A Fast (2, 2^m)-Threshold Secret Sharing Scheme Using m Linearly Independent Binary Vectors

Fast (k, n)-threshold secret sharing schemes with exclusive-OR 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 only XOR operation at distribution and restoration processes. In these cases for the number of shares n, target data must be equally divided into individual n_p-1 pieces where np 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 have proposed, this leads to general constructions of (2, p + 1)-threshold secret sharing schemes.In this paper, we use m-dimensional vector spaces over Z₂ on having bases that meet certain conditions in order to construct proposed methods. This paper defines a new notion "2-propagation bases set" as a bases set to be used in the configuration. In order to guarantee the existence of (2; 2^m)- threshold secret sharing schemes, we also treat the presence of the m-dimensional bases.