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.
Highlights
Large integer factorization is one of the basic issues in number theory
Any progress in large integer factorization will attract the attention of the cryptography community
Whether the security of RSA is equivalent to integer factorization has not been proven theoretically since there is no proof that breaking RSA requires large integer factorization
Summary
Large integer factorization is one of the basic issues in number theory. The Pisano period of the product of two prime numbers will be studied in-depth and associated with large integer factorization. It can be seen that the subsequence, {Fkd(m)+j(m)}, k ≥ 0, 0 ≤ j ≤ d(m) − 1, of the Pisano sequence is a geometric series modulo m for a fixed j, with Fj(m) as the first term and t as the common ratio: Fkd(m)+j(m) ≡ Fj(m)tk−1 (mod m). Definition: For the integer sequence {ωn}, an integer m greater than 1, an integer s greater than 0, a nonnegative integer n0 and an integer c, if gcd(m, c) = 1, n ≥ n0, ωn+s ≡ cωn (mod m), the minimum positive integer s satisfying the above equation is called the constraint period of.
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