Proving the primality of an integer N (see primality proving algorithm) is easy if \(N-1\) can be factored: N is a prime number if and only if the multiplicative group of invertible elements (ℤ/Nℤ)* is cyclic of order \(N-1\) (see modular arithmetic). To prove that an integer g is a generator of (ℤ/Nℤ)* and hence that the group is cyclic, it suffices to check that \(g^{N-1}\equiv 1 \bmod N\) and \(g^{(N-1)/q}\not\equiv 1\bmod N\) for all prime factors q of \(N-1\). (It is quite easy to find a generator, or to prove that none exists, given the prime factors.)
The above method is the converse of Fermat's Little Theorem. However, it is rare that \(N-1\) is easy to factor. Less rare is the case where \(N-1\) has a large prime cofactor C, in which case the primality of \(N-1\) can be proven in the same way, modulo the assumption that Ccan be proven prime in turn. This approach of primality cannot succeed to prove the primality of all numbers in reasonable time. Other approaches have...
This is a preview of subscription content, log in via an institution.
References
Adleman, L.M. and M.-D.A. Huang (1996). “Primality testing and Abelian varieties over finite fields.” Proc. of International Workshop TYPES'96, Lecture Notes in Math, vol. 1512, eds. E. Gimenez and C. Paulin-Mohring. Springer-Verlag, Berlin.
Adleman, L.M., C. Pomerance, and R.S. Rumely (1983). “On distinguishing prime numbers from composite numbers.” Ann. Math. (2), 117 (1), 173–206.
Atkin, A.O.L. (1986). Manuscript. Lecture Notes of a Conference, Boulder CO.
Atkin, A.O.L. and F. Morain (1993). “Elliptic curves and primality proving.” Math. Comp., 61 (203), 29–68.
Bosma, W. and M.-P. van der Hulst (1990). “Primality proving with cyclotomy.” PhD Thesis, Universiteit van Amsterdam.
Cohen, H. and A.K. Lenstra (1987). “Implementation of a new primality test.” Math. Comp., 48 (177), 103–121.
Cohen, H. and H.W. Lenstra, Jr. (1984). “Primality testing and Jacobi sums.” Math. Comp., 42 (165), 297–330.
Franke, J., T. Kleinjung, F. Morain, and T. Wirth. Proving the primality of very large numbers with fastecpp. In D. Buell, editor, Algorithmic Number Theory, volume 3076 of Lecture Notes in Computer Science, pages 194–207. Springer, Berlin, 2004. 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 2004.
Goldwasser, S. and J. Kilian (1986). “Almost all primes can be quickly certified.” Proc. 18th STOC, Berkeley, CA. ACM, New York, 316–329.
Goldwasser, S. and J. Kilian (1990). “Primality testing using elliptic curves.” J. ACM, 46 (4), 450–472.
Lenstra, A.K. and H.W. Lenstra, Jr. (1996). “Algorithms in number theory.” Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, ed. J. van Leeuwen. North Holland, Amsterdam, Chapter 12, 674–715.
Mihăilescu, P. (1997). “Cyclotomy of rings and primality testing.” Diss. ETH no. 12278, Swiss Federal Institute of Technology Zürich.
Morain, F. (2005). Implementing the asymptotically fast version of the elliptic curve primality proving algorithm. Available at http://www.lix.polytechnique.fr/Labo/Francois.Morain/
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 International Federation for Information Processing
About this entry
Cite this entry
Morain, F. (2005). Elliptic Curves for Primality Proving. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_139
Download citation
DOI: https://doi.org/10.1007/0-387-23483-7_139
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23473-1
Online ISBN: 978-0-387-23483-0
eBook Packages: Computer ScienceReference Module Computer Science and Engineering