Advances in composite integer factorization
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The document proposes a new integer factorization method. It begins by reviewing existing factorization methods like trial division and Pollard's rho algorithm. It then introduces the proposed method, which estimates the prime factors p and q of a composite integer n as the primes nearest to 1/2(log(n)). The method is demonstrated on some examples and is argued to provide better efficiency than existing algorithms. While an improvement, the author acknowledges the method remains similar to direct search. Further analyzing trends in prime differences is suggested to seek an even faster factorization algorithm.
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.2, 2013
Advances in composite integer factorization
Aldrin W. Wanambisi1* Shem Aywa2, Cleophas Maende3, Geoffrey Muchiri Muketha4
1. School of Pure and Applied Science, Mount Kenya University, P.O box 342-00100, Thika, Kenya.
2. Dept of Mathematics, Masinde Muliro University of Science and Technology, P.O Box 150-50100,
Kakamega, Kenya.
3. School of Post graduate studies, Mount Kenya University, P.O box 342-00100, Thika, Kenya.
4. Dept of Computer Science, Masinde Muliro University of Science and Technology, P.O Box 150-50100,
Kakamega, Kenya.
* E-mail of the corresponding author: wawanambisi@gmail.com
Keywords: Integer factorization, Prime number, logarithm.
Abstract
In this research we propose a new method of integer factorization. Prime numbers are the building blocks of
arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given
integer is prime or and its prime factors [1]. This fact is the basis behind many of the cryptosystems currently in use.
1.0 Introduction
There are no known algorithms which can factor arbitrary large integers efficiently. Probabilistic algorithms such as
the Pollard rho and Pollard p-1 algorithm are in most cases more efficient than the trial division and Fermat
factorization algorithms. However, probabilistic algorithms can fail when given certain prime products: for example,
Pollard's rho algorithm fails for N = 21 [6]. Integer factorization algorithms are an important subject in mathematics,
both for complexity theory, and for practical purposes such as data security on computers [3].
2.0 Basic Concepts
An integer is prime if it has no positive divisors other than 1 and itself. An integer greater than or equal to 2
that is not prime is composite. An integer is composite if and only if it has factors a and b such that
and If n > 1 then there is a prime p such that p | n where p | n denotes p divides n [8].
2.1 Prime factorization algorithms
Many algorithms have been devised for determining the prime factors of a given number (a process called prime
factorization). They vary quite a bit in sophistication and complexity [1], [2]. It is very difficult to build a
general-purpose algorithm for this computationally "hard" problem, so any additional information that is known about
the number in question or its factors can often be used to save a large amount of time.
86](https://image.slidesharecdn.com/advancesincompositeintegerfactorization-130401080557-phpapp02/85/Advances-in-composite-integer-factorization-1-320.jpg)
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.2, 2013
The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method,
all possible factors are systematically tested using trial division to see if they actually divide the given number. It is
practical only for very small numbers.
The fastest-known fully proven deterministic algorithm is the Pollard-Strassen method [6].
2.2 Integer factorization
The factorization of a number into its constituent primes, also called prime decomposition. Given a positive integer n
≥ 2, the prime factorization is written
where the pi s are the k prime factors, each of order αi. Each factor piαi is called a primary. Prime factorization can be
performed in Mathematica using the command Factor Integer[n], which returns a list of pairs.
Through his invention of the Pratt certificate, Pratt (1975) [4] became the first to establish that prime factorization lies
in the complexity class NP.
The number of digits in the prime factorization of n = 1, 2, ..., are 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, (Sloane's A050252). [4]
In general, integer factorization is a difficult problem, and many sophisticated integer factorization algorithms have
been devised for special types of numbers.
Interestingly, prime numbers p equal to 1 (mod 4) can always by factored into Gaussian primes in the form
where the real and imaginary parts are inverted in the two parts, while prime numbers equal to 3 (mod 4) cannot be
factored into Gaussian primes. This is directly related to Fermat's 4n+1 theorem.
2.3 Examples of factorization algorithms
2.3.1 Trial Division
A brute- force method of finding a divisor of an integer by simply plugging in one or a set of integers and seeing if
they divide . Repeated application of trial division to obtain the complete prime factorization of a number is called
87](https://image.slidesharecdn.com/advancesincompositeintegerfactorization-130401080557-phpapp02/85/Advances-in-composite-integer-factorization-2-320.jpg)
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.2, 2013
direct search factorization. An individual integer being tested is called a trial divisor. [1]
2.3.2 Direct search factorization
Direct search factorization is the simplest (and most simple-minded) prime factorization algorithm. It consists of
searching for factors of a number by systematically performing trial divisions usually using a sequence of increasing
numbers. Multiples of small primes are commonly excluded to reduce the number of trial divisors, but just including
them is sometimes faster than the time required to exclude them. Direct search factorization is very inefficient, and can
be used only with fairly small numbers.
When using this method on a number , only divisors up to (where is the floor function) need to be tested.
This is true since if all integers less than this had been tried, then
In other words, all possible factors have had their cofactors already tested. It is also true that, when the smallest prime
factor of is , then its cofactor (such that ) must be prime. To prove this, suppose that the
smallest p is . If , then the smallest value and could assume is . But then
,
which cannot be true. Therefore, must be prime, so [1].
3.0 Results
3.1 Introduction
In this section, we focus on the aspect of integer factorization. The so far proposed algorithms have proved not to be
efficient namely trial method, the direct search method, the fermat factorization method and GNFS [2]. These
algorithms do not run polynomial times.
88](https://image.slidesharecdn.com/advancesincompositeintegerfactorization-130401080557-phpapp02/85/Advances-in-composite-integer-factorization-3-320.jpg)
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.2, 2013
3.3 The proposed algorithm
The fundamental theorem of arithmetic states that every positive integer can be written uniquely as a product of
primes, when the primes in the product are written in non decreasing order. The fundamental theorem of arithmetic
implies that any composite integer can be factored. Let n be a composite integer a product of two primes p and q which
are not necessary equal but close as is in the RSA [3]. Now clearly the logarithm of the two primes is approximately ½
(log n). After the approximate logarithm has been obtained the nearest prime can be determined through direct search.
For example 21, first 1/2 (log 21) = 07563, the nearest primes with this 3 and 7. For Blum integers, which has
extensively been used in the domain of cryptography, are integers with form , where p and q are different
primes both ≡ 3 mod 4 and are odd integers. These integers can be divided two types:
1. , hence , the actual values of p and q can be estimated from primes
nearest to the integer equivalent to .
2. , where at least one of is greater than 1, hence
similarly the and can be estimated as .
This estimation algorithm reduces the number of steps that can be used determine the prime factors of composite
integers. The table below shows some composite integers and the prime factors based on the estimation algorithm:
Table 3.0
Integer ½ (log n) (4 d.p) p q
21 0.7563 3 7
568507 2.8774 751 757
7064963 6.8491 2657 2659
31945104 7.5044 5651 5653
4.0 Conclusion
The above algorithm can be used to factor large integer with relatively better efficiency compared to the existing
algorithms. Though it could be argued the algorithm is more like direct search but a number of steps are significantly
reduced. I believe that a study in trends in differences between consecutive primes is the way to go in seeking a much
faster algorithm to factor composite integers. A closer look at differences between primes reveals the different
89](https://image.slidesharecdn.com/advancesincompositeintegerfactorization-130401080557-phpapp02/85/Advances-in-composite-integer-factorization-4-320.jpg)
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.2, 2013
categories of primes for example twin primes, primes 3 (mod 4) have their differences as 2, 4 etc. If these
differences can be linked to the products obtained when such primes are multiplied then the number of steps required to
factorize such will be greatly reduced. The differences between any two primes which are close but not necessarily
equal are always in the units place
Authors' contributions
All authors contributed to the conceptualisation of the paper. Wanambisi A.W. did the initial review, the selection of
abstracts, and the identification of papers to be included in the final review. All authors contributed to the assessment
of papers. All authors reviewed the results of the analysis. Wanambisi drafted the manuscript, and all authors
contributed to its completion.
Acknowledgements
Thanks to those who have been instrumental in the success of this research: The Masinde Muliro University of
Science and Technology, the adviser, for participating in this research study and for their support of this study
References
[1]. Connelly B. “Integer Factorization Algorithms”. December 7, 2004
[2]. "General number field sieve." From Wikipedia, an online encyclopedia. November 13, 2004.
Available: http://en.wikipedia.org/wiki/GNFS
[3]. Wesstein, Eric W. "RSA Encryption." From Mathworld, an online encyclopedia. April, 2001.
Available: http://mathworld.wolfram.com/RSAEncryption.html
[4]. "Integer factorization . Difficulty and complexity." From Wikipedia, an online encyclopedia.
October 30, 2004. Available: http://en.wikipedia.org/wiki/Integer_factorization
[5]. Weisstein, Eric W. "Fermat, Pierre de." From MathWorld, an online encyclopedia.
Available: http://scienceworld.wolfram.com/biography/Fermat.html
[6]. Weisstein, Eric W. "Pollard Rho Factorization." From MathWorld, an online encyclopedia.
December 28, 2002. Available: http://mathworld.wolfram.com/PollardRhoFactorizationMethod.html
[7]. Weisstein, Eric W. "Brent's Factorization Method." From MathWorld, an online encyclopedia.
December 28, 2002. Available: http://mathworld.wolfram.com/BrentsFactorizationMethod.html
[8] Hefferon J. Elementary Number Theory, December, 2003
90](https://image.slidesharecdn.com/advancesincompositeintegerfactorization-130401080557-phpapp02/85/Advances-in-composite-integer-factorization-5-320.jpg)
Advances in composite integer factorization
- 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.2, 2013 Advances in composite integer factorization Aldrin W. Wanambisi1* Shem Aywa2, Cleophas Maende3, Geoffrey Muchiri Muketha4 1. School of Pure and Applied Science, Mount Kenya University, P.O box 342-00100, Thika, Kenya. 2. Dept of Mathematics, Masinde Muliro University of Science and Technology, P.O Box 150-50100, Kakamega, Kenya. 3. School of Post graduate studies, Mount Kenya University, P.O box 342-00100, Thika, Kenya. 4. Dept of Computer Science, Masinde Muliro University of Science and Technology, P.O Box 150-50100, Kakamega, Kenya. * E-mail of the corresponding author: [email protected] Keywords: Integer factorization, Prime number, logarithm. Abstract In this research we propose a new method of integer factorization. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or and its prime factors [1]. This fact is the basis behind many of the cryptosystems currently in use. 1.0 Introduction There are no known algorithms which can factor arbitrary large integers efficiently. Probabilistic algorithms such as the Pollard rho and Pollard p-1 algorithm are in most cases more efficient than the trial division and Fermat factorization algorithms. However, probabilistic algorithms can fail when given certain prime products: for example, Pollard's rho algorithm fails for N = 21 [6]. Integer factorization algorithms are an important subject in mathematics, both for complexity theory, and for practical purposes such as data security on computers [3]. 2.0 Basic Concepts An integer is prime if it has no positive divisors other than 1 and itself. An integer greater than or equal to 2 that is not prime is composite. An integer is composite if and only if it has factors a and b such that and If n > 1 then there is a prime p such that p | n where p | n denotes p divides n [8]. 2.1 Prime factorization algorithms Many algorithms have been devised for determining the prime factors of a given number (a process called prime factorization). They vary quite a bit in sophistication and complexity [1], [2]. It is very difficult to build a general-purpose algorithm for this computationally "hard" problem, so any additional information that is known about the number in question or its factors can often be used to save a large amount of time. 86
- 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.2, 2013 The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method, all possible factors are systematically tested using trial division to see if they actually divide the given number. It is practical only for very small numbers. The fastest-known fully proven deterministic algorithm is the Pollard-Strassen method [6]. 2.2 Integer factorization The factorization of a number into its constituent primes, also called prime decomposition. Given a positive integer n ≥ 2, the prime factorization is written where the pi s are the k prime factors, each of order αi. Each factor piαi is called a primary. Prime factorization can be performed in Mathematica using the command Factor Integer[n], which returns a list of pairs. Through his invention of the Pratt certificate, Pratt (1975) [4] became the first to establish that prime factorization lies in the complexity class NP. The number of digits in the prime factorization of n = 1, 2, ..., are 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, (Sloane's A050252). [4] In general, integer factorization is a difficult problem, and many sophisticated integer factorization algorithms have been devised for special types of numbers. Interestingly, prime numbers p equal to 1 (mod 4) can always by factored into Gaussian primes in the form where the real and imaginary parts are inverted in the two parts, while prime numbers equal to 3 (mod 4) cannot be factored into Gaussian primes. This is directly related to Fermat's 4n+1 theorem. 2.3 Examples of factorization algorithms 2.3.1 Trial Division A brute- force method of finding a divisor of an integer by simply plugging in one or a set of integers and seeing if they divide . Repeated application of trial division to obtain the complete prime factorization of a number is called 87
- 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.2, 2013 direct search factorization. An individual integer being tested is called a trial divisor. [1] 2.3.2 Direct search factorization Direct search factorization is the simplest (and most simple-minded) prime factorization algorithm. It consists of searching for factors of a number by systematically performing trial divisions usually using a sequence of increasing numbers. Multiples of small primes are commonly excluded to reduce the number of trial divisors, but just including them is sometimes faster than the time required to exclude them. Direct search factorization is very inefficient, and can be used only with fairly small numbers. When using this method on a number , only divisors up to (where is the floor function) need to be tested. This is true since if all integers less than this had been tried, then In other words, all possible factors have had their cofactors already tested. It is also true that, when the smallest prime factor of is , then its cofactor (such that ) must be prime. To prove this, suppose that the smallest p is . If , then the smallest value and could assume is . But then , which cannot be true. Therefore, must be prime, so [1]. 3.0 Results 3.1 Introduction In this section, we focus on the aspect of integer factorization. The so far proposed algorithms have proved not to be efficient namely trial method, the direct search method, the fermat factorization method and GNFS [2]. These algorithms do not run polynomial times. 88
- 4. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.2, 2013 3.3 The proposed algorithm The fundamental theorem of arithmetic states that every positive integer can be written uniquely as a product of primes, when the primes in the product are written in non decreasing order. The fundamental theorem of arithmetic implies that any composite integer can be factored. Let n be a composite integer a product of two primes p and q which are not necessary equal but close as is in the RSA [3]. Now clearly the logarithm of the two primes is approximately ½ (log n). After the approximate logarithm has been obtained the nearest prime can be determined through direct search. For example 21, first 1/2 (log 21) = 07563, the nearest primes with this 3 and 7. For Blum integers, which has extensively been used in the domain of cryptography, are integers with form , where p and q are different primes both ≡ 3 mod 4 and are odd integers. These integers can be divided two types: 1. , hence , the actual values of p and q can be estimated from primes nearest to the integer equivalent to . 2. , where at least one of is greater than 1, hence similarly the and can be estimated as . This estimation algorithm reduces the number of steps that can be used determine the prime factors of composite integers. The table below shows some composite integers and the prime factors based on the estimation algorithm: Table 3.0 Integer ½ (log n) (4 d.p) p q 21 0.7563 3 7 568507 2.8774 751 757 7064963 6.8491 2657 2659 31945104 7.5044 5651 5653 4.0 Conclusion The above algorithm can be used to factor large integer with relatively better efficiency compared to the existing algorithms. Though it could be argued the algorithm is more like direct search but a number of steps are significantly reduced. I believe that a study in trends in differences between consecutive primes is the way to go in seeking a much faster algorithm to factor composite integers. A closer look at differences between primes reveals the different 89
- 5. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.2, 2013 categories of primes for example twin primes, primes 3 (mod 4) have their differences as 2, 4 etc. If these differences can be linked to the products obtained when such primes are multiplied then the number of steps required to factorize such will be greatly reduced. The differences between any two primes which are close but not necessarily equal are always in the units place Authors' contributions All authors contributed to the conceptualisation of the paper. Wanambisi A.W. did the initial review, the selection of abstracts, and the identification of papers to be included in the final review. All authors contributed to the assessment of papers. All authors reviewed the results of the analysis. Wanambisi drafted the manuscript, and all authors contributed to its completion. Acknowledgements Thanks to those who have been instrumental in the success of this research: The Masinde Muliro University of Science and Technology, the adviser, for participating in this research study and for their support of this study References [1]. Connelly B. “Integer Factorization Algorithms”. December 7, 2004 [2]. "General number field sieve." From Wikipedia, an online encyclopedia. November 13, 2004. Available: http://en.wikipedia.org/wiki/GNFS [3]. Wesstein, Eric W. "RSA Encryption." From Mathworld, an online encyclopedia. April, 2001. Available: http://mathworld.wolfram.com/RSAEncryption.html [4]. "Integer factorization . Difficulty and complexity." From Wikipedia, an online encyclopedia. October 30, 2004. Available: http://en.wikipedia.org/wiki/Integer_factorization [5]. Weisstein, Eric W. "Fermat, Pierre de." From MathWorld, an online encyclopedia. Available: http://scienceworld.wolfram.com/biography/Fermat.html [6]. Weisstein, Eric W. "Pollard Rho Factorization." From MathWorld, an online encyclopedia. December 28, 2002. Available: http://mathworld.wolfram.com/PollardRhoFactorizationMethod.html [7]. Weisstein, Eric W. "Brent's Factorization Method." From MathWorld, an online encyclopedia. December 28, 2002. Available: http://mathworld.wolfram.com/BrentsFactorizationMethod.html [8] Hefferon J. Elementary Number Theory, December, 2003 90