On Polynomial Secret Sharing Schemes

Abstract

Nearly all secret sharing schemes studied so far are linear or multi-linear schemes. Although these schemes allow to implement any monotone access structure, the share complexity, SC, may be suboptimal - there are access structures for which the gap between the best known lower bounds and best known multi-linear schemes is exponential. 
There is growing evidence in the literature, that non-linear schemes can improve share complexity for some access structures, with the work of Beimel and Ishai (CCC '01) being among the first to demonstrate it. This motivates further study of non linear schemes.
We initiate a systematic study of polynomial secret sharing schemes (PSSS), where shares are (multi-variate) polynomials of secret and randomness vectors ~s,~r respectively over some finite field 𝔽_q. Our main hope is that the algebraic structure of polynomials would help obtain better lower bounds than those known for the general secret sharing. Some of the initial results we prove in this work are as follows. 

On share complexity of polynomial schemes. First we study degree (at most) 1 in randomness variables ~r (where the degree of secret variables is unlimited). We have shown that for a large subclass of these schemes, there exist equivalent multi-linear schemes with O(n) share complexity overhead. Namely, PSSS where every polynomial misses monomials of exact degree c≥ 2 in ~s and 0 in ~r, and PSSS where all polynomials miss monomials of exact degree ≥ 1 in ~s and 1 in ~r. This translates the known lower bound of Ω(n^{log(n)}) for multi linear schemes onto a class of schemes strictly larger than multi linear schemes, to contrast with the best Ω(n²/log(n)) bound known for general schemes, with no progress since 94'. An observation in the positive direction we make refers to the share complexity (per bit) of multi linear schemes (polynomial schemes of total degree 1). We observe that the scheme by Liu et. al obtaining share complexity O(2^{0.994n}) can be transformed into a multi-linear scheme with similar share complexity per bit, for sufficiently long secrets. For the next natural degree to consider, 2 in ~r, we have shown that PSSS where all share polynomials are of exact degree 2 in ~r (without exact degree 1 in ~r monomials) where 𝔽_q has odd characteristic, can implement only trivial access structures where the minterms consist of single parties.
Obtaining improved lower bounds for degree-2 in ~r PSSS, and even arbitrary degree-1 in ~r PSSS is left as an interesting open question.

On the randomness complexity of polynomial schemes. We prove that for every degree-2 polynomial secret sharing scheme, there exists an equivalent degree-2 scheme with identical share complexity with randomness complexity, RC, bounded by 2^{poly(SC)}. For general PSSS, we obtain a similar bound on RC (preserving SC and 𝔽_q but not degree). So far, bounds on randomness complexity were known only for multi linear schemes, demonstrating that RC ≤ SC is always achievable. Our bounds are not nearly as practical as those for multi-linear schemes, and should be viewed as a proof of concept. If a much better bound for some degree bound d=O(1) is obtained, it would lead directly to super-polynomial counting-based lower bounds for degree-d PSSS over constant-sized fields. Another application of low (say, polynomial) randomness complexity is transforming polynomial schemes with polynomial-sized (in n) algebraic formulas C(~s,~r) for each share, into a degree-3 scheme with only polynomial blowup in share complexity, using standard randomizing polynomials constructions.