Topic
Bilinear map
About: Bilinear map is a research topic. Over the lifetime, 584 publications have been published within this topic receiving 14027 citations. The topic is also known as: bilinear mapping.
Papers published on a yearly basis
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Papers
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TL;DR: In this paper, the Radon-Nikodym theorem is generalized to the case of strongly bounded vector measures, which is a generalization of a result due to E. Leonard and K. Sundaresan.
Abstract: The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space X to have the property that bounded additive X-valued maps on o-algebras be strongly bounded are presented, namely, X can contain no copy of /„. The next two sections treat the Jordan decomposition for measures with values in Z.|-spaces on c0(r) spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of c0. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space X has an equivalent very smooth norm (in particular, a Fréchet differentiable normithenitsdualhas the Radon-Nikodym property. Consequently, a C(H) space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper. The present paper contains results on various aspects of the general theory of vector-valued measures. It proceeds in four sections which are unrelated to each other except for their general relationship to the topic of the title. A brief outline of the results of each section is presented below—a more complete discussion of the sections is delayed (largely because of their disconnected nature) until the sections themselves. §1 is concerned with the theory of strongly bounded vector measures. The main result of this section (Theorem 1.1) provides criteria for a Banach space X to possess the property that every X-valued bounded additive map with values in X be strongly bounded. This theorem sharpens the classical Pettis theorem on weakly countably additive set functions and allows a sharpening of several other related results. §2 is concerned with the Jordan decomposition of vector measures with values in a Banach lattice. The results of this section are necessarily meager: not much is possible. Our most precise results are in case the range space is an abstract £space or c0. A few remarks are also made concerning the range of certain vector measures. §3 deals with the integrability of certain scalar functions with respect to a vector measure. Utilizing the series representation of a scalar function and its integral, a result of D. R. Lewis is generalized. Also, a criterion for integrability Received by the editors February 5, 1973 and, in revised form, May 25, 1973. AMS (MOS) subject classifications (1970). Primary 46B05.
1,786 citations
Posted Content•
TL;DR: In this paper, a Hierarchical Identity Based Encryption (HIBE) scheme is presented, where the ciphertext consists of just three group elements and decryption requires only two bilinear map computations, regardless of the hierarchy depth.
Abstract: We present a Hierarchical Identity Based Encryption (HIBE) system where the ciphertext consists of just three group elements and decryption requires only two bilinear map computations, regardless of the hierarchy depth. Encryption is as ecient as in other HIBE systems. We prove that the scheme is selective-ID secure in the standard model and fully secure in the random oracle model. Our system has a number of applications: it gives very ecient forward secure public key and identity based cryptosystems (with short ciphertexts), it converts the NNL broadcast encryption system into an ecient public key broadcast system, and it provides an ecient mechanism for encrypting to the future. The system also supports limited delegation where users can be given restricted private keys that only allow delegation to bounded depth. The HIBE system can be modified to support sublinear size private keys at the cost of some ciphertext expansion.
1,076 citations
13 Apr 2008
TL;DR: In this article, a general methodology for constructing very simple and efficient non-interactive zero-knowledge proofs and noninteractive witness-indistinguishable proofs that work directly for groups with a bilinear map, without needing a reduction to Circuit Satisfiability is presented.
Abstract: Non-interactive zero-knowledge proofs and non-interactive witnessindistinguishable proofs have played a significant role in the theory of cryptography. However, lack of efficiency has prevented them from being used in practice. One of the roots of this inefficiency is that non-interactive zeroknowledge proofs have been constructed for general NP-complete languages such as Circuit Satisfiability, causing an expensive blowup in the size of the statement when reducing it to a circuit. The contribution of this paper is a general methodology for constructing very simple and efficient non-interactive zero-knowledge proofs and non-interactive witness-indistinguishable proofs that work directly for groups with a bilinear map, without needing a reduction to Circuit Satisfiability.
Groups with bilinear maps have enjoyed tremendous success in the field of cryptography in recent years and have been used to construct a plethora of protocols. This paper provides non-interactive witness-indistinguishable proofs and non-interactive zero-knowledge proofs that can be used in connection with these protocols. Our goal is to spread the use of non-interactive cryptographic proofs from mainly theoretical purposes to the large class of practical cryptographic protocols based on bilinear groups.
940 citations
27 Jun 2016
TL;DR: This paper proposed a latent embedding model for learning a compatibility function between image and class embeddings, in the context of zero-shot classification, which augments the state-of-the-art bilinear compatibility model by incorporating latent variables.
Abstract: We present a novel latent embedding model for learning a compatibility function between image and class embeddings, in the context of zero-shot classification. The proposed method augments the state-of-the-art bilinear compatibility model by incorporating latent variables. Instead of learning a single bilinear map, it learns a collection of maps with the selection, of which map to use, being a latent variable for the current image-class pair. We train the model with a ranking based objective function which penalizes incorrect rankings of the true class for a given image. We empirically demonstrate that our model improves the state-of-the-art for various class embeddings consistently on three challenging publicly available datasets for the zero-shot setting. Moreover, our method leads to visually highly interpretable results with clear clusters of different fine-grained object properties that correspond to different latent variable maps.
929 citations
30 Oct 2006
TL;DR: This work provides a new scheme that is proven secure in the plain public-key model, meaning requires nothing more than that each signer has a (certified) public key, and is provenSecure in the random-oracle model under a standard (not bilinear map related) assumption.
Abstract: A multi-signature scheme enables a group of signers to produce a compact, joint signature on a common document, and has many potential uses. However, existing schemes impose key setup or PKI requirements that make them impractical, such as requiring a dedicated, distributed key generation protocol amongst potential signers, or assuming strong, concurrent zero-knowledge proofs of knowledge of secret keys done to the CA at key registration. These requirements limit the use of the schemes. We provide a new scheme that is proven secure in the plain public-key model, meaning requires nothing more than that each signer has a (certified) public key. Furthermore, the important simplification in key management achieved is not at the cost of efficiency or assurance: our scheme matches or surpasses known ones in terms of signing time, verification time and signature size, and is proven secure in the random-oracle model under a standard (not bilinear map related) assumption. The proof is based on a simplified and general Forking Lemma that may be of independent interest.
705 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 2 |
| 2024 | 1 |
| 2023 | 2 |
| 2022 | 7 |
| 2021 | 12 |
| 2020 | 13 |