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International Symposium on Information Theory and its Applications

Fourier-based Verifiable Function Secret Sharing

Takeshi Koshiba

Published Oct 24, 2020• 1 citation
OpenAlex

Abstract

Function secret sharing is a cryptographic protocol for secure distribution of a secret function. While the notion of function secret sharing is a conceptual extension of secret sharing schemes, it faces difficulties in its constructions. We demonstrate that if we consider function secret sharing schemes for Fourier basis functions then we can enjoy the linear properties of linear secret sharing schemes to construct function secret sharing schemes. In this paper, we construct a verifiable function secret sharing by using Fourier basis functions, where players can verify if the dealer is cheating.

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Function secret sharing is a cryptographic protocol for secure distribution of a secret function. While the notion of function secret sharing is a conceptual extension of secret sharing schemes, it faces difficulties in its constructions. We demonstrate that if we consider function secret sharing schemes for Fourier basis functions then we can enjoy the linear properties of linear secret sharing schemes to construct function secret sharing schemes. In this paper, we construct a verifiable function secret sharing by using Fourier basis functions, where players can verify if the dealer is cheating.
Abstract. We present a general treatment of all non-cryptographic (i.e., information-theoretically secure) linear verifiable-secret-sharing (VSS) and distributed-commitment (DC) schemes, based on an underlying secret sharing scheme, pairwise checks between players, complaints, and accusations of the dealer. VSS and DC are main building blocks for unconditional secure multi-party computation protocols. This general approach covers all known linear VSS and DC schemes. The main theorem states that the security of a scheme is equivalent to a pure linear-algebra condition on the linear mappings (e.g. described as matrices and vectors) describing the scheme. The security of all known schemes follows as corollaries whose proofs are pure linear-algebra arguments, in contrast to some hybrid arguments used in the literature. Our approach is demonstrated for the CDM DC scheme, which we generalize to be secure against mixed adversary settings (some curious and some dishonest players), and for the classical BGW VSS scheme, for which we show that some of the checks between players are superfluous, i.e., the scheme is not optimal. More generally, our approach, establishing the minimal conditions for security (and hence the common denominator of the known schemes), can lead to the design of more efficient VSS and DC schemes for general adversary structures. 1
Cheating immune secret sharing in the unconditionally secure case was investigated. Secret sharing is a k-cheating immune if any k cheaters are not better off than a participant which follows the protocol honestly. k-cheating immune secret sharing scheme under the unconditionally secure would be given. The obtained function of the secret sharing scheme is k-resilient and satisfies k-strengthened propagation criterion. Furthermore, its degree is high. So it has better cryptographic properties than that of the known results.
In a k-threshold secret sharing scheme, a dealer who holds a secret s distributes parts of this secret (the shares) to n players; If k or more of these players pool their shares, they are able to determine s, but if less than k of them pool their shares, they cannot infer any information on s. In 1979, Shamir introduced such a scheme, based on polynomial interpolation in a finite field. This scheme is widely used in other cryptographic protocols, because it is simple, elegant and above all information-theoretically secure. There are a few proofs of the scheme's security, but to our knowledge, none of them is entirely based on the information-theoretical entropy function introduced by Shannon in 1948. We propose such a demonstration.
This work proposes a [Formula: see text]-dimensional quantum multi-secret sharing scheme with a cheat-detection mechanism. The dealer creates multiple secrets and distributes the shares of these secrets using multi-access structures and a monotone span program. The dealer detects the cheating of each participant using the black box’s cheat-detection mechanism. To detect the participants’ deceit, the dealer distributes secret shares’ shadows derived from a randomly invertible matrix [Formula: see text] to the participants, stored in the black box. The black box identifies the participant’s deceitful behavior during the secret recovery phase. Only honest participants authenticated by the black box acquire their secret shares to recover the multiple secrets. After the black box cheating verification, the participants reconstruct the secrets by utilizing the unitary operations and quantum Fourier transform. The proposed protocol is reliable in preventing attacks from eavesdroppers and participants. The scheme’s efficiency is demonstrated in different noise environments: dit-flip noise, [Formula: see text]-phase-flip noise and amplitude-damping noise, indicating its robustness in practical scenarios. The proposed protocol provides greater versatility, security and practicality.
This paper proposed a verifiable quantum secret sharing protocol based on entanglement swapping.The dealer communicated with the participants one by one.He chose the detection pattern or information pattern by uniform distribution,until he could make sure the information to the participant was safe,and then communicated with the next participant.It could prevent not only the external eavesdropping but also the internal cheating,and guaranteed the correction and security of information shared for verifying.