This seminar on number-theoretical algorithms and their applications was the fifth on this topic at Dagstuhl over a period of more than 10 years. This time we attracted a record number of 54 participants from 14 countries.

One of the major goals of these seminars has been to broaden interactions between number theory and other areas. For instance, there has been an effort to bring together people developing the theory of efficient algorithms with people actually writing software. There has also been continuing interest in cryptography, and almost a third of the talks were on algebraic curves, most with an eye to applications in cryptography. Since elliptic curves in cryptography seem to be mainly objects of hardware implementations, nowadays, the focus is on higher genus curves and related more sophisticated mathematical objects.

This time we also had a major new topic: algorithmic K-theory which has been rapidly developing over the last few years. Not surprisingly seven talks were given on this subject, several alone on (algorithmic aspects of) logarithmic class groups.

Most of the other talks focused on more classical topics of algorithmic algebraic number theory, with half a dozen on various aspects of solving Diophantine equations. Among the variety of problems considered we just mention the computation of Picard groups and Drinfeld modules, but also quantum computing of unit groups. Several talks were on problems related to the development of number theoretical software.

The variety of topics was stimulating to the audience. The reaction of the participants was very positive and we believe that we succeeded in having an effective meeting that was able to appeal to a broad audience. We made sure to allow for adequate breaks between sessions, and there were many opportunities for discussions that the participants took advantage of. The pleasant atmosphere of Schloss Dagstuhl once again contributed to a very productive meeting.