The 2018 International Symposium on Information Theory and Its Applications (ISITA2018)
An algebraic interpretation of the XOR-based Secret Sharing Schemes
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, np+1)-threshold secret sharing schemes where np is a prime. The later proposal realizes (2,m(m + 1)/2)-threshold secret sharing schemes for small positive integer m. In this paper, we attempt an algebraic interpretation of previous methods, especially we introduce representations of m- dimensional vector spaces over Z2 on having bases that meet certain conditions. Moreover 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 Fp[X]/f(X) where f(X) is a primitive polynomial, these functionalities lead to general constructions of (2, 2^m)-threshold secret sharing schemes for all integers m.