Almost Optimal Cheating-Detectable (2, 2, n) Ramp Secret Sharing Scheme

In this research, we consider a strong ramp secret sharing scheme that can detect cheating. A cheating-detectable (k, L, n) ramp secret sharing scheme has been studied so far, and a strong ramp secret sharing scheme which achieves lower bounds on the size of shares and random number used in encoding (i. e., share generation), and the successful cheating probability of impersonation attack has been presented. Now a challenging task is to achieve the lower bound on the successful cheating probability of substitution attack. In this paper, we present a strong (2, 2, n) ramp secret sharing scheme that almost achieves the lower bound on the successful cheating probability of substitution attack. Although the parameter is limited to (2, 2, n) type ramp scheme, the proposed scheme is the first to almost achieve the lower bound. Moreover the proposed scheme also achieves other lower bounds such as those on the size of shares and random number used in encoding, and the successful cheating probability of impersonation attack. To achieve the lower bound on the successful cheating probability of substitution attack, we take a unique strategy to construct the scheme. Most existing works present generic type verification functions that can be applied to arbitrary linear and strong (k, L, n) ramp secret sharing schemes. On the other hand, our proposed verification function (one of those which we call limited type verification functions) cannot be applied to arbitrary strong (k, L, n) ramp schemes but applied to a special case of strong (2, 2, n) ramp schemes with certain property, which make it possible to (almost) achieve the lower bound on the successful cheating probability of substitution attack. Fortunately, we can prove that, for any secret (S-1, S-2) is an element of GF(p(m))(2), there exist sufficiently many strong (2, 2, n) ramp schemes with which the proposed scheme is proven to be secure. Further, we suggest that it is difficult to generalize our strategy to more generalized parameter because of the difficulty to find conditions of strong (k, L, n) ramp schemes with which any limited type verification functions guarantee security against substitution attacks.