A (t, n, δ) secret sharing scheme with cheating detection property (SSCD) is a t-out-of-n threshold secret sharing scheme that has the following additional property; the probability that any t malicious players can successfully cheat (without being caught) an honest player by opening forged shares and causing the honest player to reconstruct the wrong secret is at most δ. There are two flavors of security for such schemes known as OKS and CDV. The lower bound on share sizes for an OKS-secure SSCD scheme is known, and concrete schemes in which share sizes are equal to (or almost the same as) the lower bound have been proposed, albeit with some limitations. We first present a OKS-secure scheme with share sizes only one bit longer than its existing lower bound. Our construction is free from any special requirements. We next present a CDV-secure SSCD scheme, where a stronger form of cheating is allowed. The share size of our CDV-secure scheme is also one bit longer than the existing lower bound.
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