# American Institute of Mathematical Sciences

November  2014, 8(4): 427-436. doi: 10.3934/amc.2014.8.427

## Some remarks on primality proving and elliptic curves

 1 Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States

Received  January 2014 Published  November 2014

We give an overview of a method for using elliptic curves with complex multiplication to give efficient deterministic polynomial time primality tests for the integers in sequences of a special form. This technique has been used to find the largest proven primes $N$ for which there was no known significant partial factorization of $N-1$ or $N+1$.
Citation: Alice Silverberg. Some remarks on primality proving and elliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 427-436. doi: 10.3934/amc.2014.8.427
##### References:
 [1] A. Abatzoglou, A CM elliptic curve framework for deterministic primality proving on numbers of special form, Ph.D thesis, University of California at Irvine, 2014. [2] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, available online at http://primes.utm.edu/primes/page.php?id=106847 [3] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, available online at http://primes.utm.edu/primes/page.php?id=117544 [4] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, Deterministic elliptic curve primality proving for a special sequence of numbers, in Algorithmic Number Theory, Math. Sci. Publ., 2013, 1-20. doi: 10.2140/obs.2013.1.1. [5] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, Math. Comp., to appear. [6] M. Agrawal, N. Kayal and N. Saxena, Primes is in P, Ann. Math., 160 (2004), 781-793. doi: 10.4007/annals.2004.160.781. [7] A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp., 61 (1993), 29-68. doi: 10.1090/S0025-5718-1993-1199989-X. [8] W. Bosma, Primality Testing with Elliptic Curves, Doctoraalscriptie Report, University of Amsterdam 85-12, 1985, available online at http://www.math.ru.nl/ bosma/pubs/PRITwEC1985.pdf [9] D. V. Chudnovsky and G. V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization tests, Adv. Appl. Math., 7 (1986), 385-434. doi: 10.1016/0196-8858(86)90023-0. [10] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Second edition, Springer, New York, 2005. [11] R. Denomme and G. Savin, Elliptic curve primality tests for Fermat and related primes, J. Number Theory, 128 (2008), 2398-2412. doi: 10.1016/j.jnt.2007.12.009. [12] S. Goldwasser and J. Kilian, Almost all primes can be quickly certified, in STOC '86 - Proc. 18th Annual ACM Symp. Theory Computing, 1986, 316-329. doi: 10.1145/12130.12162. [13] S. Goldwasser and J. Kilian, Primality testing using elliptic curves, J. ACM, 46 (1999), 450-472. doi: 10.1145/320211.320213. [14] D. M. Gordon, Pseudoprimes on elliptic curves, in Théorie des nombres, de Gruyter, Berlin, 1989, 290-305. [15] B. H. Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Springer, Berlin, 1980. [16] B. H. Gross, Minimal models for elliptic curves with complex multiplication, Compositio Math., 45 (1982), 155-164. [17] B. H. Gross, An elliptic curve test for Mersenne primes, J. Number Theory, 110 (2005), 114-119. doi: 10.1016/j.jnt.2003.11.011. [18] A. Gurevich and B. Kunyavskiĭ, Primality testing through algebraic groups, Arch. Math. (Basel), 93 (2009), 555-564. doi: 10.1007/s00013-009-0065-9. [19] A. Gurevich and B. Kunyavskiĭ, Deterministic primality tests based on tori and elliptic curves, Finite Fields Appl., 18 (2012), 222-236. doi: 10.1016/j.ffa.2011.07.011. [20] M. Kida, Primality tests using algebraic groups, Exper. Math., 13 (2004), 421-427. doi: 10.1080/10586458.2004.10504550. [21] H. W. Lenstra, Jr., Elliptic curves and number-theoretic algorithms, in Proc. Int. Congr. Math., Amer. Math. Soc., Providence, 1987, 99-120. [22] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. Math., 126 (1987), 649-673. doi: 10.2307/1971363. [23] H. W. Lenstra, Jr. and C. Pomerance, Primality testing with Gaussian periods, available online at http://www.math.dartmouth.edu/~carlp/aks041411.pdf, 2011. [24] C. Pomerance, Primality testing: variations on a theme of Lucas, Congr. Numer., 201 (2010), 301-312. [25] K. Rubin and A. Silverberg, Point counting on reductions of CM elliptic curves, J. Number Theory, 129 (2009), 2903-2923. doi: 10.1016/j.jnt.2009.01.020. [26] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, Math. Comp., 44 (1985), 483-494. doi: 10.2307/2007968. [27] A. Silverberg, Group order formulas for reductions of CM elliptic curves, in Proc. Conf. Arith. Geom. Crypt. Coding Theory, Amer. Math. Soc., Providence, 2010, 107-120. doi: 10.1090/conm/521/10277. [28] H. Stark, Counting points on CM elliptic curves, Rocky Mountain J. Math., 26 (1996), 1115-1138. doi: 10.1216/rmjm/1181072041. [29] Y. Tsumura, Primality tests for $2^p + 2^{\frac{p+1}{2}} + 1$ using elliptic curves, Proc. Amer. Math. Soc., 139 (2011), 2697-2703. doi: 10.1090/S0002-9939-2011-10839-6. [30] A. Wong, Primality Test Using Elliptic Curves with Complex Multiplication by $\mathbbQ(\sqrt{-7})$, Ph.D thesis, University of California at Irvine, 2013.

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##### References:
 [1] A. Abatzoglou, A CM elliptic curve framework for deterministic primality proving on numbers of special form, Ph.D thesis, University of California at Irvine, 2014. [2] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, available online at http://primes.utm.edu/primes/page.php?id=106847 [3] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, available online at http://primes.utm.edu/primes/page.php?id=117544 [4] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, Deterministic elliptic curve primality proving for a special sequence of numbers, in Algorithmic Number Theory, Math. Sci. Publ., 2013, 1-20. doi: 10.2140/obs.2013.1.1. [5] A. Abatzoglou, A. Silverberg, A. V. Sutherland and A. Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, Math. Comp., to appear. [6] M. Agrawal, N. Kayal and N. Saxena, Primes is in P, Ann. Math., 160 (2004), 781-793. doi: 10.4007/annals.2004.160.781. [7] A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp., 61 (1993), 29-68. doi: 10.1090/S0025-5718-1993-1199989-X. [8] W. Bosma, Primality Testing with Elliptic Curves, Doctoraalscriptie Report, University of Amsterdam 85-12, 1985, available online at http://www.math.ru.nl/ bosma/pubs/PRITwEC1985.pdf [9] D. V. Chudnovsky and G. V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization tests, Adv. Appl. Math., 7 (1986), 385-434. doi: 10.1016/0196-8858(86)90023-0. [10] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Second edition, Springer, New York, 2005. [11] R. Denomme and G. Savin, Elliptic curve primality tests for Fermat and related primes, J. Number Theory, 128 (2008), 2398-2412. doi: 10.1016/j.jnt.2007.12.009. [12] S. Goldwasser and J. Kilian, Almost all primes can be quickly certified, in STOC '86 - Proc. 18th Annual ACM Symp. Theory Computing, 1986, 316-329. doi: 10.1145/12130.12162. [13] S. Goldwasser and J. Kilian, Primality testing using elliptic curves, J. ACM, 46 (1999), 450-472. doi: 10.1145/320211.320213. [14] D. M. Gordon, Pseudoprimes on elliptic curves, in Théorie des nombres, de Gruyter, Berlin, 1989, 290-305. [15] B. H. Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Springer, Berlin, 1980. [16] B. H. Gross, Minimal models for elliptic curves with complex multiplication, Compositio Math., 45 (1982), 155-164. [17] B. H. Gross, An elliptic curve test for Mersenne primes, J. Number Theory, 110 (2005), 114-119. doi: 10.1016/j.jnt.2003.11.011. [18] A. Gurevich and B. Kunyavskiĭ, Primality testing through algebraic groups, Arch. Math. (Basel), 93 (2009), 555-564. doi: 10.1007/s00013-009-0065-9. [19] A. Gurevich and B. Kunyavskiĭ, Deterministic primality tests based on tori and elliptic curves, Finite Fields Appl., 18 (2012), 222-236. doi: 10.1016/j.ffa.2011.07.011. [20] M. Kida, Primality tests using algebraic groups, Exper. Math., 13 (2004), 421-427. doi: 10.1080/10586458.2004.10504550. [21] H. W. Lenstra, Jr., Elliptic curves and number-theoretic algorithms, in Proc. Int. Congr. Math., Amer. Math. Soc., Providence, 1987, 99-120. [22] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. Math., 126 (1987), 649-673. doi: 10.2307/1971363. [23] H. W. Lenstra, Jr. and C. Pomerance, Primality testing with Gaussian periods, available online at http://www.math.dartmouth.edu/~carlp/aks041411.pdf, 2011. [24] C. Pomerance, Primality testing: variations on a theme of Lucas, Congr. Numer., 201 (2010), 301-312. [25] K. Rubin and A. Silverberg, Point counting on reductions of CM elliptic curves, J. Number Theory, 129 (2009), 2903-2923. doi: 10.1016/j.jnt.2009.01.020. [26] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, Math. Comp., 44 (1985), 483-494. doi: 10.2307/2007968. [27] A. Silverberg, Group order formulas for reductions of CM elliptic curves, in Proc. Conf. Arith. Geom. Crypt. Coding Theory, Amer. Math. Soc., Providence, 2010, 107-120. doi: 10.1090/conm/521/10277. [28] H. Stark, Counting points on CM elliptic curves, Rocky Mountain J. Math., 26 (1996), 1115-1138. doi: 10.1216/rmjm/1181072041. [29] Y. Tsumura, Primality tests for $2^p + 2^{\frac{p+1}{2}} + 1$ using elliptic curves, Proc. Amer. Math. Soc., 139 (2011), 2697-2703. doi: 10.1090/S0002-9939-2011-10839-6. [30] A. Wong, Primality Test Using Elliptic Curves with Complex Multiplication by $\mathbbQ(\sqrt{-7})$, Ph.D thesis, University of California at Irvine, 2013.
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