>
+Initialize <math>b_0=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor-P_{i-1}}{\sqrt{Q_i}}\right\rfloor,P_0=b_0\sqrt{Q_i}+P_{i-1},Q_0=\sqrt{Q_i},Q_1=\frac{kN-P_0^2}{Q_0}</math>
-   <li>[http://www.promo-grimpe.com/spip.php?article124/ http://www.promo-grimpe.com/spip.php?article124/]</li>
+Repeat
-   <li>[http://1.ts.cn/home.php?mod=space&uid=32877&do=blog&quickforward=1&id=6064303 http://1.ts.cn/home.php?mod=space&uid=32877&do=blog&quickforward=1&id=6064303]</li>
+<math>b_i=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor+P_{i-1}}{Q_i}\right\rfloor,P_i=b_iQ_i-P_{i-1},Q_{i+1}=Q_{i-1}+b_i(P_{i-1}-P_i)</math>
- </ul>
+until <math>P_{i+1}=P_i.</math>
+Then if <math>f=\gcd(N,P_i)</math> is not equal to <math>1</math> and not equal to <math>N</math>, then <math>f</math> is a non-trivial factor of <math>N</math>.   Otherwise try another value of <math>k</math>.
+Shanks' method has time complexity <math>O(\sqrt[4]{N})</math>.
+Stephen S. McMasters (see link in External Link section) wrote
+a more detailed discussion of the mathematics of Shanks' method,
+together with a proof of its correctness.
+==Example==
+N = 11111, k = 1
+P<sub>0</sub> = 105 Q<sub>0</sub> = 1 Q<sub>1</sub> = 86
+P<sub>1</sub> = 67 Q<sub>1</sub> = 86 Q<sub>2</sub> = 77
+P<sub>2</sub> = 87 Q<sub>2</sub> = 77 Q<sub>3</sub> = 46
+P<sub>3</sub> = 97 Q<sub>3</sub> = 46 Q<sub>4</sub> = 37
+P<sub>4</sub> = 88 Q<sub>4</sub> = 37 Q<sub>5</sub> = 91
+P<sub>5</sub> = 94 Q<sub>5</sub> = 91 Q<sub>6</sub> = 25
+Here Q<sub>6</sub> is a perfect square
+P<sub>0</sub> = 104 Q<sub>0</sub> = 5 Q<sub>1</sub> = 59
+P<sub>1</sub> = 73 Q<sub>1</sub> = 59 Q<sub>2</sub> = 98
+P<sub>2</sub> = 25 Q<sub>2</sub> = 98 Q<sub>3</sub> = 107
+P<sub>3</sub> = 82 Q<sub>3</sub> = 107 Q<sub>4</sub> = 41
+P<sub>4</sub> = 82
+Here P<sub>3</sub> = P<sub>4</sub>
+gcd(11111, 82) = 41, which is a factor of 11111.
+==References==
+*{{cite book
+ | author = D. A. Buell
+| title = Binary Quadratic Forms
+| publisher = Springer-Verlag
+| year = 1989
+| isbn=0-387-97037-1 }}
+*{{cite book
+| author = D. M. Bressoud|authorlink=David Bressoud
+| title = Factorisation and Primality Testing
+| publisher = Springer-Verlag
+| year = 1989
+| isbn=0-387-97040-1 }}
+* {{cite book
+| last=Riesel | first=Hans | authorlink=Hans Riesel
+| title=Prime numbers and computer methods for factorization
+| publisher=Birkhauser
+|edition = 2nd
+| year=1994
+| isbn=0-8176-3743-5 }}
+==External links==
+*[http://cadigweb.ew.usna.edu/~wdj/mcmath/TridentFinal.pdf 2005, Stephen S. McMath]{{dead link|date=August 2012}}
+*[http://www.usna.edu/Users/math/wdj/mcmath/shanks_analysis.pdf Analysis and Improvement of the Continued Fraction Method of Factorization, Daniel Shanks (transcribed by S.McMath 2004)]
+*[http://www.usna.edu/Users/math/wdj/mcmath/shanks_squfof.pdf SQUFOF Notes, Daniel Shanks (transcribed by S. McMath 2004)]
+*[http://homes.cerias.purdue.edu/~ssw/squfof.pdf Square Form Factorisation, Jason Gower and Samuel Wagstaff]
+{{number theoretic algorithms}}
+[[Category:Integer factorization algorithms]]

Recessional velocity: Difference between revisions

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Recessional velocity: Difference between revisions

Revision as of 23:34, 19 August 2013

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