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A good range of problems on trees can be described by the following general setting: Given a bilinear map ∗ : R d × R d → R d and a vector s ∈ R d , we need to estimate the largest possible absolute value g(n) of an entry over all vectors obtained from applying n - 1 applications of ∗ to n instances of s. When the coefficients of ∗ are nonnegative and the entries of s are positive, the value g(n) is known to follow a growth rate λ = lim n → ∞ g (n) n . In this article, we prove that for such ∗ and s there exist nonnegative numbers r , r ′ and positive numbers a , a ′ so that for every n, a n - r λ n ≤ g (n) ≤ a ′ n r ′ λ n. While proving the upper bound, we actually also provide another approach in proving the limit λ itself. The lower bound is proved by showing a certain form of submultiplicativity for g(n). Corollaries include a lower bound and an upper bound for λ , which are followed by a good estimation of λ when we have the value of g(n) for an n large enough. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) is a new technology that has drawn interest from both industry and academia in the modern digital era. The IoT provides a platform for any device to connect and share data over the Internet. Because of the diverse IoT environment and resource-constrained IoT devices, this nascent technology has numerous security challenges. Due to the lack of fine-grained access control and flexibility, traditional cryptographic algorithms such as private key encryption and public-key encryption are not suitable for providing access control mechanisms. ABE (Attribute-Based Encryption) is a new cryptographic approach that not only provides security but also fine-grained access control. This paper provides a comprehensive survey of ABE and its various forms, such as key policy ABE and ciphertext policy ABE, based on access structure, multi-authority, and hierarchy features. Further access structures used in ABE and applications of the ABE technique in IoT are also discussed. A comparison of different ABE schemes is presented regarding features, efficiency, and security. In addition, numerous schemes are analyzed to uncover obstacles, issues, and gaps that need to be investigated further. [ABSTRACT FROM AUTHOR]

Aiming at the problem that current schemes mostly let the latter signer to verify the multiple signatures of the previous signer, and the message is signed and then passed to the next signer, leading to the efficiency problem of long overall signature time.A sequential aggregate signature scheme based on bilinear pairs was proposed.The aggregate signatures of documents were generated by multiple users in a certain order, and only the final signature was needed to be verified to confirm the correct order of signatures and the legitimacy of multiple user signatures.The complexity of verifying the multi-user sequential signature was effectively reduced and the offline verification of the authenticity of signature was realized when the user was offline or in a delay-tolerant network with limited node caching capacity and network resources.It is shown that the proposed scheme is existential unforgeability against chosen-message attacks under adversary adaptive selection messages in the random oracle model.

Smart homes are a special use-case of the IoT paradigm, which is becoming more and more important in our lives. Although sensors, devices and applications make our daily lives easier, they often collect our sensitive data, which may lead to security problems (e.g., hacked devices, botnets, etc.). In several cases, the appropriate security mechanisms are missing within the devices. Therefore, security measures have become a central topic in the field of IoT. The most essential requirements are secure user–device authentication and confidentiality of transferred sensitive data. Passwords are the most widely used factors in various areas, such as user authentication, key establishment, and also secret sharing. Password-based protocols that are resistant to typical threats, such as offline dictionary, man-in-the-middle and phishing attacks, generate new session keys. The major aim of these solutions is to guarantee high-level security, even if a user applies a single low-entropy human memorable password for all their accounts. We introduce a threshold and password-based, distributed, mutual authenticated key agreement with key confirmation protocol for a smart home environment. The proposed protocol is a scalable and robust scheme, which forces the adversary to corrupt l - 1 smart home devices, where l is the threshold, in order to perform an offline dictionary attack. The protocol is designed to achieve password-only setting, and end-to-end security if the chosen IoT devices are also authenticated besides the user. We also provide a security analysis of the protocol in AVISPA. We apply the on-the-fly model checker and the constraint-logic-based attack searcher to perform protocol verification for bounded numbers of sessions. We show that the proposed protocol provides session key secrecy and mutual authentication of the user and the device manager. Since efficiency is a crucial aspect, we implemented our protocol to measure the computation and communication costs and demonstrate that our solution is appropriate and eligible for smart homes. [ABSTRACT FROM AUTHOR]

Let A be an algebra over a field F with char(F) 1= 2. If A is generated as an algebra by [[A,A], [A,A]], then for every skew-symmetric bilinear map F: A×A X, where X is an arbitrary vector space over F, the condition that F(x2, x) = 0 for all x A implies that F(xy, z) + F(zx, y) + F(yz, x) = 0 for all x, y, z A. This is applicable to the question of whether A is zero Lie product determined and is also used in proving that a Jordan homomorphism from A onto a semiprime algebra B is the sum of a homomorphism and an antihomomorphism. [ABSTRACT FROM AUTHOR]

Elliptic curve is a set of two variable points on polynomials of degree 3 over a field acted by an addition operation that forms a group structure. The motivation of this study is the mathematics behind that elliptic curve to the applicability within a cryptosystem. Nowadays, pairings bilinear maps on elliptic curves are popular to construct cryptographic protocol pairings that help to transform a discrete logarithm problem on an elliptic curve into a discrete logarithm problem in finite fields. The purpose of this paper is to introduce elliptic curves and bilinear pairings on elliptic curves based on pairing cryptography. Also, this investigation serves as a basis for guiding anyone interested to understand one of the applications of group theory in cryptosystems.

Abstract With increasing application of cloud computing and big data technologies, a large amount of personal information is stored on the Internet, which raises the issue of privacy leakage. To protect people’s data privacy, this paper firstly presents a new anonymous Identify-Based Encryption (IBE) scheme and gives the proof of its security under the Bilinear Diffie–Hellman Security Assumption. Then, by introducing the anonymous IBE scheme into anonymous communication fields, this paper introduces a new lightweight anonymous communication model for cloud computing, which guarantees the anonymity of system users and the security of messages in small groups. Our analysis shows that, the proposed communication model cannot only reduce memory consumption and improve message transmission efficiency, but also effectively resist traffic-analysis attacks, node eavesdropping, and finally achieve secure anonymous communication in cloud computing.

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Compact E-cash is the first scheme which can withdraw 2 coins within (1) operations and then store them in() bits. Because of its high efficiency, a lot of research has been carried out on its basis, but no previous research pay attention to the privacy of payees and in some cases, payees have the same privacy requirement as payers. We propose a double-blind compact E-cash scheme, which means that both the payer and the payee can keep anonymous while spending. In our scheme, the payer and the bank cannot determine whether the payees of two different transactions are the same one and connect the payee with transactions related to him, in this way, the privacy of the payee is protected. And our protocols disconnect the received coin from previous transaction, then, the coin can be transferred into an unspent coin which belongs to the payee. The proposed e-cash scheme is secure within y-DDHI and LRSW assumption. [ABSTRACT FROM AUTHOR]

Pairings are particular bilinear maps, and they have been defined based on elliptic curves whichare abelian groups. In cryptography and security problems use these pairings. Mrabet et al. proposedpairings from a tensor product of groups in 2013. Also Mahalanobis et al. proposed bilinear cryptographyusing groups of nilpotency class two in 2017. In this paper, I develop a novel idea of a bilinear cryptosystemusing Lie algebras from p-groups. First the researcher proposes pairing on Lie algebras from elliptic curves,and then pairings that can be constructed on Lie algebras from some of the non-abelian p-groups.

In the big data background, data privacy becomes more and more important when data leakage and other security events occur more frequently. As one of the key means of privacy protection, anonymous communication attracts large attention. Aiming at the problems such as low efficiency of message forwarding, high communication delay and abusing of anonymity, this paper presents an identity-based traceable anonymous communication model by adding a preprocessing phase, modifying the ciphertext structure and increasing the controllability of anonymity. Firstly, a new identity-based signature algorithm is proposed, and its security is proved via existential unforgeability against chosen-message attacks (EU-CMA). The signature algorithm is further applied to the anonymous communication model to implement the controllability of revocable anonymity. Secondly, by adding a preprocessing Setup phase, the operations of identifications distribution and user authentication are launched before the anonymous communication phase starts, and this practice significantly improves the efficiency of the anonymous communication model. Finally, by adding the hash value of the message and the user identification as the message authentication code, we design a new ciphertext structure, which can efficiently guarantee the integrity of the ciphertext. Performance analysis and simulation results show that the proposed anonymous communication model has high message forwarding efficiency and better security and controllability of anonymity. [ABSTRACT FROM AUTHOR]

A triple receiver public key cryptosystem was proposed.In the cryptosystem,a sender encrypted a message and sent to three receivers,while the three receivers were able to decrypt the message with their own private keys.Based on bilinear map,two triple receiver public key encryption schemes with different security were constructed.If the gap bilinear Diffie-Hellman (GBDH) problem and the computational Diffie-Hellman (CDH) problem were proved formally to be intractable,then the two schemes proposed were semantically secure against chosen-plaintext attacks and against adaptive chosen ciphertext attacks respectively.The proposed scheme only added an exponential operation and a hash operation,and constructed three independent receivers which had a high efficiency.Analyses show that proposed scheme can improve the security of TLS protocol and apply to hierarchical public key cryptosystems.

A triple receiver public key cryptosystem was proposed. In the cryptosystem, a sender encrypted a message and sent to three receivers, while the three receivers were able to decrypt the message with their own private keys. Based on bilinear map, two triple receiver public key encryption schemes with different security were constructed. If the gap bilinear Diffie-Hellman (GBDH) problem and the computational Diffie-Hellman (CDH) problem were proved formally to be intractable, then the two schemes proposed were semantically secure against chosen-plaintext attacks and against adaptive chosen ciphertext attacks respectively. The proposed scheme only added an exponential operation and a hash operation, and constructed three independent receivers which had a high efficiency. Analyses show that proposed scheme can improve the security of TLS protocol and apply to hierarchical public key cryptosystems.

Introduction Let , and be Banach spaces and be a bilinear mapping. In 1951 Arens found two extension for as and from into . The mapping is the unique extension of such that from into is continuous for every , but the mapping is not in general continuous from into unless . Thus for all the mapping is continuous if and only if is Arens regular. Regarding as a Banach , the operation extends to and defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space becomes a Banach algebra. Material and methods The constructions of the two Arens multiplications in lead us to definition of topological centers for with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive, -algebra is Arens regular. In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras. Results and discussion We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity. Conclusion The following conclusions were drawn from this research. if and only if the bilinear mapping is Arens regular. A bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact. if and only if the bilinear mapping is Arens regular. Assume that has approximate identity. Then is Arens regular if and only if is reflexive../files/site1/files/62/9Abstract.pdf

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:X\times Y\to Z$, depended on a natural number $n$ and a cardinal number $\kappa$; which is called $n$-factorization property of level $\kappa$. Then we study the relation between $n$-factorization property of level $\kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~\cite{neu1} and ~\cite{neu}. Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to the Ghahramani-Lau conjecture is raised.

We study the sampling design with fixed sample size from a geometric point of view. The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. It is possible to study them inside of linear spaces provided with a quadratic and linear metric. We define particular random quantities whose logically possible values are all logically possible samples of a given size. In particular, we define random quantities which are complementary to the Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations. We use the α-criterion of concordance introduced by Gini in order to identify it. We innovatively apply to probability this statistical criterion.

Smart meters have been widely applied in the smart grid, and they enable two-way communication in the smart grid. User's electricity consumption data and other data are transmitted between the entities. It is necessary to ensure the security of this two-way communication. Several authentication schemes have been proposed to solve this problem. Recently, Mahmood et al. proposed an authentication scheme for the smart grid. However, we find that their scheme cannot provide the perfect forward secrecy and private key privacy as they have claimed. An improved version by Abbasinezhad is found to be subject to replay attack, too. In this paper, a bilinear map pairing-based authentication and key establish scheme is proposed, which can withstand the aforementioned attacks and achieves more security features, private key privacy, perfect forward privacy, and message integrity. We designed a simpler registration scheme, which implements the same functionalities, while the computation cost is reduced. We also conducted a formal security analysis of the proposed scheme, and the result shows that the proposed scheme is secure. Our simulation results show that the proposed scheme has a comparable communication cost and computation cost.

With increasing application of cloud computing and big data technologies, a large amount of personal information is stored on the Internet, which raises the issue of privacy leakage. To protect people's data privacy, this paper firstly presents a new anonymous Identify-Based Encryption (IBE) scheme and gives the proof of its security under the Bilinear Diffie–Hellman Security Assumption. Then, by introducing the anonymous IBE scheme into anonymous communication fields, this paper introduces a new lightweight anonymous communication model for cloud computing, which guarantees the anonymity of system users and the security of messages in small groups. Our analysis shows that, the proposed communication model cannot only reduce memory consumption and improve message transmission efficiency, but also effectively resist traffic-analysis attacks, node eavesdropping, and finally achieve secure anonymous communication in cloud computing. [ABSTRACT FROM AUTHOR]

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulas for evaluating bilinear maps over finite fields, such as Strassen or Karatsuba formulas. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulas. We apply in particular our approach to the short product modulo X 5 and the circulant product modulo (X 5 − 1). Moreover, we are able to prove that there is essentially only one optimal decomposition of the product of 3 × 2 by 2 × 3 matrices up to the action of some group of automorphisms. [ABSTRACT FROM AUTHOR]

Let V be a linear space of arbitrary dimension and over an arbitrary base field F , endowed with a bilinear map f : V × V → V. A basis B = { v i } i ∈ I of V is an f -basis if for any i , j ∈ I we have that f (v i , v j) ∈ F v k for some k ∈ I. We associate to any triplet (V , f , B) an adequate graph (V , E). By arguing on this graph we show that V decomposes as a direct sum of strongly f -invariant linear subspaces, each one being associated to one connected component of (V , E). Also the B -semisimplicity and the B -simplicity of V are characterized in terms of the associated graph. [ABSTRACT FROM AUTHOR]

Mobile wallet is a very convenient means of mobile payment to allow the clients to conduct the payment via their mobile devices. To reduce the computation burden of resourcesconstraint mobile devices, a few mobile wallet protocols with outsourced verification in cloud computing were proposed. But in some of the protocols, there exist the risk of a colluding attack of the customer and the untrusted cloud server. In this paper, we propose an improved protocol, in which the payment information is protected by Hash function and random number. The malicious customer and cloud server cannot change the payment information to conduct a collusion forgery attack to defraud the merchant. The security analysis indicates that the proposed improved protocol can enhance the security in terms of correctness, unforgeability and traceability without increasing the computational burden. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra such that its (2n)th dual for some (n = 1) with first Arens product is m-weakly amenable for some (m > 2n). We introduce some conditions by which if m is odd [even], then A is weakly [2-weakly] amenable. [ABSTRACT FROM AUTHOR]