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The authors introduce the notion of an arithmetic codex, or codex for short. Codices encompass several well-established notions from cryptography (various types of arithmetic secret sharing schemes, which all enjoy additive as well as multiplicative properties) and from algebraic complexity (bilinear complexity of multiplication in algebras) in a single mathematical framework. Arithmetic secret sharing schemes have important applications to secure multiparty computation and even to two-party cryptography. Interestingly, several recent applications to two-party cryptography rely crucially on the existence of certain asymptotically good schemes. It is intriguing that their construction requires asymptotically good towers of algebraic function fields over a finite field: no elementary (probabilistic) constructions are known in these cases.
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