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Chinese Journal of Electronics

A Verifiable Multi-Secret Sharing Scheme Based on Short Integer Solution

Fulin Li, Jiayun Yan, Shixin Zhu, Hang Hu

Published May 1, 2023• 8 citations
Publisher

Abstract

With the possible birth of the quantum computer, traditional secret sharing schemes have been unable to meet security requirements. We proposed a new verifiable multi-secret sharing scheme based on the short integer solution problem. By utilizing a symmetric binary polynomial,$k$secrets and secret shares can be generated, and then we convert the secret shares into binary string on$\mathbb{Z}_{q}$, which can be identified by one-way anti-collision hash function on the lattice, so that multiple secrets can be reconstructed safely. The advantages mainly focus on verifiability without interaction in the distribution phase and less memory requirement. In a secret sharing scheme, verifiability prevents the dealer to share the wrong shares and forces the participants to submit their shares correctly. Meanwhile, the interaction can be reduced, which means the security is improved. In a multi-secret sharing scheme, releasing the public values is inevitable, this paper has less public values and less size of shares per secret size to reduce the pressure of memory consumption in the proper parameters. In the end, it can also effectively resist the quantum attack.

DOI
10.23919/cje.2021.00.062

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With the possible birth of the quantum computer, traditional secret sharing schemes have been unable to meet security requirements. We proposed a new verifiable multi-secret sharing scheme based on the short integer solution problem. By utilizing a symmetric binary polynomial,kkksecrets and secret shares can be generated, and then we convert the secret shares into binary string onZq\mathbb{Z}_{q}Zq​, which can be identified by one-way anti-collision hash function on the lattice, so that multiple secrets can be reconstructed safely. The advantages mainly focus on verifiability without interaction in the distribution phase and less memory requirement. In a secret sharing scheme, verifiability prevents the dealer to share the wrong shares and forces the participants to submit their shares correctly. Meanwhile, the interaction can be reduced, which means the security is improved. In a multi-secret sharing scheme, releasing the public values is inevitable, this paper has less public values and less size of shares per secret size to reduce the pressure of memory consumption in the proper parameters. In the end, it can also effectively resist the quantum attack.
In this paper, we construct a lattice based (t, n) threshold multi-stage secret sharing (MSSS) scheme according to Ajtai's construction for one-way functions. In an MSSS scheme, the authorized subsets of participants can recover a subset of secrets at each stage while other secrets remain undisclosed. In this paper, each secret is a vector from a t-dimensional lattice and the basis of each lattice is kept private. A t-subset of n participants can recover the secret(s) using their assigned shares. Using a lattice based one-way function, even after some secrets are revealed, the computational security of the unrecovered secrets is provided against quantum computers. The scheme is multi-use in the sense that to share a new set of secrets, it is sufficient to renew some public information such that a new share distribution is no longer required. Furthermore, the scheme is verifiable meaning that the participants can verify the shares received from the dealer and the recovered secrets from the combiner, using public information.
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.
Abstract Based on the d-dimensional Greenberger–Horne–Zeilinger (GHZ) state, this paper proposes a novel multi-party quantum secret sharing scheme with the characteristics of identity authentication and dual-threshold. In the scheme, the dealer first constructs an asymmetric binary polynomial and decomposes it into two unary polynomials, then divides the participants into two groups, and generates secret shares and corresponding polynomial pairs for each participant. The dealer prepares a three-qudit GHZ state and uses two of them to perform dual-line transmission between the two groups of participants. The participants perform specific unitary transformations on the transmitted particles. Finally, the recovery agent collects and measures the information particles to recover the secret. The scheme uses a hash function and a session key to detect fraudulent behavior and identify specific participants to achieve identity authentication. Security analysis shows that the scheme can effectively resist intercept-resend attacks, entangle-measure attacks, dishonest participant attacks and Trojan horse 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.
A verifiable quantum secret sharing protocol using quantum walks on a lattice is proposed in this paper. The protocol utilizes generalized Bell states as the initial quantum resource, integrating the quantum walk evolution operator U with the global shift operator GabG_{ab}Gab​ to facilitate joint encrypted transmission among multiple participants. A bidirectional identity authentication mechanism based on hash functions is introduced to ensure the legitimacy of all communication parties, which effectively resists impersonation attacks. Random decoy particles are randomly prepared in either the ZdZ_{d}Zd​ or XdX_{d}Xd​ basis and inserted into the transmission sequence, enabling real-time eavesdropping detection on the quantum channel. In the secret reconstruction phase, a public verification mechanism is designed to allow participants to verify the correctness of the recovered secret and identify internal dishonest behaviors. Security analysis demonstrates that the protocol achieves information-theoretic security, resisting typical external attacks such as intercept-resend, measure-resend, and entangle-measure attacks, as well as collusion attacks from dishonest internal participants. Compared with existing quantum-walk-based secret sharing schemes, the proposed protocol maintains high qubit efficiency while supporting both identity authentication and public verifiability, thus possessing stronger practicality and security.
We present a new framework for constructing practically efficient publicly verifiable secret sharing (PVSS) with non-interactive dealers , in that the dealer may go offline after sending a single message, and is not involved in the share verification process. We use identity-based encryption (IBE) and commitments as the main ingredients and avoid expensive zero-knowledge proofs. By instantiating them with post-quantum secure schemes, a lattice-based IBE and a hash-based commitment, we obtain our first construction: a post-quantum secure PVSS with non-interactive dealers that outperforms the prior lattice-based practical construction by Gentry et al. [Eurocrypt 2022]. However, to enable the aggregation of PVSS transcripts (which facilitates many additional applications such as secure voting), we propose our second construction by replacing hash-based commitments with Pedersen's homomorphic commitments. While this does not achieve full-fledged post-quantum security (as Pedersen's scheme is not quantum safe), it still provides privacy against a post-quantum adversary. We prove the security of this construction in a new model, which we call long-lasting security. This model guarantees that the protocol is secure in the present (pre-quantum era) while maintaining privacy in the long term (post-quantum era). This new model is of independent interest for constructing efficient schemes that are resilient to the harvest-now-decrypt-later line of attacks. In this model, we propose a blockchain-compatible secure voting scheme using our PVSS scheme. Our PVSS schemes demonstrate practical efficiency: our post-quantum PVSS shares a secret among 1024 receivers in 667 ms and verifies the dealing in < 5 ms, and communicates 4.4MB, overall whereas the state of the art [Gentry et al., Eurocrypt 2022] communicates 251MB and requires 2060s for dealing the secret.