Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Enge, Andreas (2013)
Publisher: Chapman and Hall/CRC
Journal: Handbook of Finite Fields
Languages: English
Types: Part of book or chapter of book
Subjects: elliptic curves, [INFO.INFO-CR] Computer Science [cs]/Cryptography and Security [cs.CR], cryptology, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]

Classified by OpenAIRE into

ACM Ref: ComputingMilieux_MISCELLANEOUS
International audience
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • September 1998. Available at http://grouper.ieee.org/groups/1363/private/ x9-62-09-20-98.zip.
    • [2] ANSI. Key agreement and key transport using elliptic curve cryptography. Working Draft American National Standard: Public Key Cryptography for the Financial Services Industry X9.63-199x, American National Standards Institute, January 1999. Available at http://grouper.ieee.org/groups/1363/private/ x9-63-01-08-99.zip.
    • [3] R. Balasubramanian and Neal Koblitz. The improbability that an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm. J. Cryptology, 11(2):141{145, 1998.
    • [4] Paulo S. L. M. Barreto, Steven D. Galbraith, Colm O'hEigeartaigh, and Michael Scott. E cient pairing computation on supersingular abelian varieties. Designs, Codes and Cryptography, 42:239{271, 2007.
    • [5] Juliana Belding, Reinier Broker, Andreas Enge, and Kristin Lauter. Computing Hilbert class polynomials. In Alf van der Poorten and Andreas Stein, editors, Algorithmic Number Theory | ANTS-VIII, volume 5011 of Lecture Notes in Computer Science, pages 282{295, Berlin, 2008. Springer-Verlag.
    • [6] Mihir Bellare and Phillip Rogaway. Minimizing the use of random oracles in authenticated encryption schemes. In Yongfei Han, Tatsuaki Okamoto, and Sihan Qing, editors, Information and Communications Security, volume 1334 of Lecture Notes in Computer Science, pages 1{16, Berlin, 1997. Springer-Verlag.
    • [7] I. F. Blake, G. Seroussi, and N. P. Smart. Elliptic curves in cryptography, volume 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2000. Reprint of the 1999 original.
    • [8] Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart. Advances in Elliptic Curve Cryptography, volume 317 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.
    • [9] Dan Boneh, Eu-Jin Goh, and Kobbi Nissim. Evaluating 2-DNF formulas on ciphertexts. In Joe Kilian, editor, Theory of Cryptography | TCC 2005, volume 3378 of Lecture Notes in Computer Science, pages 325{341, Berlin, 2005. Springer-Verlag.
    • [10] A. Bostan, F. Morain, B. Salvy, and E. Schost. Fast algorithms for computing isogenies between elliptic curves. Mathematics of Computation, 77(263):1755{1778, 2008.
    • [11] Friederike Brezing and Annegret Weng. Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr., 37(1):133{141, 2005.
    • [12] Daniel R. L. Brown. Generic groups, collision resistance, and ECDSA. Designs, Codes and Cryptography, 35:119{152, 2005.
    • [15] Henri Cohen. A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1993.
    • [16] Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, and Frederik Vercauteren, editors. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006.
    • [17] Jean-Marc Couveignes and Thierry Henocq. Action of modular correspondences around CM points. In Claus Fieker and David R. Kohel, editors, Algorithmic Number Theory | ANTS-V, volume 2369 of Lecture Notes in Computer Science, pages 234{243, Berlin, 2002. Springer-Verlag.
    • [18] Jean-Marc Couveignes and Jean-Gabriel Kammerer. The geometry of ex tangents to a cubic curve and its parameterizations. Journal of Symbolic Computation, 47:266{281, 2012.
    • [19] Claus Diem. The GHS attack in odd characteristic. Journal of the Ramanujan Mathematical Society, 18(1):1{32, 2003.
    • [20] Noam D. Elkies. Elliptic and modular curves over nite elds and related computational issues. In Computational perspectives on number theory (Chicago, IL, 1995), volume 7 of AMS/IP Stud. Adv. Math., pages 21{76. Amer. Math. Soc., Providence, RI, 1998.
    • [21] Andreas Enge. The complexity of class polynomial computation via oating point approximations. Mathematics of Computation, 78(266):1089{1107, 2009.
    • [22] Andreas Enge. Computing modular polynomials in quasi-linear time. Mathematics of Computation, 78(267):1809{1824, 2009.
    • [23] Reza Rezaeian Farashahi. Hashing into Hessian curves. To appear in Africacrypt, 2011.
    • [24] David Mandell Freeman. Converting pairing-based cryptosystems from compositeorder groups to prime-order groups. In Henri Gilbert, editor, Advances in Cryptology | EUROCRYPT 2010, volume 6110 of Lecture Notes in Computer Science, pages 44{61, Berlin, 2010. Springer-Verlag.
    • [25] David Freemann, Michael Scott, and Edlyn Teske. A taxonomy of pairing-friendly elliptic curves. Journal of Cryptology, 23(2):224{280, 2010.
    • [26] G. Frey and H.-G. Ruck. A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Math. Comp., 62:865{874, 1994.
    • [27] Gerhard Frey. Applications of arithmetical geometry to cryptographic constructions. In Dieter Jungnickel and Harald Niederreiter, editors, Finite Fields and Applications | Proceedings of The Fifth International Conference on Finite Fields and Applications Fq5, held at the University of Augsburg, Germany, August 2{6, 1999, pages 128{161, Berlin, 2001. Springer-Verlag.
    • [28] D. Fu and J. Solinas. IKE and IKEv2 authentication using the elliptic curve digital signature algorithm (ECDSA). RFC 4754, Internet Engineering Task Force, 2007. http://www.ietf.org/rfc/rfc4754.txt.
    • [29] Steven D. Galbraith, Florian Hess, and Nigel P. Smart. Extending the GHS Weil descent attack. In Lars Knudsen, editor, Advances in Cryptology | EUROCRYPT 2002, volume 2332 of Lecture Notes in Computer Science, pages 29{44, Berlin, 2002. Springer-Verlag.
    • [30] Steven D. Galbraith and Kenneth G. Paterson, editors. Pairing-Based Cryptography | Pairing 2008, volume 5209 of Lecture Notes in Computer Science, Berlin, 2008. Springer-Verlag.
    • [31] Steven D. Galbraith and Nigel P. Smart. A cryptographic application of Weil descent. In Michael Walker, editor, Cryptography and Coding, volume 1746 of Lecture Notes in Computer Science, pages 191{200, Berlin, 1999. Springer-Verlag.
    • [32] Robert Gallant, Robert Lambert, and Scott Vanstone. Improving the parallelized Pollard lambda search on binary anomalous curves. Mathematics of Computation, 69(232):1699{1705, 2000.
    • [33] P. Gaudry, F. Hess, and N. P. Smart. Constructive and destructive facets of Weil descent on elliptic curves. Journal of Cryptology, 15(1):19{46, 2002.
    • [34] P. Gaudry and F. Morain. Fast algorithms for computing the eigenvalue in the Schoof{Elkies{Atkin algorithm. In Jean-Guillaume Dumas, editor, Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computations | ISSAC MMVI, pages 109{115, New York, 2006. ACM Press.
    • [35] Pierrick Gaudry. Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. Journal of Symbolic Computation, 44(12):1690{1702, 2009.
    • [36] Damien Giry and Jean-Jacques Quisquater. Bluekrypt cryptographic key length recommendation, 2011. v26.0, April 18, http://www.keylength.com/.
    • [38] David Harvey. Kedlaya's algorithm in larger characteristic. Int. Math. Res. Not. IMRN, 2007(22):Art. ID rnm095, 29, 2007.
    • [39] Florian Hess. Pairing lattices. In S. D. Galbraith and K. Paterson, editors, PairingBased Cryptography | Pairing 2008, volume 5209 of Lecture Notes in Computer Science, pages 18{38, Berlin, 2008. Springer-Verlag.
    • [40] Florian Hess, Nigel P. Smart, and Frederik Vercauteren. The eta pairing revisited. IEEE Transactions on Information Theory, 52(10):4595{4602, 2006.
    • [41] Thomas Icart. How to hash into elliptic curves. In Shai Halevi, editor, Advances in Cryptology | CRYPTO 2009, volume 5677 of Lecture Notes in Computer Science, pages 303{316, Berlin, 2009. Springer-Verlag.
    • [42] IEEE. Standard speci cations for public key cryptography. Standard P1363-2000, Institute of Electrical and Electronics Engineering, 2000. Draft D13 available at http://grouper.ieee.org/groups/1363/P1363/draft.html.
    • [43] Antoine Joux and Vanessa Vitse. Cover and decomposition index calculus on elliptic curves made practical | Application to a seemingly secure curve over Fp6. To appear in Eurocrypt 2012, http://eprint.iacr.org/2011/020.pdf, 2011.
    • [44] Marc Joye, Atsuko Miyaji, and Akira Otsuka, editors. Pairing-Based Cryptography | Pairing 2010, volume 6487 of Lecture Notes in Computer Science, Berlin, 2010. Springer-Verlag.
    • [45] Jean-Gabriel Kammerer, Reynald Lercier, and Guenael Renault. Encoding points on hyperelliptic curves over nite elds in deterministic polynomial time. In Marc Joye, Atsuko Miyaji, and Akira Otsuka, editors, Pairing-Based Cryptography | Pairing 2010, volume 6487 of Lecture Notes in Computer Science, pages 278{297, Berlin, 2010. Springer-Verlag.
    • [46] Neal Koblitz. Elliptic curve cryptosystems. 48(177):203{209, 1987.
    • [47] Arjen K. Lenstra and Eric R. Verheul. Selecting cryptographic key sizes (extended abstract). In Hideki Imai and Yuliang Zheng, editors, Public Key Cryptography | 3rd International Workshop on Practice and Theory in Public Key Cryptosystems PKC 2000, volume 1751 of Lecture Notes in Computer Science, pages 446{465, Berlin, 2000. Springer-Verlag.
    • [49] Alfred J. Menezes, Tatsuaki Okamoto, and Scott A. Vanstone. Reducing elliptic curve logarithms to logarithms in a nite eld. IEEE Trans. Inform. Theory, 39(5):1639{1646, 1993.
    • [50] Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. Handbook of applied cryptography. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, 1997. With a foreword by Ronald L. Rivest.
    • [51] Jean-Francois Mestre. Lettre adressee a Gaudry et Harley. http://www.math. jussieu.fr/~mestre/lettreGaudryHarley.ps, December 2000.
    • [52] P. Mihailescu, F. Morain, and E. Schost. Computing the eigenvalue in the Schoof{ Elkies{Atkin algorithm using abelian lifts. In C. W. Brown, editor, Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation | ISSAC 2007, pages 285{292, New York, 2007. Association for Computing Machinery.
    • [53] Victor S. Miller. Use of elliptic curves in cryptography. In Hugh C. Williams, editor, Advances in Cryptology | CRYPTO '85, volume 218 of Lecture Notes in Computer Science, pages 417{426, Berlin, 1986. Springer-Verlag.
    • [54] Gary L. Mullen and Daniel Panario, editors. Handbook of Finite Fields. Discrete Mathematics and Its Applications. Chapman and Hall/CRC, Boca Raton, 2013.
    • [55] NIST. Digital signature standard (DSS). Federal Information Processing Standards Publication 186-3, National Institute of Standards and Technology, July 2009.
    • [56] Bart Preneel et al. NESSIE security report. Technical Report D20-v2, New European Schemes for Signatures, Integrity, and Encryption, 2003.
    • [57] Hans-Georg Ruck. A note on elliptic curves over nite elds. Mathematics of Computation, 49(179):301{304, July 1987.
    • [58] Takakazu Satoh. The canonical lift of an ordinary elliptic curve over a nite eld and its point counting. J. Ramanujan Math. Soc., 15(4):247{270, 2000.
    • [59] Takakazu Satoh and Kiyomichi Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul., 47(1):81{92, 1998.
    • [60] Rene Schoof. Elliptic curves over nite elds and the computation of square roots mod p. Mathematics of Computation, 44(170), April 1985.
    • [61] I. A. Semaev. Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p. Math. Comp., 67(221):353{356, 1998.
    • [62] Hovav Shacham and Brent Waters, editors. Pairing-Based Cryptography | Pairing 2009, volume 5671 of Lecture Notes in Computer Science, Berlin, 2009. SpringerVerlag.
    • [63] Andrew Shallue and Christiaan E. van de Woestijne. Construction of rational points on elliptic curves over nite elds. In Florian Hess, Sebastian Pauli, and Michael Pohst, editors, Algorithmic Number Theory | ANTS-VII, volume 4076 of Lecture Notes in Computer Science, pages 510{524, Berlin, 2006. Springer-Verlag.
    • [64] M. Skalba. Points on elliptic curves over nite elds. Acta arithmetica, 117(3):293{ 301, 2005.
    • [65] N. P. Smart. The discrete logarithm problem on elliptic curves of trace one. J. Cryptology, 12(3):193{196, 1999.
    • [66] N. P. Smart. The exact security of ECIES in the generic group model. In Bahram Honary, editor, Cryptography and Coding, volume 2260 of Lecture Notes in Computer Science, pages 73{84, Berlin, 2001. Springer-Verlag.
    • [67] Nigel Smart et al. ECRYPT II yearly report on algorithms and keysizes (2009- 2010). Technical Report D.SPA.13, European Network of Excellence in Cryptology II, 2010.
    • [68] Jacques Stern, David Pointcheval, John Malone-Lee, and Nigel Smart. Flaws in applying proof methodologies to signature schemes. In Moti Yung, editor, Advances in Cryptology | CRYPTO 2002, volume 2442 of Lecture Notes in Computer Science, pages 93{110, Berlin, 2002. Springer-Verlag.
    • [69] Andrew V. Sutherland. Genus 1 point counting in essentially quartic time and quadratic space, September 2010. Slides, http://math.mit.edu/~drew/NYU0910. pdf.
    • [70] Andrew V. Sutherland. Genus 1 point-counting record modulo a 5000+ digit prime, July 2010. Posting to the Number Theory List, http: //listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1007&L=nmbrthry& T=0&F=&S=&P=287.
    • [71] Tsuyoshi Takagi, Tatsuaki Okamoto, Eiji Okamoto, and Takeshi Okamoto, editors. Pairing-Based Cryptography | Pairing 2007, volume 4575 of Lecture Notes in Computer Science, Berlin, 2007. Springer-Verlag.
    • [72] Alberto Tonelli. Bemerkung uber die Au osung quadratischer Congruenzen. Nachrichten von der Konigl. Gesellschaft der Wissenschaften und der GeorgAugusts-Universitat zu Gottingen, pages 344{346, 1891.
    • [73] Frederic Vercauteren. Optimal pairings. IEEE Transactions on Information Theory, 56(1):455{461, 2010.
    • [74] Eric R. Verheul. Evidence that XTR is more secure than supersingular elliptic curve cryptosystems. Journal of Cryptology, 17(4):277{296, 2004.
    • [75] William C. Waterhouse. Abelian varieties over nite elds. Annales Scienti ques de l'Ecole Normale Superieure, 4e Serie, 2:521{560, 1969.
    • [76] Michael J. Wiener and Robert J. Zuccherato. Faster attacks on elliptic curve cryptosystems. In Sta ord Tavares and Henk Meijer, editors, Selected Areas in Cryptography | SAC '98, volume 1556 of Lecture Notes in Computer Science, pages 190{100, Berlin, 1999. Springer-Verlag.
  • No related research data.
  • No similar publications.

Share - Bookmark

Published in

  • Handbook of Finite Fields

Funded by projects


Cite this article