Abstract
Suppose that we want to factor integer where \(N=pq\), p, q are two distinct odd primes. Then we can reduce the problem of integer factorization to computing the generators of the Mordell-Weil group of \(E_{Nr}:y^{2}=x^{3}-Nrx\), where r is a suitable integer with \((r,N)=1\). We consider the family of elliptic curves \(E_{Nr}\). As r varies in \(\mathbb {Z}\), we get two results. Firstly, we estimate the probability of \(r_{E_{Nr}}\ge 1\). Secondly, we estimate probability that the points on \(E_{Nr}\) can factor N. Finally we conduct some experiments to illustrate our method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Birch, B.J., Stephens, N.M.: The parity of the rank of the Mordell-Weil group. Topology 5(4), 295–299 (1966)
Birch, B.J., Swinnerton-Dyer, H.: Notes on elliptic curves. II. J. Für Die Rne Und Angewandte Mathematik 1963(212), 7–25 (1963)
Burhanuddin, I.A., Huang, M.: Factoring integers and computing elliptic curve rational points. Bsc Computer Science (2012)
Burhanuddin, I.A., Huang, M.: On the equation \(y^{2}=x^{3}-pqx\). J. Numbers 2014(193), 1–5 (2014)
Cohen, H.: Number Theory: Volume I: Tools and Diophantine Equations. Springer, New York (2008). https://doi.org/10.1007/978-0-387-49923-9
Cremona, J.: The elliptic curve database for conductors to 130000. In: International Algorithmic Number Theory Symposium (2006)
Dokchitser, T.: Notes on the parity conjecture. In: Elliptic Curves, Hilbert Modular Forms and Galois Deformations. Springer, Basel (2010). https://doi.org/10.1007/978-3-0348-0618-3_5
Kane, D.M., Thorne, J.A.: On the \(\phi \)-selmer groups of the elliptic curves \(y^{2} = x^{3}-dx\). Math. Proc. Cambridge Philos. Soc. 163(01), 1–23 (2016)
Li, X.M., Zeng, J.X.: On the elliptic curve \(y^{2} = x^{3}-2rdx\) and factoring integers. Sci. China 2014(04), 719–728 (2014)
Miller, R.L.: Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one. LMS J. Comput. Math. (2013)
Pan, Z., Li, X.: Elliptic curve and integer factorization. In: International Conference on Information Security and Cryptology (2021)
Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM, vol. 106. Springer, New York (2009). https://doi.org/10.1007/978-0-387-09494-6
Washington, L.C.: Elliptic Curves: Number Theory and Cryptography, 2nd edn. CRC Press, Inc. (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Pan, Z., Li, X. (2021). A Method of Integer Factorization. In: Stănică, P., Mesnager, S., Debnath, S.K. (eds) Security and Privacy. ICSP 2021. Communications in Computer and Information Science, vol 1497. Springer, Cham. https://doi.org/10.1007/978-3-030-90553-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-90553-8_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-90552-1
Online ISBN: 978-3-030-90553-8
eBook Packages: Computer ScienceComputer Science (R0)