The torsion-limit for algebraic function fields and its application to arithmetic secret sharing
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This work deals with "MPC-friendly" linear secret sharing schemes (LSSS), a mathematical primitive upon which secure multi-party computation (MPC) can be based and which was introduced by Cramer, Damgaard and Maurer (EUROCRYPT 2000). Chen and Cramer proposed a special class of such schemes that is constructed from algebraic geometry and that enables efficient secure multi-party computation over fixed finite fields (CRYPTO 2006). We extend this in four ways. First, we propose an abstract coding-theoretic framework in which this class of schemes and its (asymptotic) properties can be cast and analyzed. Second, we show that for every finite field ${\mathbb F}_q$, there exists an infinite family of LSSS over ${\mathbb F}_q$ that is asymptotically good in the following sense: the schemes are "ideal," i.e., each share consists of a single ${\mathbb F}_q$-element, and the schemes have t-strong multiplication on n players, where the corruption tolerance $\frac{3t}{n-1}$ tends to a constant 驴(q) with 0 驴(q) n tends to infinity. Moreover, when $|{\mathbb F}_q|$ tends to infinity, 驴(q) tends to 1, which is optimal. This leads to explicit lower bounds on $\widehat{\tau}(q)$, our measure of asymptotic optimal corruption tolerance. We achieve this by combining the results of Chen and Cramer with a dedicated field-descent method. In particular, in the ${\mathbb F}_2$-case there exists a family of binary t-strongly multiplicative ideal LSSS with $\frac{3t}{n-1}\approx 2.86\%$ when n tends to infinity, a one-bit secret and just a one-bit share for every player. Previously, such results were shown for ${\mathbb F}_q$ with q 驴 49 a square. Third, we present an infinite family of ideal schemes with t-strong multiplication that does not rely on algebraic geometry and that works over every finite field ${\mathbb F}_q$. Its corruption tolerance vanishes, yet still $\frac{3t}{n-1}= \Omega(1/(\log\log n)\log n)$. Fourth and finally, we give an improved non-asymptotic upper bound on corruption tolerance.