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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:2169-3536
2169-3536
DOI:10.1109/ACCESS.2019.2953755