The Arithmetic Codex

Ronald Cramer (CWI, Amsterdam & Mathematical Institute, Leiden University)

Friday 9 December 2011, 15.00, b-it 1.25 (cosec meeting room)

We define the notion of an arithmetic codex (or codex, for short).

This notion encompasses as well as generalizes, in a single mathematical framework, all known types of specialized secret sharing schemes from the area of secure multi-party computation, i.e., the so-called (strongly) multiplicative linear secret sharing schemes. It also captures bilinear multiplication algorithms for field extensions as a special case.

Our notion is not merely a unification for its own sake. First, it casts these schemes in terms of a dedicated ``representation'' of algebras, thereby bringing the relevant mathematical structure to the surface. Second, it identifies novel types of special secret sharing schemes. And, third, there are novel cryptographic applications.

Besides presenting some elementary examples and giving an overview of the basic theory and the main applications, we discuss a construction of arithmetic secret sharing schemes based on a novel algebraic-geometric paradigm that we also introduce. Based on joint works with Nacho Cascudo (CWI, Amsterdam) and Chaoping Xing (NTU, Singapore).