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Secret Sharing, Zero Sum Sets, and Hamming Codes

Author

Listed:
  • 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)

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.

Suggested Citation

  • Selda Çalkavur & Patrick Solé, 2020. "Secret Sharing, Zero Sum Sets, and Hamming Codes," Mathematics, MDPI, vol. 8(10), pages 1-7, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1644-:d:418071
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