International Association
for Cryptologic Research

We introduce the notion of {\\em arithmetic codex}, or {\\em codex} for short.

It encompasses several well-established notions from cryptography (arithmetic secret sharing schemes, i.e., enjoying additive as well as multiplicative properties) and algebraic complexity theory (bilinear complexity of multiplication) in a natural mathematical framework.

Arithmetic secret sharing schemes have important applications to secure multiparty computation and even to {\\em two}-party cryptography. Interestingly, several recent applications to two-party cryptography rely crucially on the existing results on ``{\\em asymptotically good} families\'\' of suitable such schemes. Moreover, the construction of these schemes requires asymptotically good towers of function fields over finite fields: no elementary (probabilistic) constructions are known in these cases. Besides introducing the notion, we discuss some of the constructions, as well as some limitations.