Abstract. In this paper, we show that there exists a &cheater identifiable (k, n) threshold secret sharing scheme such as follows for cheating probability e > 0. If k 2 31 + 1, then 1. Just k participants are enough to identify who are cheaters.2. 1x1 is independent of n. That is, 1x1 = lSl(l/e)(t+2), where S denotes the set of secrets and V, denotes the set of shares of a participant Pi, respectively.(Previously, no schemes were known which satisfy both requirements.)Further, we present a lower bound on lvtl for our model and for the model of Tompa and Woll. Our bound for the TW model is much more tight than the previous bound.

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For the last two decades the notion and implementations of proxy signatures have been used to allow transfer of digital signing power within some context (in order to enable flexibility of signers within organizations and among entities). On the other hand, various notions of the key-evolving signature paradigms (forward-secure, key-insulated, and intrusion-resilient signatures) have been suggested in the last few years for protecting the security of signature schemes, localizing the damage of secret key exposure. In this work we relate the various notions via direct and concrete security reductions that are tight. We start by developing the first formal model for fully hierarchical proxy signatures, which, as we point out, also addresses vulnerabilities of previous schemes when self-delegation is used. Next, we prove that proxy signatures are, in fact, equivalent to key-insulated signatures. We then use this fact and other results to establish a tight hierarchy among the key-evolving notions, showing that intrusion-resilient signatures and key-insulated signatures are equivalent, and imply forward-secure signatures. We also introduce other relations among extended notions. Besides the importance of understanding the relationships among the various notions that were originally designed with different goals or with different system configuration in mind, our findings imply new designs of schemes. For example, many proxy signatures have been presented without formal model and proofs, whereas using our results we can employ the work on key-insulated schemes to suggest new provably secure designs of proxy signatures schemes.

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Abstract. In Crypto'95, Kurosawa, Obana and Ogata proposed a kout-of-n secret sharing scheme capable of identifying up to t cheaters with probability 1 − under the condition t ≤ (k − 1)/3 . The size of share |Vi| of the scheme satisfies |Vi| = |S|/ t+2 , which was the most efficient scheme known so far. In this paper, we propose new k-out-ofn secret sharing schemes capable of identifying cheaters. The proposed scheme possesses the same security parameters t, as those of Kurosawa et al.. The scheme is surprisingly simple and its size of share is |Vi| = |S|/ , which is much smaller than that of Kurosawa et al. and is almost optimum with respect to the size of share; that is, the size of share is only one bit longer than the existing bound. Further, this is the first scheme which can identify cheaters, and whose size of share is independent of any of n, k and t. We also present schemes which can identify up to (k − 2)/2 , and (k−1)/2 cheaters whose sizes of share can be approximately written by |Vi| ≈ (n · (t + 1) · 2 3t−1 · |S|)/ and |Vi| ≈ ((n · t · 2 3t ) 2 · |S|)/ 2 , respectively. The number of cheaters that the latter two schemes can identify meet the theoretical upper bound.