  In 1968, Liu described the problem of securing documents in a shared secret
project. In an 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
an efficient $k$-out-of-$n$ secret sharing scheme based on Lagrange's
interpolation. Liu and Shamir also claimed that the minimal solution using
physical locks is clearly impractical and exponential in the number of
participants. In this paper we relax some implicit assumptions in their claim
and propose an optimal physical solution to the problem of Liu that uses
physical padlocks, but the number of padlocks is not greater than the number of
participants. Then, we show that no device can do better for $k$-out-of-$n$
threshold padlock systems as soon as $k\geq{\sqrt{2n}}$, which holds true in
particular for Liu's example. More generally, we derive bounds required to
implement any threshold system and prove a lower bound of
$\mathcal{O}{\log(n)}$ padlocks for any threshold larger than $2$. For instance
we propose an optimal scheme reaching that bound for $2$-out-of-$n$ threshold
systems and requiring less than $2\log_2(n)$ padlocks. We also discuss more
complex access structures, a wrapping technique, and other sublinear
realizations like an algorithm to generate $3$-out-of-$n$ systems with
$2.5\sqrt{n}$ padlocks. Finally we give an algorithm building $k$-out-of-$n$
threshold padlock systems with only $\mathcal{O}{\log(n)^{k-1}}$ padlocks.
Apart from the physical world, our results also show that it is possible to
implement secret sharing over small fields.
