Editing Secret sharing (section)

===Blakley's scheme===
Two [[Parallel (geometry)|nonparallel]] lines in the same [[plane (mathematics)|plane]] intersect at exactly one point. Three nonparallel planes in space intersect at exactly one point. More generally, any ''n'' nonparallel [[dimension|{{nowrap|(''n'' − 1)}}-dimensional]] [[hyperplane]]s intersect at a specific point. The secret may be encoded as any single coordinate of the point of intersection.  If the secret is encoded using all the coordinates, even if they are random, then an insider (someone in possession of one or more of the [[dimension|{{nowrap|(''n'' − 1)}}-dimensional]] [[hyperplane]]s) gains information about the secret since he knows it must lie on his plane.  If an insider can gain any more knowledge about the secret than an outsider can, then the system no longer has [[information theoretic security]].  If only one of the ''n'' coordinates is used, then the insider knows no more than an outsider (i.e., that the secret must lie on the ''x''-axis for a 2-dimensional system).  Each player is given enough information to define a hyperplane; the secret is recovered by calculating the planes' point of intersection and then taking a specified coordinate of that intersection.

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{| border="0" cellspacing="2px" style="margin-left: auto; margin-right:auto;" width="600px"
| [[File:Secretsharing 1.svg|250px|One share]]
| [[File:Intersecting Planes 2.svg|250px|Two shares intersecting on a line]]
| [[File:Secretsharing 3-point.svg|250px|Three shares intersecting at a point]]
|-
| colspan="3" | ''Blakley's scheme in three dimensions: each share is a [[plane (mathematics)|plane]], and the secret is the point at which three shares intersect. Two shares are insufficient to determine the secret, although they do provide enough information to narrow it down to the [[line (mathematics)|line]] where both planes intersect.''
|}
</div>

Blakley's scheme is less space-efficient than Shamir's; while Shamir's shares are each only as large as the original secret, Blakley's shares are ''t'' times larger, where ''t'' is the threshold number of players. Blakley's scheme can be tightened by adding restrictions on which planes are usable as shares. The resulting scheme is equivalent to Shamir's polynomial system.