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SIAM Journal on Discrete Mathematics

The Detection of Cheaters in Thershold Schemes

Ernest F. Brickell, Douglas R. Stinson

Published Nov 1, 1991• 43 citations
Publisher

Abstract

Informally, a $( t,w )$-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key (or secret), but no subset of fewer than t participants can determine the key. Presented in this paper is an unconditionally secure threshold scheme in which any cheating participant can be detected and identified with high probability by any honest participant, even if the cheater is in coalition with other participants. Also given is a construction that will detect with high probability a dealer who distributes inconsistent shadows (shares) to the honest participants. The scheme is not perfect; a set of $t - 1$ participants can rule out at most $1 + \begin{pmatrix} {w - t + 1} \\ {t - 1} \end{pmatrix}$ possible keys, given the information they have. In this scheme, the key will be an element of GF$( q )$ for some prime power q. Hence q can be chosen large enough so that the amount of information obtained by any $t - 1$ participants is negligible.

DOI
10.1137/0404044

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Informally, a (t,w)( t,w )(t,w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key (or secret), but no subset of fewer than t participants can determine the key. Presented in this paper is an unconditionally secure threshold scheme in which any cheating participant can be detected and identified with high probability by any honest participant, even if the cheater is in coalition with other participants. Also given is a construction that will detect with high probability a dealer who distributes inconsistent shadows (shares) to the honest participants. The scheme is not perfect; a set of t−1t - 1t−1 participants can rule out at most 1+(w−t+1t−1)1 + \begin{pmatrix} {w - t + 1} \\ {t - 1} \end{pmatrix}1+(w−t+1t−1​) possible keys, given the information they have. In this scheme, the key will be an element of GF(q)( q )(q) for some prime power q. Hence q can be chosen large enough so that the amount of information obtained by any t−1t - 1t−1 participants is negligible.
Quantum secret sharing holds a significant place in quantum cryptography. In this paper, a (w, t, n)-weighted threshold dynamic quantum secret sharing scheme with cheating identification is constructed based on the CRT for a polynomial ring. In our scheme, on the premise of not altering the private shares of the old participants, the dynamic update of the participants can be realized without other participants’ help, and the shared secret will not be changed. During the secret reconstruction phase, via executing the quantum state digital signature, the proposed scheme not only can timely detect the existence of the cheating behavior, but also once exists, definitely can identify one cheater and eliminate him from the system, making the scheme more secure. Moreover, our scheme enables secret sharing among participants with different authorities, which enhances the practicability of the scheme. Finally, the security analysis shows our scheme not only is resistant to a range of typical external attacks but can detect forgery, collusion, denial and dishonest revoked participant attacks.
A verifiable (t, n) threshold quantum secret sharing scheme with sequential communication was proposed recently. In this work, we analyze its security and then give two new participant attacks. Using the first participant attack, the first participant can obtain the dealer's secrets by himself with nonzero probability without being detected. Using the second participant attack, a dishonest participant can gain access to the dealer's secrets by himself in the secret reconstruction phase while he can make the other participants recover false secrets instead of the real ones without being detected. Furthermore, we present an effective way to prevent these attacks.
Abstract Threshold quantum secret sharing is a typical method for quantum secret sharing (QSS) schemes. In this paper, we propose a verifiable t , n threshold QSS scheme based on the d -dimensional Bell state and asymmetric binary polynomial. In this scheme, the dealer encodes the secret using the asymmetric binary polynomial and generates the corresponding secret share for each participant. Then, the dealer prepares the d -dimensional Bell state, and the participants perform corresponding unitary operations on the particles in transmission to recover the secret. With the hash function and the session key pairs, not only the cheating behavior of the dishonest participants can be detected, but also the specific cheaters can be identified. Furthermore, we consider the case when no less than t participants cooperate to recover the secret, which makes the proposed scheme more practical. Analyses show that the scheme can resist forgery attack, collusion attack and other common attacks.
We present DQ-(5,11), a dealer-based (5,11) threshold quantum secret sharing (QSS) scheme where any 5 of the 11 participants can find a dealer's secret and 4 or fewer get nothing about it. A key point is that, due to the Cleve-Gottesman-Lo no-cloning bound, there is only a (k,n) threshold scheme for any quantum secret if n < 2k. Since 11≥10, is a (5,11) quantity, the threshold scheme is impossible for quantum secrets, and in any (5,11) quantum secret sharing scheme there is a classical secret that is encrypted in a way that is protected by quantum communication. DQ-(5,11) is a combination of Shamir's polynomial sharing over GF13 and a single-qudit reconstruction in dimension d=13. Each of the five cooperating participants in DQ-(5,11) performs one phase operation with the share of the quantum secret and a zero-sum mask so that the intermediate states will not have any partial sum. Decoy qudits from two mutually unbiased bases detect eavesdropping, and a dealer-published commitment lets participants check the reconstructed value. We achieve correctness of all 462 subsets of the network, information-theoretic secrecy against coalitions of up to four insiders and a requirement for the success of intercept-resend attacks to be (d+1)/2d)m for m decoys (8.6×10-9) for m=30). We use numerical data to verify the deterministic reconstruction for all 2,310 runs and show the effect of depolarizing noise (79.1% success at 5% noise at each stage) and channel loss. Furthermore, we compare our orbital-angular-momentum and time-bin qudit model to Trojan-horse countermeasures and other schemes.
Abstract Threshold quantum secret sharing is a cryptography technique for dividing and reconstructing secret information. The dealer shares a secret based on quantum mechanics principles, and a subset of participants cooperate to recover that secret. For example, in k , n quantum threshold scheme and n , n quantum threshold scheme, the former can only reconstruct the secret when at least k shares are collected. The latter requires all n shares to be collected to reconstruct the secret. However, in real scenarios, the priority of participant identities may vary. In order to meet the actual needs of participants with different rights, this paper proposes a w , ω , n quantum secret sharing (QSS) scheme. When the sum of participants’ weights w is greater than or equal to a threshold ω, they can cooperate to reconstruct the secret. In order to achieve this, this paper uses the entangled state of single particles and constructs a verifiable QSS scheme based on the Chinese remainder theorem. Significantly, in case of a dishonest participant, the dealer can delete the participant before recovering the secret to ensure the security of the communication. This scheme is more flexible than other threshold protocols. At the same time, it reduces the communication cost of the participants. In addition, security analysis shows that the scheme resists typical external and internal attacks.