# Reference for additive secret sharing

Regarding the secret sharing schemes, I am wondering if there are any academic references for the Additive Secret Sharing and its variations. Is there any textbook or article which has specifically discussed this scheme for secret sharing? When and where has it been formally introduced for the first time?

• Can you please give a quick overview / non-academic reference to the scheme? I for example don't know right to what exactly you are referring. – SEJPM Apr 10 '19 at 11:39
• @SEJPM: I assume the OP is referring to the "trivial" unconditionally secure $n$-out-of-$n$ secret sharing scheme obtained by taking any finite abelian group (e.g. the integers modulo $m$), issuing $n-1$ of the shareholders a random group element $a_i$, and issuing the last shareholder the share $a_n = s - \sum_{i=1}^{n-1} a_i$. Reconstructing the secret $s$ then just requires summing the shares. Most commonly, of course, this is done bitwise over the integers modulo 2, such that addition and subtraction are both XOR. – Ilmari Karonen Apr 10 '19 at 13:13
• Anyway, I would've guessed that either Blakley or Shamir would've mentioned this scheme in their respective 1979 papers, but it seems that neither of them actually did. The general result that any subset of $n-1$ out of the $n$ shares chosen as above are independent must surely have been known and described before (at least in the modulo 2 case), but presumably not linked to the term "secret sharing", which AFAIK was essentially coined by Shamir. – Ilmari Karonen Apr 10 '19 at 13:34