How to Share a Secret, Infinitely

Ilan Komargodski, Moni Naor, Eylon Yogev

Research output: Contribution to journalArticlepeer-review


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 access structure. The best known example is the k-threshold access structure, where the qualified subsets are those of size at least k. When k = 2 and there are n parties, there are schemes for sharing an ℓ-bit secret in which the share size of each party is roughly max{ℓ, logn} bits, and this is tight even for secrets of 1 b. In these schemes, the number of parties n 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 t th party arriving the smallest possible share as a function of t. Our main result is such a scheme for the k-threshold access structure and 1-bit secrets where the share size of party t is (k-1)·logt+poly(k)·o(logt). Fork = 2 we observe an equivalence to prefix codes and present matching upper and lower bounds of the form log t + log log t + log log log t + O(1). Finally, we show that for any access structure there exists such a secret sharing scheme with shares of size 2 t-1 .
Original languageEnglish
Pages (from-to)4179-4190
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number6
Early online date4 Dec 2017
StatePublished - Jun 2018


  • Secret sharing
  • evolving access structure
  • prefix code
  • threshold

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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