Bilinear Maps Research Articles

Zero product preserving bilinear operators acting in sequence spaces

Consider a couple of sequence spaces and a product function $-$ a canonical bilinear map associated to the pointwise product $-$ acting in it. We analyze the class of "zero product preserving" bilinear operators associated with this product, that are defined as the ones that are zero valued in the couples in which the product equals zero. The bilinear operators belonging to this class have been studied already in the context of Banach algebras, and allow a characterization in terms of factorizations through $\ell^r(\mathbb{N})$ spaces. Using this, we show the main properties of these maps such as compactness and summability.

A verifiable ranked ciphertext retrieval scheme based on bilinear mapping

SummaryIt is prevalent nowadays for data owners to outsource their data to the cloud. Most of the traditional searchable schemes are proposed under the honest and curious model, lacking verification of integrity and correctness of the retrieval results. Because the cloud server is not completely trustworthy and there may be improper execution of the users' retrieval requests, it is necessary to verify the retrieval results returned by the cloud server. In order to address these issues, we propose a verifiable ranked ciphertext retrieval scheme (VRCRS) based on bilinear mapping. For the purpose of ranking the keywords and improving the security of ciphertext retrieval scheme, we utilize the lucene search engine toolkit to score keywords, and improve the traditional inverted index structure for building a secure inverted index structure. By adding dummy words into the dictionary of keywords, the proposed scheme can resist the keywords statistics attack of malicious server. To ensure the correctness and integrity of returned ciphertext, we apply bilinear mapping to generate validation tags for keywords. The theoretical analysis and experimental results show that the proposed scheme is secure and efficient. VRCRS can actually identify the illegal behavior of cloud server such as tampering and forgery.

A ciphertext-policy Attribute based encryption scheme for wireless body area networks based on ECC

The Internet of Things (IoT) based healthcare system is one of the prominent approaches to overcome the issue of the increasing burden of healthcare costs in India. Wireless Body Area Networks (WBANs) are the key enabler in this scenario. It provides continuous monitoring of the patient’s health condition remotely, by making the real-time health data of the patient available to healthcare professionals through cloud/public channel. On the other side, the practical implementation of this service is not possible if the issue of data security is not considered. Thus, in this paper, we have proposed a secure framework for WBAN using Elliptic Curve Cryptography based Ciphertext-Policy Attribute Based Encryption (CPABE) without bilinear pairing operations. The proposed CPABE is secured under Elliptic Curve Decisional Diffie-Hellman assumption and also has a feature of user/attribute revocation. We have evaluated the lightweight feature of the proposed CPABE by comparing it with other existing ABE schemes for WBAN. The result shows that our scheme outperforms the existing schemes for WBAN in terms of keys-ciphertext size and computation overhead.

Low-Rank Pairwise Alignment Bilinear Network For Few-Shot Fine-Grained Image Classification

Deep neural networks have demonstrated advanced abilities on various visual classification tasks, which heavily rely on the large-scale training samples with annotated ground-truth. However, it is unrealistic always to require such annotation in real-world applications. Recently, Few-Shot learning (FS), as an attempt to address the shortage of training samples, has made significant progress in generic classification tasks. Nonetheless, it is still challenging for current FS models to distinguish the subtle differences between fine-grained categories given limited training data. To filling the classification gap, in this paper, we address the Few-Shot Fine-Grained (FSFG) classification problem, which focuses on tackling the fine-grained classification under the challenging few-shot learning setting. A novel low-rank pairwise bilinear pooling operation is proposed to capture the nuanced differences between the support and query images for learning an effective distance metric. Moreover, a feature alignment layer is designed to match the support image features with query ones before the comparison. We name the proposed model Low-Rank Pairwise Alignment Bilinear Network (LRPABN), which is trained in an end-to-end fashion. Comprehensive experimental results on four widely used fine-grained classification datasets demonstrate that our LRPABN model achieves the superior performances compared to state-of-the-art methods.

Second-Order Pooling for Graph Neural Networks.

Graph neural networks have achieved great success in learning node representations for graph tasks such as node classification and link prediction. Graph representation learning requires graph pooling to obtain graph representations from node representations. It is challenging to develop graph pooling methods due to the variable sizes and isomorphic structures of graphs. In this work, we propose to use second-order pooling as graph pooling, which naturally solves the above challenges. In addition, compared to existing graph pooling methods, second-order pooling is able to use information from all nodes and collect second-order statistics, making it more powerful. We show that direct use of second-order pooling with graph neural networks leads to practical problems. To overcome these problems, we propose two novel global graph pooling methods based on second-order pooling; namely, bilinear mapping and attentional second-order pooling. In addition, we extend attentional second-order pooling to hierarchical graph pooling for more flexible use in GNNs. We perform thorough experiments on graph classification tasks to demonstrate the effectiveness and superiority of our proposed methods. Experimental results show that our methods improve the performance significantly and consistently.

A Quadratic and Linear Metric Characterizing the Sampling Design with Fixed Sample Size Considered From a Geometric Viewpoint

The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. We innovatively define univariate and bivariate random quantities whose logically possible values are samples of a given size in order to obtain the first-order and second-order inclusion probabilities by means of their coherent previsions. We consider linear maps connected with univariate random quantities as well as bilinear maps connected with bivariate random quantities. The covariance of two univariate random quantities that are the components of a bivariate random quantity has been expressed by means of two bilinear maps. We show that a univariate random quantity denoted by S is complementary to the univariate Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations that we innovatively define. We use the α-criterion of concordance introduced byGini in order to identify it. It is a statistical criterion that we innovatively apply to probability.

Bilinear operator multipliers into the trace class

Given Hilbert spaces H1,H2,H3, we consider bilinear maps defined on the cartesian product S2(H2,H3)×S2(H1,H2) of spaces of Hilbert-Schmidt operators and valued in either the space B(H1,H3) of bounded operators, or in the space S1(H1,H3) of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras Mi⊂B(Hi), i=1,2,3, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→B(H1,H3) by the membership of a natural symbol of u to the von Neumann algebra tensor product M1⊗‾M2op⊗‾M3. In the case when M2 is injective, we characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→S1(H1,H3) by a weak factorization property, which extends to the bilinear setting a famous description of bimodule linear mappings going back to Haagerup, Effros-Kishimoto, Smith and Blecher-Smith. We make crucial use of a theorem of Sinclair-Smith on completely bounded bilinear maps valued in an injective von Neumann algebra, and provide a new proof of it, based on Hilbert C⁎-modules.

Hybrid conditional privacy-preserving authentication scheme for VANETs

In existing authentication schemes for vehicular ad hoc networks (VANETs), the public key infrastructure (PKI)-based pseudonym certificate, identity-based encryption and group signature technology are usually used to preserve the security and privacy. However, these schemes face some challenges, e.g., the time-consuming certificate revocation list (CRL) checking, identity revocation issue and the computation overhead of group signature, respectively. To cope with these challenges, we propose a hybrid conditional privacy-preserving authentication protocol based on the PKI certificate and identity-based signature. In our scheme, the trust authority (TA) assigns the unique long-term certificate for every registered node. Only the vehicle with valid certificate can apply the anonymous short term identity from the current RSU to sign safety-related message. The identity-based signature avoids the CRL checking and the complex bilinear paring operation. When vehicle is compromised, TA can easily revoke its identity by the only long-term certificate. To further enhance efficiency, vehicle can verify the received messages by the single or batch authentication manner. In addition, we optimize the authentication process to avoid vehicle verifying the repeated and unnecessary signatures. Compared with the current schemes, the simulation result shows that our authentication protocol can effectively reduce verified messages in the case of meeting the requirement of security and privacy.

Outsourced ciphertext-policy attribute-based encryption with partial policy hidden

Attribute-based encryption is an efficient and flexible fine-grained access control scheme. However, how to realize the attribute privacy concerns in the access policy and optimize the heavy computing overhead have been not adequately addressed. First, in view of the open-access policies formulated by data owners in the cloud environment and the linear growth of bilinear pairing operations with the number of attributes in the decryption process, a verifiable outsourced attribute-based encryption with partial policy hidden scheme is proposed, in which the attribute name of access policy can be sent while attribute value involving sensitive information can be hidden, so nobody can infer information from the access policy. Second, the bilinear pairing operation and modular power operation are outsourced to the cloud server, then users only need to perform constant exponential operation to decrypt. In addition, the proposed scheme is based on the composite order bilinear group and satisfies full secure under the standard model. Finally, compared with other schemes in term of function and performance, it shows that this scheme is more efficient and suitable for resource-constrained mobile devices in outsourcing environment.

Unbounded inner product functional encryption from bilinear maps

Inner product functional encryption (IPFE) is one class of functional encryption supporting only inner product functionality. All previous IPFE schemes are bounded schemes, meaning that the vector length that can be handled in the scheme is fixed in the setup phase. In this paper, we propose the first unbounded IPFE schemes, in which we do not have to fix the lengths of vectors in the setup phase and can handle (a priori) unbounded polynomial lengths of vectors. Our first scheme is private-key based and fully function hiding. That is, secret keys hide the information of the associated function. Our second scheme is public-key based and provides adaptive security in the indistinguishability based security definition. Both our schemes are based on SXDH, which is a well-studied standard assumption, and secure in the standard model. Furthermore, our schemes are quite efficient, incurring an efficiency loss by only a small constant factor from previous bounded function hiding schemes.

Functional encryption with application to machine learning: simple conversions from generic functions to quadratic functions

Functional encryption (FE) and predicate encryption (PE) can be utilized in deploying and executing machine learning (ML) algorithms to improve efficiency. However, most of existing FE and PE algorithms only consider generic functions. Actually, quadratic-functions-based FE and PE can be used to further reduce the computation costs significantly. In this paper, we present a functional encryption scheme for quadratic functions from those for generic functions. In our constructions, ciphertexts are associated with a pair of vectors $(\mathsf {x},\mathsf {y})\in \mathbb {Z}^{n}_{q}\times \mathbb {Z}^{m}_{q}$ , private keys are associated with a quadratic function, and the decryption of ciphertexts CT(x,y) with a private key skF, where F is a n × m-dimensional matrix, recovers $(\mathsf {x})^{\top }\mathsf {F}\mathsf {y}\in \mathbb {Z}_{q}$ . Compared with Baltico et al.’s FEs for quadratic functions (at Crypto 2017), our schemes could obtain almost the same ciphertexts size of $O((n+m)\log q)$ as their schemes (in contrast to O(n) in Baltico et al.’s schemes), and the computation for quadratic functions in our scheme does not rely on bilinear maps, while their schemes must rely on this assumption. In particular, our schemes under the standard assumptions achieve adaptive security, while Baltico et al.’s scheme only obtains selective security. Moreover, beyond the MDDH and GGM assumptions, our schemes allow for instantiations under standard assumptions such as LWE, LPN, and etc.

Multidomain security authentication for the Internet of things

SummaryWith the rapid development of Internet of Things (IoT) information technology, the IoT has become a key infrastructure for telemedicine, smart home, and intelligent transportation. One of the key technologies for these applications is information sharing and interoperation among multiple domains. However, the security and privacy issues of multidomain interaction face severe security challenges. Aiming at these problems, an certificateless multidomain authentication technology is proposed for IoT. Bilinear mapping and short signature technology are used to realize mutual authentication among entities in different domains, which protects secure data sharing and secure interoperability among domains. Certificateless multidomain authentication can avoid inherent security risks of key escrow in existing identity‐based authentication. And it also solves complex certificate management and network bottlenecks problems in traditional certificate‐based authentication. The proof and analysis show that the proposed scheme has good security and performance, it supports anonymous authentication among entities, and it is also suitable for large‐scale distributed network security alliance authentication mechanism.

Right Semi-Tensor Product for Matrices Over a Commutative Semiring

This paper generalizes the right semi-tensor product for real matrices to that for matrices over an arbitrary commutative semiring, and investigates its properties. This product is defined for any pair of matrices satisfying the matching-dimension condition. In particular, the usual matrix product and the scalar multiplication are its special cases. The right semi-tensor product turns out to be an associative bilinear map that is compatible with the transposition and the inversion. The product also satisfies certain identity-like properties and preserves some structural properties of matrices. We can convert between the right semi-tensor product of two matrices and the left semi-tensor product using commutation matrices. Moreover, certain vectorizations of the usual product of matrices can be written in terms of the right semi-tensor product.

Identity Based Data Sharing and Profile Matching using Probabilistic Key in Cloud

Cloud technology is popular since the utilizations and their information are expanding hugely step by step. It gives usefulness to overseeing data information in a conveyed and pervasive way supporting many platform. Data sharing and security problem are the principle issue to the increasing use of health care, since health information is delicate. However issues, for example, risks of privacy exposure ,versatility in key administration, adaptable access and effective uses denial always been the most significant difficulties for accomplishing good- grained cryptographically upheld information access control. Right now structure a protected information move and profile coordinating plan with probabilistic open key encryption in correspondence test to diminish the size of open key which helps in decreasing the capacity and furthermore time productive. Our probabilistic open key encryption plot when executed in a bilinear gathering, it can check whether two figure writings are scrambled of a similar message. Strangely, in encryption or decoding system bilinear mapping isn't required in Public Key Encryption conspire except if when individuals need to do in fairness test between two figure messages that might be produced utilizing diverse open keys. Additionally the profile coordinating component dependent on Probabilistic Public Key Encryption with fairness test( PPKEET) causes patient to discover companions in a security saving way with age of trapdoors and structure social connections as indicated by the desire. The security examination and trial results demonstrated that our plan is ensuring the information and giving right medicine in cloud.

Enhanced Security Identity-Based Privacy-Preserving Authentication Scheme Supporting Revocation for VANETs

Existing conditional privacy-preserving communication protocols based on a tamper-proof device (TPD) in vehicular ad hoc networks (VANETs) face the challenges of master key compromise and vehicle revocation. To address these challenges, we propose a new identity-based anonymous authentication scheme. In this scheme, the master key of the system will not be directly installed in the TPD; hence, it is impossible for an adversary to obtain the system master key from the TPD. To generate the private key of the vehicle, an additional secret is required, and this secret is provided by the roadside unit (RSU). Therefore, revoking a malicious vehicle from the VANETs is efficient, since the RSU only needs to stop generating the up-to-date secret for the vehicle. In addition, the signature verification algorithm of the scheme involves no bilinear pairing operations, making the verification process more efficient. The security of the scheme is proved to be equivalent to the elliptic curve discrete logarithm assumption in the random oracle. The experimental results show that the authentication of the scheme is more efficient, and the communication cost has been diminished.

Nonsingular bilinear maps revisited

AbstractA bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

An Efficient Certificateless Aggregate Signature Scheme for Blockchain-Based Medical Cyber Physical Systems.

Different from the traditional healthcare field, Medical Cyber Physical Systems (MCPS) rely more on wireless wearable devices and medical applications to provide better medical services. The secure storage and sharing of medical data are facing great challenges. Blockchain technology with decentralization, security, credibility and tamper-proof is an effective way to solve this problem. However, capacity limitation is one of the main reasons affecting the improvement of blockchain performance. Certificateless aggregation signature schemes can greatly tackle the difficulty of blockchain expansion. In this paper, we describe a two-layer system model in which medical records are stored off-blockchain and shared on-blockchain. Furthermore, a multi-trapdoor hash function is proposed. Based on the proposed multi-trapdoor hash function, we present a certificateless aggregate signature scheme for blockchain-based MCPS. The purpose is to realize the authentication of related medical staffs, medical equipment, and medical apps, ensure the integrity of medical records, and support the secure storage and sharing of medical information. The proposed scheme is highly computationally efficient because it does not use bilinear maps and exponential operations. Many certificateless aggregate signature schemes without bilinear maps in Internet of things (IoT) have been proposed in recent years, but they are not applied to the medical field, and they do not consider the security requirements of medical data. The proposed scheme in this paper has high computing and storage efficiency, while meeting the security requirements in MCPS.

Biprojectivity and Biflatness of Bi-amalgamated Banach Algebras

Let A, X be two Banach algebras and let X be an algebraic Banach A-module equipped with a bounded bilinear map $$\Theta :X\times X\rightarrow A$$ which is compatible with the A-module operations of X. Then the $$\ell ^1$$ -direct sum $$A\times X$$ endowed with the multiplication $$\begin{aligned} (a,x)(b,y)=(ab+\Theta (x,y),ay+xb+xy) \quad (a,b\in A, x, y\in X) \end{aligned}$$ is a Banach algebra, denoted by $$A\boxtimes _\Theta X$$ and will be called a bi-amalgamated Banach algebra. Many known Banach algebras such as (generalized) module extension Banach algebras, Lau product Banach algebras, generalized matrix Banach algebras have this general framework. The main aim of this paper is to investigate biprojectivity and biflatness of $$A\boxtimes _\Theta X$$ . Our results extend several results in the literature and provide simple direct proofs for some known results. In particular, we characterize the biprojectivity and biflatness of certain classes of the module extension Banach algebras and generalized matrix Banach algebras. Some unsolved questions are also included.

Multi-component generalizations of mKdV equation and nonassociative algebraic structures

Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg–de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie–Jordan algebras [Speciality of Lie–Jordan algebras, J. Algebra 237 (2001) 621–636].

Server-Aided Revocable Attribute-Based Encryption from Lattices

Attribute-based encryption (ABE) can support a fine-grained access control to encrypted data. When the user’s secret-key is compromised, the ABE system has to revoke its decryption privileges to prevent the leakage of encrypted data. Although there are many constructions about revocable ABE from bilinear maps, the situation with lattice-based constructions is less satisfactory, and a few efforts were made to close this gap. In this work, we propose the first lattice-based server-aided revocable attribute-based encryption (SR-ABE) scheme and thus the first such construction that is believed to be quantum resistant. In the standard model, our scheme is proved to be secure based on the hardness of the Learning With Errors (LWE) problem.