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We know that Blakley's scheme is less space-efficient than Shamir's and that they both work based on pretty different aspects (Shamir = polynomial-based; Blakley = hyperplane-based). Looking at how both work, my gut feeling tells me Shamir’s scheme seems to be more robust than Blakley’s because – for example – if an insider can gain any more knowledge about the secret than an outsider can, then Blakley's system no longer has information theoretic security.

But gut feelings rarely tend to represent actual facts and I might be missing something in relation to Shamir’s scheme, which is why I’m asking: Are there any practical security differences between Shamir's Secret Sharing Scheme and Blakley's Secret Sharing Scheme? Am I correct to think that Shamir’s scheme is more robust than Blakley’s?

It would be nice (but optional) if you could add one or more relevant references to your answer.

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  • $\begingroup$ While trying to answer this question, I ran into the complication that there seem to be multiple descriptions of "Blakley's secret sharing" around, and it's not at all obvious to me that they're all equivalent or even equally secure. Wikipedia's summary is too vague to be of any use. The slides linked from this answer only show an example of the 3-out-of-3 scheme, and I'm not too sure even that's accurate. ... $\endgroup$ – Ilmari Karonen Jan 28 '17 at 16:14
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    $\begingroup$ ... This paper by E.F. Brickell has a very nice-looking geometric description, but it looks quite different from the other descriptions I've seen. And of course there's Blakley's original paper, but it's pretty hard to follow, its proofs are incomplete, and it appears to describe a scheme that is even less efficient than the other descriptions I've seen (e.g. only reconstructing a set of $t^2+t$ candidate secrets, for a threshold $t$ scheme). $\endgroup$ – Ilmari Karonen Jan 28 '17 at 16:15
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They are both perfectly (information theoretical) secure, as you mentioned, Shamir's Secret Sharing Scheme is more space efficient. When Blakley's scheme is optimized by using finite fields, it eventually turns into Shamir's Secret Sharing Scheme. They are equal from a security perspective.

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    $\begingroup$ "When Blakley's scheme is optimized by using finite fields", actually, the original Blakley paper is already on a finite field. $\endgroup$ – poncho Jan 28 '17 at 15:05

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