Secret Sharing, Zero Sum Sets, and Hamming Codes

ÇALKAVUR S., Sole P.

MATHEMATICS, vol.8, no.10, pp.1-7, 2020 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 8 Issue: 10
  • Publication Date: 2020
  • Doi Number: 10.3390/math8101644
  • Journal Name: MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Page Numbers: pp.1-7
  • Kocaeli University Affiliated: Yes

Abstract

A (t,n)-secret sharing scheme is a method of distribution of information among n participants such that any t1 of them can reconstruct the secret but any t-1 cannot. A ramp secret sharing scheme is a relaxation of that protocol that allows that some (t-1)-coalitions could reconstruct the secret. In this work, we explore some ramp secret sharing schemes based on quotients of polynomial rings. The security analysis depends on the distribution of zero-sum sets in abelian groups. We characterize all finite commutative rings for which the sum of all elements is zero, a result of independent interest. When the quotient is a finite field, we are led to study the weight distribution of a coset of shortened Hamming codes.