How to Share a Secret, Infinitely

Secret sharing schemes allow a dealer to distribute a secret piece of information among several parties such that only qualified subsets of parties can reconstruct the secret. The collection of qualified subsets is called an <italic>access structure</italic>. The best known example is the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-threshold access structure, where the qualified subsets are those of size at least <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">$k=2$ </tex-math></inline-formula> and there are <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> parties, there are schemes for sharing an <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula>-bit secret in which the share size of each party is roughly <inline-formula> <tex-math notation="LaTeX">$\max \{\ell ,\log n\}$ </tex-math></inline-formula> bits, and this is tight even for secrets of 1 b. In these schemes, the number of parties <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> must be given in advance to the dealer. In this paper, we consider the case where the set of parties is not known in advance and could potentially be infinite. Our goal is to give the <inline-formula> <tex-math notation="LaTeX">${t} {^{\mathrm{ th}}}$ </tex-math></inline-formula> party arriving the smallest possible share as a function of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>. Our main result is such a scheme for the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-threshold access structure and 1-bit secrets where the share size of party <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$(k-1)\cdot \log t + \mathsf {poly}(k)\cdot o(\log t)$ </tex-math></inline-formula>. For <inline-formula> <tex-math notation="LaTeX">$k=2$ </tex-math></inline-formula> we observe an <italic>equivalence</italic> to prefix codes and present matching upper and lower bounds of the form <inline-formula> <tex-math notation="LaTeX">$\log t + \log \log t + \log \log \log t + O(1)$ </tex-math></inline-formula>. Finally, we show that for any access structure there exists such a secret sharing scheme with shares of size <inline-formula> <tex-math notation="LaTeX">$2^{t-1}$ </tex-math></inline-formula>.

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