Efficient Ideal Threshold Secret Sharing Schemes Based on EXCLUSIVE-OR Operations

Most of secret sharing schemes have to be computed in a Galois field, such as Shamir´s scheme, which have relatively heavy computational cost. Kurihara et al. recently proposed a fast secret sharing scheme using only Exclusive-OR(XOR) operations to make shares and recover the secret. Their proposed scheme was shown to be hundreds of times faster than Shamir´s (in GF(q=2⁶⁴)) in terms of both distribution and recovery with a 4.5 MB secret when k=3 and n=11. However, some steps in their scheme still need to be improved. Their security proofs were too complex and difficult to be understood and verified intuitively. In this paper, we present a conciser, cleaner, faster scheme which is also based on XOR. Moreover, we give two geometric explanations of making shares in both our and Kurihara´s schemes respectively, which would help to easier and further understand how the shares are made in the two schemes.