Optimal threshold padlock systems.

In 1968, Liu described the problem of securing the documents in a shared secret research project. In his example, at least six out of eleven participating scientists need to be present to open the lock securing the secret documents. Shamir proposed a mathematical solution to this physical problem in 1979, by designing the first efficient k-out-of-n secret sharing scheme based on a smart usage of Lagrange's interpolation. Shamir also claimed that the minimal solution using physical padlocks is clearly impractical and exponential in the number of participants. In this paper we propose an optimal physical solution to the problem of Liu that uses physical padlocks, but the number of locks is at most equal to the number of participants. This device is optimal for k-out-of-n threshold padlock systems as soon as $k > $\sqrt$ 2n$. We also propose an optimal scheme implementing a 2-out-of-n threshold padlock system requiring only about $2 log[2](n)$ padlocks. Then we derive some lower bounds required to implement threshold systems in general. Finally, we discuss more complex access schemes together with other realizations with strictly less than n padlocks.