[Paper Review] Continued Fractions and Factoring

Michele Elia|arXiv (Cornell University)|May 26, 2019

Advanced Mathematical Identities10 references4 citations

TL;DR

This paper demonstrates that the continued fraction expansion of √N reveals a factor of composite N when the period is even, extending Legendre’s result on sum-of-squares representations. By adapting Shanks’ infrastructural method and leveraging Dirichlet’s class number-regulator product, it proposes a factoring algorithm whose complexity depends on the precision of evaluating certain integrals related to L-functions.

ABSTRACT

Legendre found that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. Similarly, it is shown here that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of a composite $N$. Shanks' infrastructural method is then revisited, and some consequences of its application to factorization by means of the continued fraction expansion of $\sqrt N$ are derived.

Motivation & Objective

• To establish a direct link between the parity of the continued fraction period of √N and the existence of a nontrivial factor of composite N.
• To extend Legendre’s classical result on sum-of-squares representations to a factoring mechanism using continued fractions.
• To develop a factoring algorithm based on Shanks’ infrastructural method, using convergents and properties of quadratic forms.
• To show that the complexity of factoring N is bounded by the accuracy required to evaluate the product h_F R_F, where h_F is the class number and R_F the regulator of Q(√N).

Proposed method

• The paper uses the continued fraction expansion of √N to generate convergents A_m/B_m via second-order recurrence relations.
• It applies the infrastructural method of Shanks by identifying quadratic forms [1, 2Ω_{τ−1}, Δ_{τ−2}] at specific positions in the sequence of convergents.
• It leverages the identity A_m B_{m−1} − A_{m−1} B_m = (−1)^{m−1} to ensure coprimality and track convergent behavior.
• It uses Dirichlet’s formula for h_F R_F = −∑_{n=1}^{⌊(D−1)/2⌋} (D/n) ln(sin(nπ/D)) to estimate the regulator and class number product.
• The algorithm performs 'giant steps' based on approximations of h_F R_F to locate forms near positions kR_F/2, followed by 'baby steps' to refine the exact location.
• When a quadratic form with discriminant D is found at a distance h_F R_F / 2^ℓ from the start, it reveals a factor of N, especially when h_F is odd or divisible by powers of 2.

Experimental results

Research questions

• RQ1 Can the continued fraction expansion of √N with even period be used to directly extract a nontrivial factor of composite N?
• RQ2 How does Shanks’ infrastructural method apply to the sequence of convergents of √N to locate quadratic forms that reveal factors?
• RQ3 What is the role of the product h_F R_F in determining the complexity of the factoring process?
• RQ4 How accurately must h_F R_F be evaluated to ensure efficient factorization using this method?
• RQ5 Can the algorithm be iterated to factor N completely, and what determines the number of iterations?

Key findings

• When the period τ of the continued fraction expansion of √N is even, a factor of N is guaranteed to appear at positions (τ−2)/2 + kτ in the sequence of convergents.
• The method identifies a quadratic form [1, 2Ω_{τ−1}, Δ_{τ−2}] at distance h_F R_F / 2 from the start of the sequence, which reveals a factor of N.
• If h_F is odd, a factor is found at distance h_F R_F / 2 or 3h_F R_F / 2; if h_F is even, the process can be iterated to locate the factor at h_F R_F / 4, etc.
• The complexity of factoring N is upper bounded by the precision required to evaluate the product h_F R_F, which can be computed via Dirichlet’s formula involving trigonometric and logarithmic integrals.
• The algorithm can be iterated to factor N completely, with each step refining the location of a quadratic form that reveals a new factor.
• The method suggests a potential polynomial-time factoring algorithm if h_F R_F can be approximated with sufficient accuracy, though the optimal trade-off between approximation precision and computational cost remains open.