Shanks' square forms factorization

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Shanks' square forms factorization is a method for integer factorization, which was devised as an improvement on Fermat's factorization method.

The success of Fermat's method depends on finding integers x, and y such that x² − y² = N, where N is the integer to be factored, and a possible improvement (noticed by Kraitchik), is to look for integers x and y such that $x^2\equiv y^2 \pmod{N}$ . Finding a pair (x,y) does not guarantee a factorisation of N, but it implies that N is a factor of x² − y² = (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 highest common factor of N and x-y will give a non-trivial factor of N.

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

[edit] Algorithm

Input: N, the integer to be factored, which must be neither a prime number nor a perfect square.

Output: a non-trivial factor of N.

The algorithm:

Initialize $P_0=\lfloor\sqrt{N}\rfloor,Q_0=1,Q_1=N-P_0^2$

Repeat

$b_i=\left\lfloor\frac{\lfloor\sqrt{N}\rfloor+P_{i-1}}{Q_i}\right\rfloor,P_i=b_iQ_i-P_{i-1},Q_{i+1}=Q_{i-1}+b_i(P_{i-1}-P_i)$

until $Q i$ is a perfect square.

Initialize $b_0=\left\lfloor\frac{\lfloor\sqrt{N}\rfloor-P_i}{\sqrt{Q_i}}\right\rfloor,P_0=b_0\sqrt{Q_i}+P_i,Q_0=\sqrt{Q_i},Q_1=\frac{N-P_0^2}{Q_0}$

Repeat

$b_i=\left\lfloor\frac{\lfloor\sqrt{N}\rfloor+P_{i-1}}{Q_i}\right\rfloor,P_i=b_iQ_i-P_{i-1},Q_{i+1}=Q_{i-1}+b_i(P_{i-1}-P_i)$

until $P i + 1 = P i$ .

Then $gcd(N, P i)$ is a non-trivial factor of $N$ .

Example:

N=11111

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.

[edit] References

D. A. Buell (1989). Binary Quadratic Forms. Springer-Verlag. ISBN 0-387-97037-1.
D. M. Bressoud (1989). Factorisation and Primality Testing. Springer-Verlag. ISBN 0-387-97040-1.