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Please send corrections and suggestions to Marc Joye (marc.joye@gemplus.com)


SOME INTERESTING REFERENCES ON ELLIPTIC CURVES...

[People] [Bibliography] [Software] [Links]

Weierstrass


PEOPLE
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Name Comments
Matthew ANDO (@) -
Oliver ATKIN (@, @) -
George BARWOOD (@) EC crypto FAQ
Renet Lovorn BENDER (@) -
Dominique BERNARDI (@) -
Dan J. BERNSTEIN (@) Papers
Daniel BLEICHENBACHER (@) Papers, Thesis
Dan BONEH (@) Papers
Thérèse BOOGAERTS (@) [Boo98]
Francis BORCEUX (@) [BQ95]
Wieb BOSMA (@) -
David M. BRESSOUD (@) [Bre89]
Johannes BUCHMANN (@) Papers
J.W.S. CASSELS (@) [Cas66], [Cas91]
Leonard S. CHARLAP (@) [CC90], [CR88]
John H. COATES (@) -
Henri COHEN (@) Papers, [Coh93]
Ian CONNELL (@) [Con96]
Jean-Marc COUVEIGNES (@) Papers, [Cou94]
Richard CRANDALL (@) -
John CREMONA (@) Papers, [Cre97]
János A. CSIRIK (@) Papers
James DAVENPORT (@) Papers
Jonathan DeMARRAIS (@) -
Erik DE WIN (@) Papers
Fred DIAMOND (@, @) -
Noam D. ELKIES (@) Papers
Andreas ENGE (@) [Eng97], [Eng99], EC page
Graham EVEREST (@) [EW99]
Stéphane FERMIGIER (@) [Fer94], EC page
Victor FLYNN (@) -
Gerhard FREY (@) Preprints
Steven D. GALBRAITH (@) [Gal96], EC page
Pierrick GAUDRY (@) [Gau95]
Shafi GOLDWASSER (@) -
Andrew GRANVILLE (@) Preprints/Reprints
Jorge GUAJARDO (@) Papers, [Gua97]
Guillaume HANROT (@) Papers, [Han97]
Erwin HESS (@) -
Robert HOFER (@) -
John JONES (@) Papers
Marc JOYE (@) Papers, [Joy95], [Joy97]
Charanjit Singh JUTLA (@) -
Burt S. KALISKI, Jr (@) [Kal88]
Hershy KISILEVSKY (@) [KM94]
Neal KOBLITZ (@) [Kob93], [Kob94], [Kob98]
Jun KOGURE (@) -
David R. KOHEL (@) Papers, [Koh96]
Kenji KOYAMA (@) -
Kenneth KRAMER (@) -
Larry LEHMAN (@) Papers
Arjen LENSTRA (@) -
Hendrik W. LENSTRA, Jr (@) -
Franck LEPREVOST (@) -
Reynald LERCIER (@) Papers, [Ler97], ZEN
Ueli MAURER (@) Papers
Barry MAZUR (@) -
Kevin McCURLEY (@, @) -
Alfred MENEZES (@) Papers, [MBG+92], [Men93], [MWZ96]
Jean-François MESTRE (@) -
Preda MIHAILESCU (@) -
Victor MILLER (@) -
J.S. MILNE (@) Papers, Course Notes, [Mil90], [Mil96], [Mil98]
Atsuko MIYAJI (@) -
Richard A. MOLLIN (@) -
Peter L. MONTGOMERY (@) CWI Reports, [Mon92]
François MORAIN (@) Papers, [Mor90]
Michael MÜLLER (@) Preprints
Volker MÜLLER (@) Papers, [Mül91], [Mül95]
Ram MURTY (@) Papers, [KM94]
Andrew ODLYZKO (@) Papers
Tatsuaki OKAMOTO (@) -
Takashi ONO (@) -
Christof PAAR (@) Papers
Sachar PAULUS (@) Preprints
Richard PINCH (@) Papers, EC page
Carl POMERANCE (@) -
Jean-Jacques QUISQUATER (@) Papers, [BQ95]
Kenneth A. RIBET (@) Papers
Matt ROBSHAW (@) -
Mike G. ROSING (@) [Ros98]
Karl RUBIN (@) Preprints, [Rub]
Hans-Georg RÜCK (@) Preprints
Dave RUSIN (@) Papers
Naoki SATOH (@) -
Takakazu SATOH (@) Papers, Preprints
Roger SCHLAFLY (@, @) -
René SCHOOF (@, @) -
Rich SCHROEPPEL (@) -
Mike SCOTT (@) -
Victor SHOUP (@, @) Papers
Joseph H. SILVERMAN (@) Preprints, [Sil86], [Sil95], [Sil97], [ST92]
Nigel P. SMART (@) Papers, [Sma98], [BSS99]
Jerome SOLINAS (@, @) -
Andreas STEIN (@) Papers
William A. STEIN (@) Papers, [Ste00], Modular forms database
Tsuyoshi TAKAGI (@) Papers
Scott A. VANSTONE (@) -
Emil VOLCHECK (@) -
José Felipe VOLOCH (@) Preprints
Joachim von zur GATHEN (@) -
Tom WARD (@) Papers, [EW99]
Hugh C. WILLIAMS (@) -
James Pate WILLIAMS, Jr (@) -
Erik V. YORK (@) [Yor92]
Noriko YUI (@) Papers
Michael E. ZIEVE (@, @) Papers
Horst-Günter ZIMMER (@) -
XianKe ZHANG (@) Papers
Robert J. ZUCCHERATO (@, @) [MWZ96], [Zuc97]

See also the Number Theorists' Home Pages by Keith Matthews.

BIBLIOGRAPHY
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1. BOOKS

[Apo90] APOSTOL, T.M.
Modular functions and Dirichlet series in number theory Graduate Texts in Mathematics 41, Springer-Verlag, 2nd edition, 1990.

[BK75] BIRCH, B., and KUYK, W. (Eds.)
Modular forms in one variable, Lecture Notes in Mathematics 476, Springer-Verlag, 1975.

[Bix98] BIX, R.
Conics and cubics: A concrete introduction to algebraic curves, Undergraduate Texts in Mathematics, Springer-Verlag, 1998.

[BSS99] BLAKE, I.F., SEROUSSI, G., and SMART, N.P.
Elliptic Curves in Cryptography, London Mathematical Society Lecture Notes Series 265, Cambridge University Press, 1999. The errata are here (PDF file).

[BQ95] BORCEUX, F., and QUISQUATER, J.-J.
Number theory and cryptology. To appear.

[Bre89] BRESSOUD, D.M.
Factorization and primality testing, Undergraduate Texts in Mathematics, Springer-Verlag, 1989.

[Cas91] CASSELS, J.W.S.
Lectures on elliptic curves, London Mathematical Society Student Texts 24, Cambridge University Press, 1991.

[CC90] CHARLAP, L.S., and COLEY, R.
An elementary introduction to elliptic curves II, CCR Expository Report No. 34, July 1990.

[CR88] CHARLAP, L.S., and ROBBINS, D.P.
An elementary introduction to elliptic curves, CRD Expository Report No. 31, December 1988.

[Coh93] COHEN, H.
A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer-Verlag, 1993. The errata are available from here.

[Con96]* CONNEL, I.
Elliptic curve handbook, Course Notes, McGill University, August 1996.

[CST97] CORNELL, G., SILVERMAN, J.H., and STEVENS, G. (Eds.)
Modular forms and Fermat's Last Theorem, Springer-Verlag, 1997.

[Cre97] CREMONA, J.
Algorithms for modular elliptic curves, Cambridge University Press, 2nd edition, 1997.

[Eng99] ENGE, A.
Elliptic curves and their applications to cryptography: An introduction, Kluwer Academic Publishers, 1999.

[EW99] EVEREST, G., and WARD, T.
Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer-Verlag, 1999.

[Ful69] FULTON, B.
Algebraic curves, Benjamin, 1969.

[Gro80] GROSS, B.H.
Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics 776, Springer-Verlag, 1980.

[Hus87] HUSEMÖLLER, D.
Elliptic curves, Graduate Texts in Mathematics 111, Springer-Verlag, 1987.

[KM94]KISILEVSKI, H., and MURTY, M., R. (Eds.)
Elliptic curves and related topics, CRM Proceedings & Lecture Notes 4, American Mathematical Society, 1994.

[Kna92] KNAPP, A.W.
Elliptic curves, Mathematical Notes 40, Princeton University Press, 1992.

[Kob93] KOBLITZ, N.
Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics 97, Springer-Verlag, 2nd edition, 1993.

[Kob94] KOBLITZ, N.
A course in number theory and cryptography, Graduate Texts in Mathematics 114, Springer-Verlag, 2nd edition, 1994.

[Kob98] KOBLITZ, N.
Algebraic aspects of cryptography, Algorithms and Computation in Mathematics 3, Springer-Verlag, 1998.

[Lan73] LANG, S.
Elliptic functions, Addison-Wesley Publishing Company, Inc., 1973.

[Lan78] LANG, S.
Elliptic curves: Diophantine analysis, Grundlehren der mathematischen Wissenschaften 231, Springer-Verlag, 1978.

[Lan87] LANG, S.
Elliptic functions, Graduate Texts in Mathematics 112, Springer-Verlag, 2nd edition, 1987.

[MM97] McKEAN, H., and MOLL, V.
Elliptic curves, Cambridge University Press, 1997.

[Men93] MENEZES, A.J.
Elliptic curve public key cryptosystems, Kluwer Academic Publishers, 1993.

[MBG+92] MENEZES, A., BLAKE, I., GAO, X., MULLIN, R., VANSTONE, S., and YAGHOOBIAN, T.
Applications of finite fields, Kluwer Academic Publishers, 1992.

[MWZ96]* MENEZES, A., WU, Y., and ZUCCHERATO, R.
An elementary introduction to hyperelliptic curves, Technical Report CORR-96-19, CACR, University of Waterloo, 1996. Also published in [Kob98, Appendix].

[Mil90]* MILNE, J.S.
Modular functions and modular forms, Course Notes, University of Michigan, Fall 1990.

[Mil96]* MILNE, J.S.
Elliptic curves, Course Notes, University of Michigan, August 1996.

[Mil98]* MILNE, J.S.
Abelian varieties, Course Notes, University of Michigan, July 1998.

[Mor93] MORENO, C. J.
Algebraic curves over finite fields, Cambridge Tracts in Mathematics 97, Cambridge University Press, 1993.

[Rie94] RIESEL, H.
Prime numbers and computer methods for factorization, Progress in Mathematics 126, Birkhäuser, 2nd edition, 1994.

[Rob73] ROBERT, A.
Elliptic curves, Lecture Notes in Mathematics 326, Springer-Verlag, 1973.

[Ros98] ROSING, M.
Implementing elliptic curve cryptography, Maning Publications Co., 1998.

[Rub]* RUBIN, K.
Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, CIME Lecture Notes.

[Sch74] SCHOENBERG, B.
Elliptic modular functions, Grundlehren der mathematischen Wissenschaften 203, Springer, 1974.

[Ser98] SERRE, J.-P.
Abelian l-adic representations and elliptic curves, Research Notes in Mathematics 7, A.K. Peters, Reprint edition (Benjamin, 1968), 1998.

[Sil86] SILVERMAN, J.H.
The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986.

[Sil95] SILVERMAN, J.H.
Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151, Springer-Verlag, 1995. The errata are available here.

[Sil97] SILVERMAN, J.H.
A friendly introduction to number theory, Prentice Hall, 1997.

[ST92] SILVERMAN, J.H., and TATE, J.
Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, 1992. The errata are available here.

[Sma98] SMART, N.
The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts 47, Cambridge University Press, 1998. The errata are here.

[Wal78] WALKER, R. J.
Algebraic curves, Springer-Verlag, Reprint edition, 1978.

(*) Available on line.


2. SURVEY ARTICLES

[Cas66] CASSELS, J.W.S.
Diophantine equations with special references to elliptic curves, J. London Math. Soc. 41 (1966), 193-291.

[Tat74] TATE, J.
The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179-206.


3. THESES

[Bea96]* BEAUREGARD, D.
Efficient algorithms for implementing elliptic curve public-key schemes, Master's Thesis, Worcester Polytechnic Institute, May 1996.

[Boo98] BOOGAERTS, T.
Étude de l'utilité des courbes elliptiques dans les systèmes cryptographiques pour cartes à puces, Master's Thesis, Université catholique de Louvain, June 1998.

[Cou94]* COUVEIGNES, J.-M.
Quelques calculs en théorie des nombres, Ph.D. Thesis, Université Bordeaux I, 1994.

[Eng97] ENGE, A.
Elliptic curves and their applications to cryptography: an elementary introduction, Master's Thesis, Universität Augsburg, 1997.

[Fer94]* FERMIGIER, S.
Arithmétique des courbes elliptiques et application des formules explicites à la théorie des représentations automorphes, Ph.D. Thesis, Université Paris 7, 1994.

[Gal96]* GALBRAITH, S.D.
Equations for modular curves, Ph.D. Thesis, Oxford University, 1996.

[Gau95] GAUDRY, P.
Courbes hyperelliptiques et cryptographie, Master's Thesis, École Polytechnique, 1995.

[Gua97]* GUAJARDO, J.
Efficient algorithms for elliptic curve cryptosystems, Master's Thesis, Worcester Polytechnic Institute, May 1997.

[Han97]* HANROT, G.
Résolution effective d'équations diophantiennes: algorithmes et applications, Ph.D. Thesis, Université Bordeaux I, 1997.

[Joy95]* JOYE, M.
Introduction élémentaire à la théorie des courbes elliptiques, Technical Report CG-1995/1, UCL Crypto Group, 1995.

[Joy97]* JOYE, M.
Security analysis of RSA-type cryptosystems, Ph.D. Thesis, Université catholique de Louvain, 1997.

[Kal88] KALISKI, B.S., Jr.
Elliptic curves and cryptography: A pseudorandom bit generator and other tools, Ph.D. Thesis, MIT, 1988.

[Koh96]* KOHEL, D.
Endomorphism rings of elliptic curves over finite fields, Ph.D. Thesis, University of California at Berkeley, 1996.

[Ler97]* LERCIER, R.
Algorithmique des courbes elliptiques dans les corps finis, Ph.D. Thesis, École polytechnique, 1997.

[Mon92]* MONTGOMERY, P.L.
A FFT extension of the elliptic curve method of factorization, Ph.D. Thesis, University of California at Los Angeles, 1992.

[Mor90]* MORAIN, F.
Courbes elliptiques et tests de primalité, Ph.D. Thesis, INRIA, 1990.

[Mül91]* MÜLLER, V.
Die Berechnung der Punktanzahl von elliptischen Kurven über endlichen Primkörpern, Master's Thesis, Universität des Saarlandes, 1991.

[Mül95]* MÜLLER, V.
Ein Algorithmus zur Bestimmung der Punktanzahl elliptischer Kurven über endlichen Körpern der Charakteristik grösser drei, Ph.D. Thesis, Universität des Saarlandes, 1995.

[Rosn98] ROSNER, M.
Elliptic curve cryptosystems on reconfigurable hardware, Master's Thesis, Worcester Polytechnic Institute, May 1998.

[Ste00]* STEIN, W.A.
Explicit approaches to modular abelian varieties, Ph.D. Thesis, University of California at Berkeley, 2000.

[Yor92] YORK, E.V.
Elliptic curves over finite fields, Master's Thesis, George Mason University, May 1992.

[Zuc97] ZUCCHERATO, R.
New applications of elliptic curves and function fields in cryptography, Ph.D. Thesis, University of Waterloo, 1997.

(*) Available on line.

SOFTWARE
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aPecs 5.x
Program written in MAPLE to perform arithmetic on elliptic curves. A version for UBASIC is also available.
Author: Ian Connel.

Elliptic Curve Calculator 5.05
Two computer packages to perform basic compuations on elliptic curves. The first is a stand-alone application which runs on any variety of Macintosh. The second is a collection of MATHEMATICA routines with extensive documentation.
Authors: Joe Silverman and Paul van Mulbregt.

mwrank (12/1/98)
This program uses 2-descent (via 2-isogeny if possible) to determine the rank of an elliptic curve E over Q, list a set of points which generate E(Q) modulo 2E(Q), and finally search for further points on the curve. For details of algorithms see [Cre97]. Two related programs are also available: findinf and ratpoint.
Author: John Cremona.

LiDIA/LC 2.0.x
LiDIA is a C++ library for computational number theory which provides a collection of highly optimized implementations of various multiprecision data types and time-intensive algorithms.
The entire LiDIA functionality can be used interactively through LiDIA's LC interpreter. LC implements a subset of C++ and provides in addition to standard programming facilities, function overloading and automatic coercions. Functions and statements are treated as ordinary objects and may be manipulated at run-time. Because of the interpreted language, LC functions can be easily transformed to C++ programs which can then be compiled.
Authors: LiDIA Group at the Darmstadt University of Technology.

Pari-GP 2.1.x
Calculator for number theory. The PARI system is a package which is capable of doing formal computations on recursive types at high speed; it is primarily aimed at number theorists, but can be used by people whose primary need is speed. It is possible to use PARI in two different ways: (1) as a library, which can be called from any upper-level language application, (2) as a sophisticated programmable calculator, named GP, which contains most of the standard control instructions of a standard language like C.
Authors: C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier.
See also the Packages Page by John Cremona.

LINKS
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In Association with Awarded best page on
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Last modified: Thu Mar 8 07:41:56 CET 2001 1