abbr. SJ GMU
ISSN 2657-5841 (printed)
Integer Factorization – Cryptology Meets Number Theory
CSIRO, Sydney, Australia,
Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland, e-mail: email@example.com
Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about behaviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient factoring algorithm is equivalent to breaking RSA.
The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern factoring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathematicians and cryptographers. Shor's quantum factorization algorithm with polynomial complexity is described and the impact on public-key encryption is discussed.
This article is an open access article distributed under a Creative Commoms Attribution (CCBY 4.0) licence
Crandall, R., Pomerance, C., 2001, Prime Numbers: A Computational Perspective, Springer.
Dattani, N.S., Bryans, N., 2014, Quantum Factorization of 56153 with only 4 Qubits, Quantum Physics, arXiv:1411.6758,
Hirvensalo, M., 2001, Quantum Computing, Natural Computing Series, Springer.
Kleinjung, T., Aoki, K., Franke, J., Lenstra, A.K., Thomé, E., Bos, J.W., Gaudry, P., Kruppa, A., Montgomery, P.L., Osvik, D.A., te Riele, H., Timofeev, A., Zimmermann, P., 2010, Factorization of a 768-bit RSA Modulus, CRYPTO’10 Proceedings of the 30th Annual Conference on Advances in Cryptology, August 15–19, Santa Barbara, CA, USA, pp. 333–350.
Knuth, D., 1997, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, 3rd ed., Addison-Wesley, Boston, MA, USA.
Lehmer, D.H., Powers, R.E., 1931, On Factoring Large Numbers, Bull. Amer. Math. Soc., vol. 37, no. 10, pp. 770–776.
Manasse, M., Lenstra, A.K., 1999, RSA Honor Roll, http://www.ontko.com/pub/rayo/primes/hr_ rsa.txt (20.08.2018).
Martin-López, E., Laing, A., Lawson, T., Alvarez, R., Zhou, Xiao-Qi, O’Brien, J.L., 2012, Experimental Realization of Shor’s Quantum Factoring Algorithm using Qubit Recycling, Nature Photonics, vol. 6, no. 11.
NIST, 2018, Post-Quantum Cryptography, https://csrc.nist.gov/Projects/Post-Quantum-Cryptography, (20.08.2018).
Pieprzyk, J., Hardjono, T., Seberry, J., 2003, Fundamentals of Computer Security, Springer.
Pomerance, C., 1996, A Tale of Two Sieves, Notices Amer. Math. Soc, vol. 43, pp.1473–1485.
Rivest, R., Shamir, A., Adleman, L., 1978, A Method for Obtaining Digital Signatures and Public Key Cryptosystems, Communications of the ACM, vol. 21, no. 2, pp. 120–126.
Shor, P.W., 1997, Polynomial-time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing 26.5, pp. 1484–1509.
Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L., Experimental Realization of Shor’s Quantum Factoring Algorithm using Nuclear Magnetic Resonance, Nature, vol. 414 no. 6866, pp.883–887.
Wagstaff, S.S. Jr., 2013, The Joy of Factoring, American Mathematical Society, Providence, RI, USA.