Professor of Mathematics
Department of Mathematics,
University of Washington,
Seattle, WA 98195, USA.
Cryptography, the analysis and construction of secret codes, has long been important to governments, and has become big business since the explosion of commerce on the Internet. In 1985, Professor Neal Koblitz, a UW number theorist, and Victor Miller, then at IBM, discovered a new system of cryptography that is much more secure than previously known systems. Their system is based on elliptic curves, which are curves defined by certain polynomial equations that play a central role in abstract number theory and algebraic geometry. Professor Koblitz continues to make major contributions to the theory of elliptic curve cryptography, and serves as an advisor to Certicom Corporation, the leading commercial provider of elliptic curve cryptographic systems.
p-adic Analysis: A Short Course on Recent Work, London Mathematical Society Lecture Note Series No. 46, Cambridge University Press, Cambridge, 1980.
p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics No. 58, Springer-Verlag, New York, 1977. Second edition, 1984.
Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Math. No. 97, Springer-Verlag, New York, 1984. Second edition, 1993.
A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987. Second edition, 1994.
Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics Vol. 3, Springer-Verlag, New York, 1998.
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, Kluwer, 2000.
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.
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.
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.
A family of jacobians suitable for discrete log cryptosystems, Advances in Cryptology - Crypto '88, Springer-Verlag, 1990, 94-99.
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,' to appear in Journal of Cryptology.
(with A. J. Menezes) Pairing-based cryptography at high security levels, in preparation.
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.
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.
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.
Three practical applications of algebraic curves, Journal of Sichuan University Natural Science Edition, Vol. 26, Special Issue, December 1989, 28-35.
Book review of A Course in Number Theory by H. E. Rose, American Scientist, March 1990.
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.
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.
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.
(with Tim Bell, Michael Fellows, Matthew Powell, Harold Thimbleby, Ian Witten) Explaining cryptographic systems, Computers & Education, Vol. 40, 2003, 199-215.
Book review of El Ultimo Secreto de Colonia Dignidad by Carlos Basso, Notices of the Amer. Math. Society, Vol. 51, 2004, pp. 32-33.