Publications
Books
- p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in
Mathematics No. 58, Springer-Verlag, 1977.
Second edition, 1984.
- p-adic Analysis: A Short
Course on Recent Work,
London Mathematical Society Lecture Note Series No. 46, Cambridge
University Press, Cambridge, 1980.
- Introduction to Elliptic Curves and Modular Forms, Graduate Texts in
Math. No. 97, Springer-Verlag, 1984. Second
edition, 1993.
- A Course in Number Theory and Cryptography, Graduate Texts in
Math. No. 114, Springer-Verlag, 1987. Second
edition, 1994.
- Algebraic Aspects of Cryptography, Algorithms and
Computation in Mathematics Vol. 3, Springer-Verlag,
1998.
- Random Curves: Journeys of a Mathematician, Springer-Verlag, 2007.
Books Edited
- Number Theory Related to Fermat's Last Theorem, Progress in Math. No.
26, Birkhauser, Boston, 1982.
- Advances in Cryptology - Crypto '96, Lec.
Notes in Computer Science No. 1109, Springer, 1996.
- Towards a Quarter-Century of Public Key Cryptography, Springer, 2000.
Videotapes
- Mathematics under hardship conditions in the Third
World, AMS-MAA Joint Lecture Series, 1992.
- The story of the proof of Fermat's Last Theorem, Undergraduate
Mathematical Sciences Colloquium, Univ. Washington, 1993.
Research Articles
- p-adic variation of the zeta-function over families of
varieties defined over finite fields, Compositio
Mathematica, Vol. 31, 1975, 119-218.
- Netselostnost periodov
parabolicheskikh form vne
kriticheskoi polosy
(Non-integrality of the periods of cusp forms outside the critical strip),
Funktsional'nyi Analiz
i ego Prilozheniya,
Vol. 9, 1975, 52-55.
- 2-adic and 3-adic ordinals of 1/j-expansion
coefficients for the weight 2 Eisenstein series, Bulletin of the London
Mathematical Society, Vol. 9, 1977, 188-192.
- Interpretation of the p-adic
log gamma function and Euler constants using the Bernoulli measure, Transactions
of the American Mathematical Society, Vol. 242, 1978, 261-269.
- Gamma function identities and elliptic differentials on
Fermat curves, Duke Mathematical Journal, Vol. 45, 1978, 87-99.
- (with D. Rohrlich) Simple
factors in the Jacobian of a Fermat curve, Canadian Journal of
Mathematics, Vol. 30, 1978, 1183-1205.
- (with A. Ogus) Algebraicity of some products of values of the gamma
function, Appendix to: P. Deligne, Valeurs de fonctions L et périodes d'intégrales,
Proceedings of Symposia in Pure Mathematics, Vol. 33 Part 2, 1979,
313-346.
- (with B. H. Gross) Gauss sums and the p-adic gamma-function, Annals of Mathematics,
Vol. 109, 1979, 569-581.
- A new proof of certain relations between p-adic L-functions and log gamma-functions, Duke
Mathematical Journal, Vol. 46, 1979, 455-468.
- q-extension
of the p-adic gamma function, Transactions
of the American Mathematical Society, Vol. 260, 1980, 449-457.
- The hypergeometric function with p-adic parameters, Proceedings of the Queen's Number
Theory Conference 1979, Queen's University, Kingston, Ontario, 1980,
319-328.
- On Carlitz's q-Bernoulli
numbers, Journal of Number Theory, Vol. 14, 1982, 332-339.
- p-adic analog of Heine's hypergeometric q-series,
Pacific Journal of Mathematics, Vol. 102, 1982, 373-383.
- q-extension
of the p-adic gamma function. II, Transactions
of the American Mathematical Society, Vol. 273, 1982, 111-129.
- The number of points on certain families of
hypersurfaces over finite fields, Compositio
Mathematica, Vol. 48, 1983, 3-23.
- p-adic eigen-functions for Kubert distributions, Canadian Journal of
Mathematics, Vol. 35, 1983, 674-686.
- p-adic integral transforms on compact subgroups of Cp, Pacific Journal of
Mathematics, Vol. 120, 1985.
- p-adic congruences and modular
forms of half integer weight, Mathematische
Annalen, Vol. 274, 1986, 199-220.
- Elliptic curve cryptosystems, Mathematics of Computation,
Vol. 48, 1987, 203-209.
- Congruences for periods of modular
forms, Duke Mathematical Journal, Vol. 54, 1987, 361-373.
- Primality of the number of points on an elliptic curve
over a finite field, Pacific Journal of Mathematics, Vol. 131,
1988, 157-165.
- Hyperelliptic cryptosystems, Journal
of Cryptology, Vol. 1, 1989, 139-150.
- A family of jacobians
suitable for discrete log cryptosystems, Advances in Cryptology -
Crypto '88, Springer-Verlag, 1990, 94-99.
- Discrete logarithms and discreet curves, in: A
Perspective in Theoretical Computer Science - Commemorative Volume for
Gift Siromoney, World Sci. Publ. Co. Ltd,
1989, 106-120.
- Jacobi sums, irreducible zeta-polynomials, and
cryptography, Canadian Math. Bulletin, Vol. 34 (2), 1991, 229-235.
- Constructing elliptic curve cryptosystems in
characteristic 2, Advances in Cryptology - Crypto '90, Springer-Verlag, 1991, 156-167.
- Elliptic curve implementation of zero-knowledge blobs, Journal
of Cryptology, Vol. 4, 1991, 207-213.
- CM-curves with good cryptographic properties, Advances
in Cryptology - Crypto '91, Springer-Verlag,
1992, 279-287.
- (with M. R. Fellows) Self-witnessing polynomial-time
complexity and prime factorization, Designs, Codes and Cryptography,
Vol. 2, 1992, 231-235.
- (with M. R. Fellows) Fixed-parameter complexity and
cryptography, in G. Cohen, Teo Mora, and O.
Moreno Eds., Applied Algebra, Algebraic Algorithms and Error-Correcting
Codes, Proc. 10th Intern. Symp., San Juan,
Puerto Rico, May 1993, Springer-Verlag,
1993, 121-131.
- (with M. R. Fellows) Combinatorially
based cryptography for children (and adults), Proc. 24th Southeastern
Intern. Conf. Combinatorics, Graph Theory and
Computing (Boca Raton, Florida, February 1993), Congressus
Numerantium, Vol. 99, 1994, 9-41.
- (with M. R. Fellows) Combinatorial cryptosystems galore!, in Finite Fields: Theory, Applications, and
Algorithms, Second Intern. Conf. Finite Fields (Las Vegas, August
1993), Contemporary Math. Vol. 168, 1994, 51-61.
- Computational algebra problems arising from combinatorial
cryptography, Proceedings of Discussion Meeting on Cryptography and
Computation (31 August - 1 September, 1995, Jawaharlal Nehru Centre
for Advanced Scientific Research, Bangalore), 12 pages.
- (with R. Balasubramanian) The
improbability that an elliptic curve has subexponential
discrete logarithm problem under the Menezes-Okamoto-Vanstone algorithm, Journal
of Cryptology, Vol. 11, 1998, 141-145.
- (with J. P. Buhler) Lattice basis reduction, Jacobi
sums, and hyperelliptic cryptosystems, Bulletin
of the Australian Mathematical Society, Vol. 57, 1998, 147-154.
- An elliptic curve implementation of the finite field
digital signature algorithm, Advances in Cryptology - Crypto '98,
Springer-Verlag, 1998, 327-337.
- (with R. G. Downey and M. R. Fellows) Techniques for
exponential parameterized reductions in vertex set problems, unpublished
but included in R. G. Downey and M. R. Fellows, Parameterized
Complexity, Springer-Verlag, 1999, 220-222.
- (with M. J. Jacobson, J. H. Silverman, A. Stein, and E.
Teske) Analysis of the xedni
calculus attack, Designs, Codes and Cryptography, Vol. 20, 2000,
41-64.
- Almost primality of group orders of elliptic curves
defined over small finite fields, Experimental Mathematics, Vol.
10, 2001, 553-558.
- (with A. J. Menezes) Obstacles to the torsion-subgroup
attack on the Decision Diffie-Hellman problem, Mathematics
of Computation, Vol. 73, 2004, 2027-2041.
- (with A. J. Menezes) Another look at `provable
security,' Journal of Cryptology, Vol. 20, 2007, 3-37; available here.
- (with A. J. Menezes) Pairing-based cryptography at high
security levels, Proceedings of the Tenth IMA International Conference
on Cryptography and Coding, Springer-Verlag,
LNCS 3796, 2005, 13-36; available here.
- (with A. J. Menezes) Another look at generic groups, Advances
in Mathematics of Communications, Vol. 1, 2007, 13-28; available here.
- (with A. J. Menezes) Another look at `provable
security'. II, Advances in Cryptology - Indocrypt
2006, Springer-Verlag, LNCS 4329, 2006,
148-175; available here.
- Another look at automated theorem-proving, Journal
of Mathematical Cryptology, Vol. 1, 2007, 385-403; available here.
- (with A. J. Menezes) Another look at non-standard
discrete log and Diffie-Hellman problems, Journal
of Mathematical Cryptology, Vol. 2, 2008, 311-326; available here.
- (with A. H. Koblitz and A. J. Menezes) Elliptic curve
cryptography: The serpentine course of a paradigm shift, Journal of
Number Theory Vol. 131, 2011, 781-814; available here.
Videoabstract available here.
- (with A. J. Menezes) Intractable assumptions in
cryptography, Proc. 9th International Conf. Finite Fields and Their
Applications, Contemporary Mathematics, Vol. 518, 2010,
279-300, available here
.
- A security weakness in composite-order pairing-based
protocols with imbedding degree k>2, available here.
- (with A. J. Menezes and I. E. Shparlinski)
Discrete logarithms, Diffie-Hellman, and
reductions, Vietnam Journal of Mathematics, Vol. 39, 2011, 267-285;
available here.
- Another look at automated theorem-proving. II, Journal
of Mathematical Cryptology, Vol. 5, 2011, 205-224; available here.
- (with A. J. Menezes) Another look at security
definitions, Advances in Mathematics of Communications, Vol. 7,
2013, 1-38; available here.
- (with A. J. Menezes) Another look at HMAC, Journal
of Mathematical Cryptology, Vol. 7, 2013, 225-251; available here.
- (with A. J. Menezes) Another look at non-uniformity, Groups,
Complexity, Cryptology, Vol. 5, No. 2, 2013, 117-139; available here.
- (with A. J. Menezes) Another look at security theorems
for 1-key nested MACs, Open Problems in Mathematics and Computational
Science, Springer-Verlag, 2014, 69-89;
available here.
- (with A. J. Menezes) The random oracle model: A
twenty-year retrospective, Designs, Codes and Cryptography, Vol.
77, 2015, 587-610; available here.
- (with S. Chatterjee, A. J. Menezes, and P. Sarkar)
Another look at tightness. II: Practical issues in cryptography, Paradigms in Cryptology – Mycrypt 2016, Springer-Verlag,
2017, 21-55, available here.
- (with M. McDonald) One bad formula can spoil
everything: A simple adjustment that would improve the UN’s Gender
Inequality Index, The Mathematical
Intelligencer, Vol. 41, No. 2, 2019, 27-34, available here.
- (with A. J. Menezes) Critical perspectives on provable
security: Fifteen years of “Another look” papers, Advances in Mathematics of
Communications, Vol. 13, 2019, 517-558, available here.
- (with D. R. L. Brown and J. LeGrow)
Cryptanalysis of `Make’, Journal
of Mathematical Cryptology, Vol. 16, 2022, 98-102, available here.
- (with P. Sarkar, S. Samajder,
and S. Singha) Concrete analysis of approximate
ideal-SIVP to decision ring-LWE reduction, Advances in Mathematics of
Communications, Vol. 18, 2024, 1216-1258, available here.
Survey Articles
- Number theory and cryptography, in M. Artin, H. Kraft, and R. Remmert,
eds., Duration and Change: Fifty Years at Oberwolfach,
Springer-Verlag, 1994, 211-236.
- Elliptic curve cryptosystems: An overview, Proceedings
of Discussion Meeting on Cryptography and Computation (31 August - 1
September, 1995, Jawaharlal Nehru Centre for Advanced Scientific Research,
Bangalore), 8 pages.
- (with A. Menezes and S. A. Vanstone) The state of
elliptic curve cryptography, Designs, Codes and Cryptography, Vol.
19, 2000, 173-194.
- A survey of number theory and cryptography, in R. P. Bambah, V. C. Dumir, and R.
J. Hans Gill, eds, Number Theory, Indian
National Science Academy, 2000, 217-239.
- Cryptography, in B. Engquist
and W. Schmid, eds., Mathematics Unlimited -
2001 and Beyond, Springer-Verlag, 2001,
749-769.
- Some number-theoretic issues in modern cryptography, Newsletter
of the Ramanujan Mathematical Society
(Madras), Vol. 11, No. 1-2, 2001, 1-14.
- Good and bad uses of elliptic curves in cryptography, Moscow
Math. Journal, Vol. 2, 2002, 693-715.
- (with A. J. Menezes) A survey of public-key
cryptosystems, SIAM Review, Vol. 46, 2004, 599-634.
- (with A. J. Menezes) Cryptocash,
cryptocurrencies, and cryptocontracts, Designs,
Codes and Cryptography, Vol. 78, 2016, 87-102, available here.
- (with A. J. Menezes) Cryptographers prepare for a
possible post-quantum future, Canadian
Math. Soc. Notes, Oct/Nov 2017, 16-17.
Other Articles
- A mathematical visit to Hanoi, The Mathematical
Intelligencer, Vol. 2, No. 1, 1979, 38-42.
- The p-adic approach to
solutions of equations over finite fields, The American Mathematical
Monthly, Vol. 87, 1980, 115-118.
- Mathematics as propaganda, in L.A. Steen, ed., Mathematics
Tomorrow, Springer-Verlag, New York, 1981.
- Why study equations over finite fields?,
Mathematics Magazine, May 1982, 144-149.
- (with Ann Hibner Koblitz)
Mathematics and the external world, an interview with A.T. Fomenko, The Mathematical Intelligencer, Vol.
8, No. 2, 1986, 8-17, 25.
- Book review of Ultrametric
Calculus: An Introduction to p-adic Analysis
by W. H. Schikhof, American Scientist,
Vol. 74, 1986, 681-682.
- Vietnam after the agony, Frontline (Madras,
India), Vol. 4, No. 14 (July 11-24, 1987), 48-56; Triumphs and problems of
science in Vietnam, Frontline, Vol. 4, No. 15 (July 25-August 7,
1987), 52-55.
- Problems that teach the obvious but difficult, The
American Mathematical Monthly, Vol. 95, 1988, 254-257.
- A tale of three equations; or the emperors have no
clothes, The Mathematical Intelligencer, Vol. 10, No. 1, 1988,
4-11; and: Reply to unclad emperors, ibid., 14-16.
- Book review of A Course in Number Theory by H.
E. Rose, American Scientist, March 1990.
- Three practical applications of algebraic curves, Journal
of Sichuan University Natural Science Edition, Vol. 26, Special Issue,
December 1989, 28-35.
- Book review of Primes of the Form x²+ny²:
Fermat, Class Field Theory, and Complex Multiplication by D. A. Cox, Mathematics
of Computation, Vol. 56, 1991, 401-403.
- Recollections of mathematics in a country under siege,
Interview with Professor Hoang Tuy, Director of
the Hanoi Mathematical Institute, The Mathematical Intelligencer,
Vol. 12, No. 3, 1990, 16-34.
- Are student ratings unfair to women?,
Association for Women in Mathematics Newsletter, Vol. 20, No. 5,
1990, 17-19.
- Mathematics under hardship conditions in the Third
World, Notices of the Amer. Math. Soc., Vol. 38, 1991, 1123-1128.
- The profit motive: the bane of mathematics education, Humanistic
Mathematics Network Journal, No. 7, 1992, 89-92.
- (with M. R. Fellows) Kid Krypto, Advances in
Cryptology - Crypto '92, Springer-Verlag,
1993, 378-396.
- Foreword to Alfred J. Menezes' book Elliptic Curve
Public Key Cryptosystems, Kluwer Publishers, 1993.
- (with M. R. Fellows and Ann Hibner
Koblitz) Cultural aspects of math education reform, Notices of the
Amer. Math. Society, Vol. 41, 1994, 5-9.
- Math majors learn from sixth grade experience, Paideia,
U.W. Office of Undergraduate Education, Spring 1994.
- Book review of Multicultural Mathematics by
David Nelson et al., American Scientist, Vol. 83
(March-April 1995), 194-195.
- The case against computers in K-13 math education
(kindergarten through calculus), The Mathematical Intelligencer,
Vol. 18, No. 1, 1996, 9-16.
- Cryptography as a teaching tool, Cryptologia,
Vol. 21, 1997, 317-326.
- Book review of Discrete Mathematics in the Schools,
edited by J. G. Rosenstein, D. S. Franzblau, and
F. S. Roberts, SIGACT News (publication of the ACM Special Interest
Group on Automata and Computability Theory), Vol. 29, No. 1, 1998, 8-12.
- (with Tim Bell, Michael Fellows, Matthew Powell, Harold
Thimbleby, and Ian Witten) Explaining
cryptographic systems to the general public, IFIP First World
Conference on Information Security Education, 1999, 221-233. Also in: Computers
and Education, Vol. 40, 2003, 199-215.
- (with Ann Hibner Koblitz) The
Kovalevskaia Fund, The Mathematical
Intelligencer, Vol. 22, No. 2, 2000, 62-65.
- (with Ann Hibner Koblitz)
Book review of In Code: A Voyage of
Mathematical Discovery by Sarah and David Flannery, London Times
Higher Education Supplement, Nov. 10, 2000.
- Essay-review of three books on cryptography, Amer.
Math. Monthly, Vol. 108, 2001, 983-988.
- Book review of Radical Equations: Math Literacy and
Civil Rights by Robert P. Moses and Charles E. Cobb Jr. (Beacon Press,
2001), Notices of the Amer. Math. Society, Vol. 49, 2002, 328-332.
- Book review of El Ultimo Secreto
de Colonia Dignidad by Carlos Basso, Notices
of the Amer. Math. Society, Vol. 51, 2004, 32-33.
- The uneasy relationship between mathematics and
cryptography, Notices of the Amer. Math. Society, Vol. 54, 2007,
972-979; available here.
- Book review of Elliptic Curves: Number Theory and
Cryptography, 2nd Edition, by L. C. Washington, SIAM Review,
Vol. 51, 2009, 230-231.
- (with A. J. Menezes) The brave new world of bodacious
assumptions in cryptography, Notices of the Amer. Math. Society,
Vol. 57, 2010, 357-365; available here.
- Secret codes and online security: A seminar for
entering students, Cryptologia, Vol. 34,
2010, 145-154.
- Book review of Decrypted Secrets: Methods and Maxims
of Cryptology, 4th Edition, by F. L. Bauer, SIAM Review, Vol.
52, 2010, 777-778.
- Interview with Professor Ngo Bao
Chau, The Mathematical Intelligencer, Vol. 33, No. 1, 2011, 46-50;
available here.
- "The Word for World is Forest": A long-range
funding source for women in math in developing countries, The
Mathematical Intelligencer, Vol. 33, No. 4, 2011, 21-24; available here.
- Crypto galore!, in H. L. Bodlaender, R. Downey, F. V. Fomin,
and D. Marx (eds.), The Multivariate Algorithmic Revolution and Beyond:
Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday,
LNCS 7370, Springer-Verlag, 2012, 39-50.
- Grothendieck's 1967 lectures in the
forest in Vietnam, The Mathematical Intelligencer, Vol. 35, No. 2,
2013, 32-34; available here.
- (with F. Verhulst and S. Krantz) Hype! An exchange of views, The
Mathematical Intelligencer, Vol. 36, No. 3, 2014, 8-13; available here.
- Why STEM majors
need the humanities, The
Chronicle of Higher Education, 6 January 2017; available here.
- An American mathematician’s perspective on
multiple-choice tests, edited version appeared in Vietnamese translation
in the magazine Tia Sáng, 20 Nov. 2016; available here.
- Time for a paradigm shift in our disciplinary culture?
(write-up of keynote talk), Paradigms
in Cryptology – Mycrypt 2016, Springer-Verlag, 2017, 11-18; PowerPoint of talk available if you
email me.
- (with A. J. Menezes) A riddle wrapped in an enigma, IEEE Security & Privacy, Vol.
14, No. 6 (Nov.-Dec. 2016), 34-42; available here.
- (with P. D. Hiêu)
Cryptology during the French and American wars in Vietnam, Cryptologia,
Vol. 41, No 6 (Dec. 2017), 491-511; available here
- El lado oscuro
del papel norteamericano
en la educación
superior en países
en desarrollo (The
dark side of the American role in higher education in developing
countries), Anales de la Academia de Ciencias
de Cuba (Annals of the Cuban
Academy of Sciences), Vol. 7, No. 2 (Dec. 2017), 105-115.
- Bitcoin: The good and the bad news about
cryptocurrency, appeared in Vietnamese translation in Tia Sáng, 20 Nov. 2017;
available here.
- Provincial universities can be key to economic
development and social equality, appeared in Vietnamese translation in Tia Sáng,
5 Dec. 2017; available here.
- A humanistic university education for all students,
appeared in Vietnamese translation in Tia
Sáng, 20 March 2018; available here.
- University admission exams, appeared in Vietnamese
translation in Zing.vn, 4 July
2018; available here.
- Challenges in mathematics education, excerpted and
translated into Vietnamese in Tia Sáng, 11 January 2019.
- Facebook's digital currency, appeared in Vietnamese
translation in Tia Sáng, 5 July 2019; available here.
- Pure and applied mathematics, appeared in Vietnamese
translation in Tia Sáng, 5 April 2021; available here.
- Getting the odds to favor success in science, appeared
in Vietnamese translation in Tia Sáng, 20 January 2022; available here.
- Quantum computing: Reality or hype?, appeared in
Vietnamese translation in Tia Sáng, 20 March 2022; available here.
- It takes more than money, appeared in Vietnamese
translation in Tia Sáng, 20 November 2022; available here.
- The public’s right to knowledge, appeared in
Vietnamese translation in Tia Sáng, 10 January 2023; available here.
- Making a bad situation worse:The dangers of AI, appeared in
Vietnamese translation in Tia Sáng, 20 June 2023; available here.
- Education reform for the modern world, appeared in
Vietnamese translation in Tia Sáng, 20 January 2024; available here.
- The hidden hazards of higher education advice and "aid" from the U.S., appeared in
Vietnamese translation in Tia Sáng, 20 February 2024; available here.
- Computers and calculators are everywhere. Does that mean that math education
needs to change?, appeared in Vietnamese translation in Tia Sáng, 20 July 2024; available here.
- Is the International Math Olympiad (IMO) a bad investment for Vietnam?,
appeared in Vietnamese translation in Tia Sáng, 5 August 2024; available here.
- When should AI be banned?,
appeared in Vietnamese translation in Tia Sáng, 20 November 2024; available here.
- Thinking outside-the-box about "Moral Hazard",
to appear in Vietnamese translation in Tia Sáng; available here.
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