TY - JOUR

T1 - How to Share a Secret, Infinitely

AU - Komargodski, Ilan

AU - Naor, Moni

AU - Yogev, Eylon

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2018/6

Y1 - 2018/6

N2 - 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, log n} 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)log t+poly(k)o(log t ). For k = 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 2t?1.

AB - 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, log n} 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)log t+poly(k)o(log t ). For k = 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 2t?1.

KW - Secret sharing

KW - evolving access structure

KW - prefix code

KW - threshold

UR - http://www.scopus.com/inward/record.url?scp=85037623199&partnerID=8YFLogxK

U2 - 10.1109/TIT.2017.2779121

DO - 10.1109/TIT.2017.2779121

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AN - SCOPUS:85037623199

SN - 0018-9448

VL - 64

SP - 4179

EP - 4190

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 6

ER -