A lattice-based changeable threshold multi-secret sharing scheme and its application to threshold cryptography

Currently, the security situation of data security and user privacy protection is increasingly serious in cloud environment. Ciphertext data storage can prevent the risk of user's privacy disclosure. But how to search keyword on ciphertext data without revealing keyword information becomes a new challenge. Searchable encryption (SE) is put forward for this reason, which can be used to realize ciphertext-search directly. In terms of multi-user data sharing, public-key encryption with keywords search (PEKS) is more widely used than symmetric searchable encryption. PEKS has been widely studied and developed by researchers in recent years. However, the existing PEKS scheme often lacks flexible access policy. Therefore, combining the advantage of policy control on attribute-based encryption (ABE), and as that the bilinear pairing related assumption is fragile in post-quantum era, lattice-based cryptography is considered as one of secure encryption technology against quantum attack. With this background, in this paper, we give a keyword-searchable ABE scheme based on the hardness of lattice problems, our scheme supports flexible attribute control policy by integrating ABE and PEKS, and the security of new scheme is proved under the learning with errors (LWE) assumption. As lattice-based cryptographic technology is currently thought to be resistant to quantum attacks, so the new scheme has stronger security in a quantum era.

SummaryThrough the Industrial Internet of Things (IIoT), the manufacturing enterprises have significantly enhanced the production efficiency. The transmission of data in IIoT over public channels has brought about information leakage issues. Therefore, it is crucial to agree on a session key among the participants. In recent years, numerous authenticated key exchange (AKE) schemes have been designed. However, those AKE schemes which adopt the certificateless approach to tackle certificate management and key escrow may remain vulnerable to quantum attacks, and other AKE schemes which are based on lattice hard problem so as to achieve the post quantum security may incur key management issues. To simultaneously address the aforementioned challenges, we propose a lightweight certificateless lattice‐based authenticated key exchange (C2LAKE) scheme. Upon the hardness assumptions of the inhomogeneous small integer solution (ISIS) and computational bilateral inhomogeneous small integer solution (CBi‐ISIS) problems, the C2LAKE scheme has been demonstrated to be secure in the eCK model and the random oracle model (ROM). Better trade‐off among security and functionality features, and communication and computation costs make the C2LAKE scheme suitable and applicable in the IIoT environment.

The proxy re-encryption allows an intermediate proxy to convert a ciphertext for Alice into a ciphertext for Bob without seeing the original message and leaking out relevant information. Unlike many prior identity based proxy re-encryption schemes which are based on the number theoretic assumptions such as large integer factorization and discrete logarithm problem. In this paper, we first propose a novel identity based proxy re-encryption scheme which is based on the hardness of standard Learning With Error(LWE) problem and is CPA secure in the standard model. This scheme can be reduced to the worst-case lattice hard problem that is able to resist attacks from quantum algorithm. The key step in our construction is that the challenger how to answer the private query under a known trapdoor matrix. Our scheme enjoys properties of the non-interactivity, unidirectionality, anonymous and so on. In this paper, we utilize primitives include G-trapdoor for lattice and sample algorithms to realize simple and efficient re-encryption.

A public key encryption scheme based on a new variant of LWE with small cipher size

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most 2^{O(sqrt{n})} nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

In this work we investigate the problem of private statistical analysis of time-series data in the distributed and semi-honest setting. In particular, we study some properties of Private Stream Aggregation (PSA), first introduced by Shi et al. 2011. This is a computationally secure protocol for the collection and aggregation of data in a distributed network and has a very small communication cost. In the non-adaptive query model, a secure PSA scheme can be built upon any key-homomorphic weak pseudo-random function as shown by Valovich 2017, yielding security guarantees in the standard model which is in contrast to Shi et al. We show that every mechanism which preserves \((\epsilon ,\delta )\)-differential privacy in effect preserves computational \((\epsilon ,\delta )\)-differential privacy when it is executed through a secure PSA scheme. Furthermore, we introduce a novel perturbation mechanism based on the symmetric Skellam distribution that is suited for preserving differential privacy in the distributed setting, and find that its performances in terms of privacy and accuracy are comparable to those of previous solutions. On the other hand, we leverage its specific properties to construct a computationally efficient prospective post-quantum protocol for differentially private time-series data analysis in the distributed model. The security of this protocol is based on the hardness of a new variant of the Decisional Learning with Errors (DLWE) problem. In this variant the errors are taken from the symmetric Skellam distribution. We show that this new variant is hard based on the hardness of the standard Learning with Errors (LWE) problem where the errors are taken from the discrete Gaussian distribution. Thus, we provide a variant of the LWE problem that is hard based on conjecturally hard lattice problems and uses a discrete error distribution that is similar to the continuous Gaussian distribution in that it is closed under convolution. A consequent feature of the constructed prospective post-quantum protocol is the use of the same noise for security and for differential privacy.

Tightly secure signature schemes from the LWE and subset sum assumptions

In security proofs of lattice based cryptography, to bound the closeness of two probability distributions is an important procedure. To measure the closeness, the Rényi divergence has been used instead of the classical statistical distance. Recent results have shown that the Rényi divergence offers security reductions with better parameters, e.g. smaller deviations for discrete Gaussian distributions. However, since previous analyses used a fixed order Rényi divergence, i.e., order two, they lost tightness of reductions. To overcome the deficiency, we adaptively optimize the orders based on the advantages of the adversary for several lattice-based schemes. The optimizations enable us to prove the security with both improved efficiency and tighter reductions. Indeed, our analysis offers security reductions with smaller parameters than the statistical distance based analysis and the reductions are tighter than that of previous Rényi divergence based analysis. As applications, we show tighter security reductions for sampling discrete Gaussian distributions with smaller precomputed tables for BLISS signatures, and variants of learning with errors (LWE) problem and small integer solution (SIS) problem called k-LWE and k-SIS.KeywordsLattice based cryptographyTight reductionRényi divergenceSampling discrete GaussianBLISSLWESIS

A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of partecipants can recover the secret, but any non-qualified subset has absolutely no information on the secret. The set of all qualified subsets defines the access structure to the secret. Sharing schemes are useful in the management of cryptographic keys and in multy-party secure protocols.We analyze the relationships among the entropies of the sample spaces from which the shares and the secret are chosen. We show that there are access structures with 4 participants for which any secret sharing scheme must give to a participant a share at least 50% greater than the secret size. This is the first proof that there exist access structures for which the best achievable information rate (i.e., the ratio between the size of the secret and that of the largest share) is bounded away from 1. The bound is the best possible, as we construct a secret sharing scheme for the above access structures which meets the bound with equality.

We introduce Attribute-based Proxy Re-Signatures (ABPRS), an extension of Proxy Re-Signatures (PRS), which allows a semi-trusted proxy to transform a signature of one entity into a signature of another, without revealing any signing key and information about the signer. The proposed primitive combines features of PRS and Attribute-based Signatures (ABS), which is useful for many applications (we show how to apply ABPRS to three such applications). We propose a unidirectional ABPRS construction based on the hardness of the Inhomogeneous Small Integer Solution (ISIS) problem in standard lattices. We show that the proposed unidirectional ABPRS construction is multi-hop, where the transformation can be performed multiple times on any signature. Furthermore, we prove that the proposed unidirectional ABPRS construction satisfies the security notion of no-signature-query unforgeability and (perfect) privacy, under the ISIS assumption. Finally, we show how to obtain a multi-hop unidirectional ABPRS scheme that satisfies a stronger security notion of co-selective unforgeability by using a generic transformation technique.

Modern cryptography pushed forward the need of having provable security. Whereas ancient cryptography was only relying on heuristic assumptions and the secrecy of the designs, nowadays researchers try to make the security of schemes to rely on mathematical problems which are believed hard to solve. When doing these proofs, the capabilities of potential adversaries are modeled formally. For instance, the black-box model assumes that an adversary does not learn anything from the inner-state of a construction. While this assumption makes sense in some practical scenarios, it was shown that one can sometimes learn some information by other means, e.g., by timing how long the computation take. In this thesis, we focus on two different areas of cryptography. In both parts, we take first a theoretical point of view to obtain a result. We try then to adapt our results so that they are easily usable for implementers and for researchers working in practical cryptography. In the first part of this thesis, we take a look at post-quantum cryptography, i.e., at cryptographic primitives that are believed secure even in the case (reasonably big) quantum computers are built. We introduce HELEN, a new public-key cryptosystem based on the hardness of the learning from parity with noise problem (LPN). To make our results more concrete, we suggest some practical instances which make the system easily implementable. As stated above, the design of cryptographic primitives usually relies on some well-studied hard problems. However, to suggest concrete parameters for these primitives, one needs to know the precise complexity of algorithms solving the underlying hard problem. In this thesis, we focus on two recent hard-problems that became very popular in post-quantum cryptography: the learning with error (LWE) and the learning with rounding problem (LWR). We introduce a new algorithm that solves both problems and provide a careful complexity analysis so that these problems can be used to construct practical cryptographic primitives. In the second part, we look at leakage-resilient cryptography which studies adversaries able to get some side-channel information from a cryptographic primitive. In the past, two main disjoint models were considered. The first one, the threshold probing model, assumes that the adversary can put a limited number of probes in a circuit. He then learns all the values going through these probes. This model was used mostly by theoreticians as it allows very elegant and convenient proofs. The second model, the noisy-leakage model, assumes that every component of the circuit leaks but that the observed signal is noisy. Typically, some Gaussian noise is added to it. According to experiments, this model depicts closely the real behaviour of circuits. Hence, this model is cherished by the practical cryptographic community. In this thesis, we show that making a proof in the first model implies a proof in the second model which unifies the two models and reconciles both communities. We then look at this result with a more practical point-of-view. We show how it can help in the process of evaluating the security of a chip based solely on the more standard mutual information metric.

In order to improve the security of Compressed Sensing, this paper put forward a new idea-Compressed Sensing Public Key Encryption (CSPKE), realized a concrete CSPKE algorithm by combing Lattice-based Cryptography with Compressed Sensing, and proved its security of indistinguishability against chosen plaintext attack (IND-CPA) on standard model based on Learning With Error (LWE) assumptions. By using the matrix characteristics of LWE, this algorithm preserved the linear structure of CS. Efficiency analysis shows that it has acceptable increment on cost comparing to CS. It can be used in many fields, such as protecting security of data acquisition process based on CS.

In this paper, we propose a new hard problem, called bilateral inhomogeneous small integer solution (Bi-ISIS), which can be seen as an extension of the small integer solution problem on lattices. The main idea is that, instead of choosing a rectangle matrix, we choose a square matrix with small rank to generate Bi-ISIS problem without affecting the hardness of the underlying SIS problem. Based on this new problem, we present two new hardness problems: computational Bi-ISIS and decisional problems. As a direct application of these problems, we construct a new lattice-based key exchange (KE) protocol, which is analogous to the classic Diffie- Hellman KE protocol. We prove the security of this protocol and show that it provides better security in case of worst-case hardness of lattice problems, relatively efficient implementations, and great simplicity.

The Learning-With-Errors (LWE) problem (and its variants including Ring-LWE and Module-LWE), whose security are based on hard ideal lattice problems, has proven to be a promising primitive with diverse applications in cryptography. For the sake of expanding sources for constructing LWE, we study the LWE problem on group rings in this work. One can regard the Ring-LWE on cyclotomic integers as a special case when the underlying group is cyclic, while our proposal utilizes non-commutative groups. In particular, we show how to build public key encryption schemes from dihedral group rings, while maintaining the efficiency of the Ring-LWE. We prove that the PKC system is semantically secure, by providing a reduction from the SIVP problem of group ring ideal lattice to the decisional group ring LWE problem. It turns out that irreducible representations of groups play important roles here. We believe that the introduction of the representation view point enriches the tool set for studying the Ring-LWE problem.

The server-aided revocable attribute-based encryption (SR-ABE) scheme was developed to improve scalability and minimize user computational overhead. This approach utilizes an untrusted, computationally potent server to execute key conversion or partial decryption operations while safeguarding users' keys. While Dong et al.'s learning with errors (LWE)based SR-ABE scheme offers theoretical efficiency gains and supports arbitrary Boolean circuits, its key generation and server authorization logic present substantial security vulnerabilities. A thorough analysis confirms that these flaws severely compromise system stability, pinpointing a critical defect in their scheme's identity and access policy design. To rectify this, we developed an optimized solution which also achieves decryption key exposure resistance (DKER).