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International Symposium on Telecommunications

Sharing secret using lattice construction

Saba Asaad, Hamidreza Amini Khorasgani, Taraneh Eghlidos, Mohammadreza Aref

Published Sep 1, 2014• 6 citations
Publisher

Abstract

In this paper, a variant of (t, n) threshold secret sharing scheme is realized using a lattice model. Due to the potential quantum attacks in the future, we need to construct a secure channels for transmitting secret shares. Such channels could be modeled by a lattice based public key cryptosystem whose security is based on the worst case hardness of lattice problems. Therefore, we propose a lattice based threshold secret sharing scheme that is compatible with the platform on which the cryptosystem is built. It is shown that the proposed scheme is correct and asymptotically perfect from information theoretic point of view.

DOI
10.1109/istel.2014.7000831

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In this paper, a variant of (t, n) threshold secret sharing scheme is realized using a lattice model. Due to the potential quantum attacks in the future, we need to construct a secure channels for transmitting secret shares. Such channels could be modeled by a lattice based public key cryptosystem whose security is based on the worst case hardness of lattice problems. Therefore, we propose a lattice based threshold secret sharing scheme that is compatible with the platform on which the cryptosystem is built. It is shown that the proposed scheme is correct and asymptotically perfect from information theoretic point of view.
In this paper, we introduce a method of threshold secret sharing scheme in which secret reconstruction is based on celebrated Babai lattice algorithm. In order to supply secure public channels for transmitting shares to parties, we need to ensure that there is no quantum threats to these channels. One solution for this problem can be utilization of lattice cryptosystems for these channels which requires designing lattice based secret sharing schemes. We indicate that our scheme is asymptotically correct. Moreover, we analyze the security of our scheme by giving a quantitative proof of security from the view point of information theory.
In this paper, we introduce a method of threshold secret sharing scheme (TSSS) in which secret reconstruction is based on Babai's nearest plane algorithm. In order to supply secure public channels for transmitting shares to parties, we need to ensure that there are no quantum threats to these channels. A solution to this problem can be utilization of lattice-based cryptosystems for these channels which requires designing lattice-based TSSSs. We investigate the effect of lattice dimension on the security and correctness of the proposed scheme. Moreover, we prove that for a fixed lattice dimension the proposed scheme is asymptotically correct. We also give a quantitative proof of security from information theoretic viewpoint.
With the development of quantum technologies, the issue of research and implementation of cryptographic primitives based on complex problems for quantum computing becomes relevant. Such cryptographic primitives are resistant to quantum cryptanalysis. Examples of problems with exponential complexity for quantum computing is lattice problems such as finding the shortest vector or finding the closest vector. One of the first and most famous quantum-resistant cryptosystems that use lattice problems as the basis of its mathematical apparatus is the Goldwasser-Goldrich-Halevi cryptosystem.A secret distribution scheme is a fundamental cryptographic primitive that allows the distribution of a secret among a set of participants while the secret recovery is possible only when all or a certain part of the participants (the threshold of participants) is authorized. Also, a necessary condition for a secret distribution scheme is the impossibility of individual participants, or groups of participants whose number is less than the threshold, to recover the secret. Variants of constructing secret sharing schemes on various mathematical models, including lattices, are currently being actively studied since they allow for security multiparty calculations and secure information dissemination by distributing the original data between different servers. It is also used for constructing compilers of schemes with protection against leakage, etc. In this paper, a new quantum-stable n-threshold secret sharing scheme for n participants, based on the Goldwasser-Goldrich-Halevi cryptosystem, is proposed.
Abstract This paper proposed a quantum secret sharing scheme over lattice-based threshold encryption: it integrates resplittable threshold public key encryption (RTPKE) and utilizes a lattice-based private key share transmission mechanism to enhance the security of the share distribution process. By leveraging the dynamic key management features of RTPKE, the scheme can flexibly adjust key distribution as required when participants change. During the quantum secret reconstruction process, the decoy particles do not need to be detected by eavesdroppers when transmitted between participants. Instead, this task is performed by a semi-trusted third party (TP), reducing resource consumption and communication overhead. The decoy particles are not chosen from a fixed quantum bases but are instead in completely random quantum bases, making it impossible for an eavesdropper to obtain the correct result through conventional measurements. The probability of the eavesdropper escaping detection after measuring any decoy particle is nearly zero. After each participant performs a unitary operation, they randomly shuffle the sequence of decoy and information particles. When the particles are transmitted back to the TP, the TP uses the sequence information recorded by the participants to restore the original order and extract the information particles, effectively defending against forgery-replay attacks and potential threats from dishonest participants.
As the most popular cryptographic scheme in the post-quantum field, lattices have received extensive attention and research. Not only do they provide quantum-resistant security, they also enable the construction of complex applications. Currently, lattice cryptography schemes based on different difficult problems have been applied in different fields. The threshold secret sharing (TSS) scheme is an important field of cryptography and has important application value and development prospects in key protection, secure multi-party computation, privacy protection, etc. However, with the rapid development of quantum computing, many existing cryptography-underlying technologies are facing huge difficulties and challenges. Therefore, post-quantum TSS has important research significance and value for the future development of cryptography. In this paper, we summarize the existing secret sharing schemes based on lattice-hard problems and the relevant applications of these schemes in the post-quantum realm. We classify existing lattice-based TSS according to different functions and introduce typical solutions. To the best of our knowledge, this is the first review paper on lattice-based TSS schemes.
In this paper, we propose a threshold increasing algorithm for a (t; n) latticebased Threshold Multi-Stage Secret Sharing (TMSSS) scheme. To realize the changeability feature, we use the zero addition protocol to construct a new (t0; n) TMSSS scheme. Therefore, the new scheme enjoys the signi cant feature of threshold changeability along with the inherited features of being multi-stage, multi-use, and veri able derived fromour previously proposed lattice-based TMSSS scheme. Furthermore, we use the improved TMSSS scheme to propose a threshold decryption algorithm for the Learning With Error (LWE) based public key encryption scheme based on the study of Lindner and Peikert. For threshold decryption, each authorized subset of participants decrypts the ciphertext partially and sends the result to the combiner. The combiner can decrypt the ciphertext using the partial decryptions. The security of both schemes is based on hardness of lattice problems, i.e. LWE and Inhomogeneous Small Integer Solution (ISIS) problems, which are believed to resist against the quantum algorithms. The proposed schemes are eficient, especially on the participants' side, making them suitable for the applications in which the participants have limited processing capacities.