[email protected] Valley Fellowship

Research Proposal

William A. Stein

The applicant proposes to develop computational tools of significant importance to high tech companies involved with public-key cryptography. Such cryptography is of foremost importance in the new frontier of electronic commerce made possible by the Internet.

In recent years there has been an enormous growth in the implementation of public-key cryptography, both in government and in industry. The level of security provided by public-key cryptography depends on the difficulty of certain mathematical problems, such as integer factorization. The security of the RSA cryptosystem, the system in most common use today, depends on our apparent inability to factor large integers. Nonetheless, through international collaboration, using a large number of workstations in parallel, specific RSA keys are regularly broken. This has necessitated an increase in key sizes, which is reflected in corresponding increases in software and hardware demands. With the advent of smart cards and electronic commerce, finding ways to keep the key size small is of great importance. One of the most promising approaches is to devise new public-key cryptosystems based on higher dimensional mathematical objects called ``abelian varieties.'' For decades abelian varieties have been studied by pure mathematicians. It is a pleasure to see that this hard work can be applied to problems in industry. There is a great deal of evidence that cryptosystems based on abelian varieties offer higher security while allowing for significantly smaller key sizes.

Our goals are two fold.

  1. To develop and implement new algorithms for computing on abelian varieties. These will be of significant importance in the development of new cryptographic protocols.
  2. To compute and disseminate via the Internet a database of abelian varieties. Both the database, and the software developed in creating it, will fill a gap in the resources currently available to those working with abelian varieties.

By developing techniques for computation on abelian varieties, the applicant will contribute to the competitiveness of high tech industries in Silicon Valley as we enter a new age of electronic commerce.