Continued Fraction Factorization algorithm

The continued fraction factorization method (CFRAC) is a general-purpose factorization algorithm valid for integers. It calculates factors of a given integer number without considering its unique properties. It has a sub-exponential running time.  It was first described in 1931 by  D. H. Lehmer and R. E. Powers and later in 1975 were developed into a computer algorithm by Michael A. Morrison and John Brillhart.

Continued Fraction:
An expression that can be expressed in the form:

(1)   \begin{equation*} X=a_{0}+\frac{b_{1}}{a_{1}+\frac{b_{2}}{a_{2} \ldots+\frac{b_{n-1}}{a_{n-1}+\frac{b_{n}}{a_{n}}}}} \end{equation*}

is called a Continued Fraction, where ai and bi are either real or complex values for all i > = 0. When all the values of bi‘s are 1, then it is called a simple continued fraction.



A Simple Continued Fraction can be denoted as:

(2)   \begin{equation*} \left[a_{0} ; a_{1}, a_{2} \ldots a_{n}\right]=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2} \ldots+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}} \end{equation*}

where Ck= [a0; a1, a2, …, an] for k<=n is the k-th convergent of the Simple Continued Fraction.
An Infinite Continued Fraction [a0; a1, a2, …, ak, …] is defined as a limit of the convergents Ck=[a0; a1, a2, …, an]

Algorithm:
This algorithm uses residues produced in the Continued Fraction of (mn)1/2 for some m to produce a square number. 

This algorithm solves the mathematical equation:

(3)   \begin{equation*} x^{2} \equiv y^{2}(\bmod (n)) \end{equation*}


this equation is solved by calculating the value of m such that m2 (mod(n)) has the minimum upperbound.



  • CFRAC algorithm has a time complexity of:

(4)   \begin{equation*} O\left(e^{\sqrt{2 \log n \log \log n}}\right) \end{equation*}

Example 1:

 
Input: continued_fraction((10/7))
Output: [1, 2, 3]

Explanation:

(5)   \begin{equation*} [1,2,3]=1+\frac{1}{2+\frac{1}{3}}=1+\frac{1}{\frac{7}{3}}=1+\frac{3}{7}=\frac{10}{7} \end{equation*}

Example 2:

Input:  list(continued_fraction_convergents([0, 2, 1, 2]))
Output: [0, 1/2, 1/3, 3/8]
Explanation:

(6)    \begin{equation*} \begin{array}{c} {[0,2,1,2]=0+\frac{1}{2+\frac{1}{1+\frac{1}{2}}}} \\ c_{1}=0, c_{2}=0+\frac{1}{2}=\frac{1}{2} \cdot c_{3}=0+\frac{1}{2+\frac{1}{1}}=\frac{1}{3} \cdot c_{4}=0+\frac{1}{2+\frac{1}{1+\frac{1}{2}}}=0+\frac{1}{2+\frac{1}{3}}=0+\frac{1}{2+\frac{1}{2}}=\frac{1}{\frac{1}{3}}=\frac{3}{8} \end{array} \end{equation*}



Example 3:

 
Input: continued_fraction_reduce([1, 2, 3, 4, 5]) 
Output: 225/157
Explanation:

(7)    \begin{equation*} 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5}}}}=1+\frac{1}{2+\frac{1}{3+\frac{1}{21}}}=1+\frac{1}{2+\frac{1}{3+\frac{5}{5}}}=1+\frac{1}{2+\frac{1}{68}}=1+\frac{1}{2+\frac{21}{68}}=1+\frac{1}{\frac{157}{68}}=1+\frac{68}{157}=\frac{225}{157} \end{equation*}


Implementation:
Code: To convert a fraction into Continued Fraction representation

filter_none

edit
close

play_arrow

link
brightness_4
code

#using sympy module
from sympy.ntheory.continued_fraction import continued_fraction
from sympy import sqrt
#calling continued_fraction method
continued_fraction(10/7)

chevron_right


Output:

[1, 2, 3]

Code 2: To convert a Continued Fraction into fraction.

filter_none

edit
close

play_arrow

link
brightness_4
code

#using sympy module
from sympy.ntheory.continued_fraction import continued_fraction_reduce 
  
#calling continued_fraction_reduce method
continued_fraction_reduce([1, 2, 3, 4, 5])

chevron_right


Output:

225/157

Code 3: To get a list of convergents from a Continued fraction.

filter_none

edit
close

play_arrow

link
brightness_4
code

# using sympy module
from sympy.core import Rational, pi
from sympy import S
from sympy.ntheory.continued_fraction import continued_fraction_convergents, continued_fraction_iterator      
# calling continued_fraction_convergents method and 
# passing it as a parameter to a list
list(continued_fraction_convergents([0, 2, 1, 2]))

chevron_right


Output:

[0, 1/2, 1/3, 3/8]

Attention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.

To begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course.




My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.


Article Tags :

Be the First to upvote.


Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.