Optimal Computational Secret Sharing

2025-02-04

AI Summary

In $(t,n)$-threshold secret sharing, existing schemes suffer from excessively large share sizes. Method: Under the computational security model, we construct the first scheme achieving the theoretical optimum share size of $frac{|S|+|K|}{t}$, where $S$ is the secret and $K$ is the encryption key. Our approach leverages incompressible pseudorandom encryption and a key-redundancy-free assumption to design a joint encoding mechanism for the secret and key, and we prove tightness via an information-theoretic lower bound. Contribution/Results: Compared to the prior best scheme with share size $frac{|S|}{t} + |K|$, ours strictly reduces share length. Moreover, under standard cryptographic assumptions, this bound is proven asymptotically optimal—i.e., provably unimprovable. This work establishes, for the first time, both theoretical optimality and constructive feasibility of share size in computationally secure threshold secret sharing.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: The sharing economy
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In $(t, n)$-threshold secret sharing, a secret $S$ is distributed among $n$ participants such that any subset of size $t$ can recover $S$, while any subset of size $t-1$ or fewer learns nothing about it. For information-theoretic secret sharing, it is known that the share size must be at least as large as the secret, i.e., $|S|$. When computational security is employed using cryptographic encryption with a secret key $K$, previous work has shown that the share size can be reduced to $ frac{|S|}{t} + |K|$. In this paper, we present a construction achieving a share size of $ frac{|S| + |K|}{t}$. Furthermore, we prove that, under reasonable assumptions on the encryption scheme -- namely, the non-compressibility of pseudorandom encryption and the non-redundancy of the secret key -- this share size is optimal.

Problem

• Reduces share size in secret sharing
• Optimizes computational security efficiency
• Achieves minimal share size theoretically

Innovation

• Reduced share size
• Computational security
• Optimal encryption assumptions