A New Construction of Secret Sharing Scheme Using the Primitive Polynomial Over Galois Fields

摘要

Fast (k, n) -threshold secret sharing schemes with XOR operations have proposed. Their methods are ideal that share size is equal to the size of the data to be distributed with the benefits that can be handled very fast for using the only XOR operations at distribution and reconstruction processes. After that, alternative methods in WAIS2013 and NBiS2013 have proposed, first method leads to general constructions of (2, n_p+1) -threshold secret sharing schemes where n_pis a prime. The later proposal realizes (2,m(m+1)/2) -threshold secret sharing schemes for small positive integer m. In this paper, we use m-dimensional vector spaces over mathbbZ₂on having bases that meet certain conditions in order to construct proposed methods proposed in NBiS2013 that has some errors of construction. So we corrects faults in NBiS2013 paper and also proposes an accurate construction by using Galois field GF(2^{m) that elements are represented in the ring F_p[X]/f(X) where f(X) is a primitive polynomial, these functionalities lead to general constructions of (2, 2m) -threshold secret sharing schemes for all integers m.}