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

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.

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(*) 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.

LINKS

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• George Barwood's elliptic curve cryptography FAQ
• Andreas Enge's elliptic curves page
• Stéphane Fermigier's elliptic curves page
• Steven Galbraith's elliptic curves page
• Franz Lemmermeyer's elliptic curve page
• Richard Pinch's elliptic curves page
• Certicom's elliptic curve cryptosystem and standards page
• SECANTS Meetings

In Association with	Awarded best page on Elliptic Curves by
	Links2Go	(since 1998/11/15)