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Some remarks on primality proving and elliptic curves
1. | Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States |
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. |
show all references
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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