Currently, I am trying to understand Blakley's secret sharing scheme. However, there seem to be multiple descriptions of it, which are very different from another, but everyone states that the scheme would be perfectly secure.
From my understanding it works like this:
- a point $S$ in $t$-dimensional space is created with one coordinate being secret and the others randomly selected
- $n$ hyperplanes through the point $S$ are created and distributed
- to reconstruct the secret, $t$ hyperplanes are required, which will intersect at the point $S$.
However, from my understanding, if someone would possess one or more hyperplanes, they would know that $S$ lies somewhere on that hyperplane (or the intersection of multiple hyperplanes)
So my question is;
- how is Blakley's scheme considered perfectly secure when an insider knows that the secret lies on his plane?