An Extension of \((2, m(m+1)/2)\)-Threshold Secret Sharing Schemes

Fast (k, n)-threshold secret sharing schemes with XOR operations have proposed by Kurihara et al. and Fujii et al. independently. Their method 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. In these cases for the number of shares n, target data must be equally divided into individual \(n_p-1\) pieces where \(n_p\) is a prime more than n. The existing methods described above are configured using the cyclic matrices with prime order. On the other hand, a new method in WAIS2013 has proposed, this leads to general constructions of \((2,p+1)\)-threshold secret sharing schemes. Moreover, we use m-dimensional vector spaces over \(\mathbb {Z}_2\) on having bases that meet certain conditions in order to construct proposed methods. Moreover, existences of \((2,2^m)\)-threshold secret sharing schemes are published (in NBiS2013) by using a new notion “2-propagation bases set” as a bases set in m-dimensional vector spaces over \(\mathbb {Z}_2\). However this construction for all parameter m is wrong, in other words, it constitutes just only \((2,m(m+1)/2)\)-SSS, not \((2,2^m)\)-SSS. This paper corrects mistakes of construction 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 an irreducible polynomial, these functionalities lead to general constructions of \((2,2^m)\)-threshold secret sharing schemes.