Secret Sharing, Zero Sum Sets, and Hamming Codes - Selda Çalkavur; Patrick Solé (Mathematics, 2020)
A <semantics>(t,n)</semantics>-secret sharing scheme is a method of distribution of information among n participants such that any <semantics>t>1</semantics> of them can reconstruct the secret but any <semantics>t−1</semantics> cannot. A ramp secret sharing scheme is a relaxation of that protocol that allows that some <semantics>(t−1)</semantics>-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.