Secret Sharing, Zero Sum Sets, and Hamming Codes

Selda Çalkavur and Patrick Solé
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Selda Çalkavur: Mathematics Department, Kocaeli University, 41380 Kocaeli, Turkey
Patrick Solé: I2M, Aix Marseille University, Centrale Marseilles, CNRS, 163 Avenue de Luminy, 13009 Marseille, France

Mathematics, 2020, vol. 8, issue 10, 1-7

Abstract: A ( t , n ) -secret sharing scheme is a method of distribution of information among n participants such that any t > 1 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.

Keywords: secret sharing; threshold scheme; ramp scheme; polynomial residue ring; zero-sum set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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