Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

An inner product space-based hierarchical key assignment/access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data controller implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data controller, the left-to-right and bottom-up policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. The parent class can obtain the keys of the child class, which means a one-way function, and the opposite direction is not allowed. Our scheme is collusion attack and privilege creep problem resistant, as well as key recovery and indistinguishability secure. The performance analysis shows that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our scheme over many others in the literature is that it needs only one operation for the derivation of the key of child classes.

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

An inner product space-based hierarchical key assignment/access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data controller implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data controller, the left-to-right and bottom-up policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. The parent class can obtain the keys of the child class, which means a one-way function, and the opposite direction is not allowed. Our scheme is collusion attack and privilege creep problem resistant, as well as key recovery and indistinguishability secure. The performance analysis shows that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our scheme over many others in the literature is that it needs only one operation for the derivation of the key of child classes.

Norms on Quotient Spaces of The 2-Inner Product Space

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

<div>An inner product space-based hierarchical access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data owner implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data owner, left-to-right and bottom-up (LRBU) policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. Our scheme is resistant to collusion attacks and privilege creep problems, as well as providing key recovery and key indistinguishability security. The performance analysis also shows us that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our key access scheme over many others in the literature is that it requires only one operation to derive the secret key of child classes securely and efficiently.</div>

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

<div>An inner product space-based hierarchical access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data owner implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data owner, left-to-right and bottom-up (LRBU) policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. Our scheme is resistant to collusion attacks and privilege creep problems, as well as providing key recovery and key indistinguishability security. The performance analysis also shows us that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our key access scheme over many others in the literature is that it requires only one operation to derive the secret key of child classes securely and efficiently.</div>

Some Properties of Fuzzy Inner Product Space

Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement of D.

A Quasi-Hilbert Space and Its Properties

This paper studies concept of a quasi-inner product space and its completeness to get and prove some properties of quasi-Hilbert spaces. The best examples of this notion are spaces where 0<p<∞.