**Mathematics of Cryptology**

September 26 - October 2,
2003

Historically,
cryptology focused on the design and cryptanalysis of encryption schemes. But,
nowadays, cryptology offers much more: signature schemes, provable security,
secret sharing, multi-party computation, zero knowledge protocols, electronic
payment schemes, electronic voting schemes, to name a few. At the same time,
the mathematics that is involved has become quite sophisticated: integer
factorization using number fields, lattices, discrete logarithms on elliptic
curves or on Jacobians of curves of higher genus, cohomological tools for point
counting, pairings, among others.

The aim
of the workshop was to help further strengthening and extending the ties
between the cryptology community and that part of the mathematics community
with an interest in cryptology, especially from algebra, number theory and
geometry.

Leading
experts from these communities were invited to give key-note lectures:

·
Jacques
Stern: When provable security meets number theory;

·
Hendrik
Lenstra: Recent developments in primality testing;

·
Rene
Schoof: Construction of Weil and Tate pairings;

·
Alan
Lauder: Computing zeta functions of hyperelliptic curves via deformations;

·
Ueli
Maurer: Probability Theory in Cryptography;

·
Gerhard
Frey: Applications of pairings in cryptography;

·
Hans
Dobbertin: Some Mathematics Behind the Design of Block Ciphers;

·
Frederik
Vercauteren: p-adic algorithms for counting points on elliptic curves;

·
Arjen
Lenstra: Integer factorization and cryptology;

·
Phong
Nguyen: New results on lattice-based cryptography;

·
Ronald
Cramer: Secure encryption from hard subgroup membership problems;

·
Berry
Schoenmakers: Homomorphic cryptosystems and applications;

·
Peter
Stevenhagen: Constructing elliptic curves with a given number of points;

·
Igor
Shparlinski: Playing 'Hide-and-Seek' in Finite Fields: Hidden Number Problem
and Its Applications;

·
Bas
Edixhoven: A possible generalisation of Schoof's algorithm.

There
were about 60 registered participants. One third came from the Netherlands, and
two thirds from Australia, Belgium, Denmark, U.K., Germany, Finland, France,
Italy, Poland, Spain, Switzerland, Turkey, and the U.S.A. Among the Dutch
participants, three came from the Ministry of Defense, and one from industry
(Cistron).

The
workshop was financed by: NWO, Lenstra (Spinoza grant), EIDMA, the Thomas Stieltjes
Institute for Mathematics, STW, and the Lorentz Center.

Our
impression is that the workshop was a big success. The lectures were of very
high quality, but the atmosphere remained quite informal. There was much
interaction between the participants. The facilities and staff of the Lorentz
Center were very much appreciated by both organizers and participants.

**S.J.** **Edixhoven**
(University Leiden, The Netherlands)

**B.** **Schoenmakers**
(University Eindhoven, The Netherlands)

**R.** **Cramer**
(Brics Aarhus, Denmark)