The Integer Factorization Algorithm With Pisano Period

Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the product of two prime numbers (or an integer multiple of it) can be derived from the two prime numbers themselves and their product, and we can th...

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Published in:IEEE access Vol. 7; pp. 167250 - 167259
Main Authors: Wu, Liangshun, Cai, H. J., Gong, Zexi
Format: Journal Article
Language:English
Published: Piscataway IEEE 2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Complexity
Cryptography
Curves
Elliptic curves
Factorization
Fibonacci
Fibonacci numbers
Fourier transforms
integer factorization
Integers
Nuclear magnetic resonance
Number theory
Pisano periods
Prime numbers
Qubit
RSA
ISSN:2169-3536, 2169-3536
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Abstract Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the product of two prime numbers (or an integer multiple of it) can be derived from the two prime numbers themselves and their product, and we can therefore decompose the two prime numbers by means of the Pisano period of their product. We reduce the computational complexity of modulo operation through the "fast Fibonacci modulo algorithm" and design a stochastic algorithm for finding the Pisano periods of large integers. The Pisano period factorization method, which is proved to be slightly better than the quadratic sieve method and the elliptic curve method, consumes as much time as Fermat's method, the continued fractional factorization method and the Pollard p-1 method on small integer factorization cases. When factoring super-large integers, the Pisano period factorization method has shown as strong performance as subexponential complexity methods; thus, this method demonstrates a certain practicability. We suggest that this paper may provide a completely new idea in the area of integer factorization problems.
AbstractList Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the product of two prime numbers (or an integer multiple of it) can be derived from the two prime numbers themselves and their product, and we can therefore decompose the two prime numbers by means of the Pisano period of their product. We reduce the computational complexity of modulo operation through the “fast Fibonacci modulo algorithm” and design a stochastic algorithm for finding the Pisano periods of large integers. The Pisano period factorization method, which is proved to be slightly better than the quadratic sieve method and the elliptic curve method, consumes as much time as Fermat's method, the continued fractional factorization method and the Pollard p-1 method on small integer factorization cases. When factoring super-large integers, the Pisano period factorization method has shown as strong performance as subexponential complexity methods; thus, this method demonstrates a certain practicability. We suggest that this paper may provide a completely new idea in the area of integer factorization problems.
Author Cai, H. J.
Gong, Zexi
Wu, Liangshun
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Cites_doi 10.1038/srep03023
10.1007/3-540-39757-4_17
10.1017/S0305004100049252
10.1016/j.ajmsc.2017.06.002
10.1080/00029890.1960.11989541
10.1007/s11433-017-9181-9
10.1038/s41598-018-36058-z
10.1007/s11433-018-9307-1
10.1007/978-1-4612-4476-9_3
10.1007/BFb0091534
10.2307/1971363
10.1109/SFCS.1994.365700
10.1126/science.359.6374.381
10.3724/SP.J.1087.2010.01026
10.1145/3207677.3278049
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References ref13
yuan (ref17) 2007; 3
ref15
gidney (ref11) 2018
ref14
zhou (ref19) 2016
wang (ref1) 2016; 43
ref10
wei (ref9) 2018
burges (ref12) 2002
ref16
pei (ref24) 2015
ref23
koblitz (ref3) 2009
ref20
ref22
ref21
chen (ref25) 1993
zhu (ref18) 1989; 23
pomerance (ref4) 1984; 209
ref8
ref7
ref6
ref5
guo (ref2) 2009; 17
References_xml – ident: ref10
  doi: 10.1038/srep03023
– volume: 209
  start-page: 169
  year: 1984
  ident: ref4
  article-title: The quadratic sieve factoring algorithm
  publication-title: Advances Cryptology
  doi: 10.1007/3-540-39757-4_17
– ident: ref5
  doi: 10.1017/S0305004100049252
– start-page: 123
  year: 1993
  ident: ref25
  publication-title: Linear Networks and Systems
– ident: ref20
  doi: 10.1016/j.ajmsc.2017.06.002
– start-page: 227
  year: 2016
  ident: ref19
  publication-title: Fibonacci-Lucas Sequences Its Application
– ident: ref16
  doi: 10.1080/00029890.1960.11989541
– year: 2009
  ident: ref3
  publication-title: A Course in Number Theory and Cryptography
– year: 2018
  ident: ref9
  article-title: Efficient universal quantum channel simulation in IBM's cloud quantum computer
  publication-title: Science China Physics Mechanics & Astronomy
  doi: 10.1007/s11433-017-9181-9
– ident: ref13
  doi: 10.1038/s41598-018-36058-z
– ident: ref14
  doi: 10.1007/s11433-018-9307-1
– ident: ref23
  doi: 10.1007/978-1-4612-4476-9_3
– ident: ref7
  doi: 10.1007/BFb0091534
– year: 2018
  ident: ref11
  article-title: Factoring with n+2 clean qubits and n-1 dirty qubits
  publication-title: arXiv 1706 07884
– volume: 17
  start-page: 29
  year: 2009
  ident: ref2
  article-title: RSA algorithm and its challenges
  publication-title: Journal of Henan College of Electrical and Mechanical
– ident: ref6
  doi: 10.2307/1971363
– volume: 43
  start-page: 24
  year: 2016
  ident: ref1
  article-title: New quantum algorithm for breaking RSA
  publication-title: Comput Sci
– ident: ref8
  doi: 10.1109/SFCS.1994.365700
– ident: ref15
  doi: 10.1126/science.359.6374.381
– ident: ref21
  doi: 10.3724/SP.J.1087.2010.01026
– ident: ref22
  doi: 10.1145/3207677.3278049
– volume: 23
  start-page: 327
  year: 1989
  ident: ref18
  article-title: Fibonacci numbers and $GL(2,F _{p})$
  publication-title: Journal of Central China Normal University (Nat Sci )
– year: 2015
  ident: ref24
  publication-title: Algorithmic Number Theory
– year: 2002
  ident: ref12
  article-title: Factoring as optimization
– volume: 3
  start-page: 119
  year: 2007
  ident: ref17
  article-title: The Periodicity of Fibonacci modulus sequence
  publication-title: Math Pract Theory
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Snippet Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the...
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StartPage 167250
SubjectTerms Algorithms
Complexity
Cryptography
Curves
Elliptic curves
Factorization
Fibonacci
Fibonacci numbers
Fourier transforms
integer factorization
Integers
Nuclear magnetic resonance
Number theory
Pisano periods
Prime numbers
Qubit
RSA
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Title The Integer Factorization Algorithm With Pisano Period
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