# Identity-Based Encryption from the Weil Pairing

19 Aug 2001-Vol. 2001, pp 213-229

TL;DR: This work proposes a fully functional identity-based encryption scheme (IBE) based on the Weil pairing that has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem.

Abstract: We propose a fully functional identity-based encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem. Our system is based on the Weil pairing. We give precise definitions for secure identity based encryption schemes and give several applications for such systems.

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TL;DR: This work proposes a fully functional identity-based encryption (IBE) scheme based on bilinear maps between groups and gives precise definitions for secure IBE schemes and gives several applications for such systems.

Abstract: We propose a fully functional identity-based encryption (IBE) scheme. The scheme has chosen ciphertext security in the random oracle model assuming a variant of the computational Diffie--Hellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic curves is an example of such a map. We give precise definitions for secure IBE schemes and give several applications for such systems.

5,110 citations

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30 Oct 2006TL;DR: This work develops a new cryptosystem for fine-grained sharing of encrypted data that is compatible with Hierarchical Identity-Based Encryption (HIBE), and demonstrates the applicability of the construction to sharing of audit-log information and broadcast encryption.

Abstract: As more sensitive data is shared and stored by third-party sites on the Internet, there will be a need to encrypt data stored at these sites. One drawback of encrypting data, is that it can be selectively shared only at a coarse-grained level (i.e., giving another party your private key). We develop a new cryptosystem for fine-grained sharing of encrypted data that we call Key-Policy Attribute-Based Encryption (KP-ABE). In our cryptosystem, ciphertexts are labeled with sets of attributes and private keys are associated with access structures that control which ciphertexts a user is able to decrypt. We demonstrate the applicability of our construction to sharing of audit-log information and broadcast encryption. Our construction supports delegation of private keys which subsumesHierarchical Identity-Based Encryption (HIBE).

4,257 citations

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22 May 2005TL;DR: In this article, a new type of identity-based encryption called Fuzzy Identity-Based Encryption (IBE) was introduced, where an identity is viewed as set of descriptive attributes, and a private key for an identity can decrypt a ciphertext encrypted with an identity if and only if the identities are close to each other as measured by the set overlap distance metric.

Abstract: We introduce a new type of Identity-Based Encryption (IBE) scheme that we call Fuzzy Identity-Based Encryption. In Fuzzy IBE we view an identity as set of descriptive attributes. A Fuzzy IBE scheme allows for a private key for an identity, ω, to decrypt a ciphertext encrypted with an identity, ω ′, if and only if the identities ω and ω ′ are close to each other as measured by the “set overlap” distance metric. A Fuzzy IBE scheme can be applied to enable encryption using biometric inputs as identities; the error-tolerance property of a Fuzzy IBE scheme is precisely what allows for the use of biometric identities, which inherently will have some noise each time they are sampled. Additionally, we show that Fuzzy-IBE can be used for a type of application that we term “attribute-based encryption”.
In this paper we present two constructions of Fuzzy IBE schemes. Our constructions can be viewed as an Identity-Based Encryption of a message under several attributes that compose a (fuzzy) identity. Our IBE schemes are both error-tolerant and secure against collusion attacks. Additionally, our basic construction does not use random oracles. We prove the security of our schemes under the Selective-ID security model.

3,610 citations

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01 Jan 2004

TL;DR: This guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment, as well as side-channel attacks and countermeasures.

Abstract: After two decades of research and development, elliptic curve cryptography now has widespread exposure and acceptance. Industry, banking, and government standards are in place to facilitate extensive deployment of this efficient public-key mechanism. Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment. In addition, the book addresses some issues that arise in software and hardware implementation, as well as side-channel attacks and countermeasures. Readers receive the theoretical fundamentals as an underpinning for a wealth of practical and accessible knowledge about efficient application. Features & Benefits: * Breadth of coverage and unified, integrated approach to elliptic curve cryptosystems * Describes important industry and government protocols, such as the FIPS 186-2 standard from the U.S. National Institute for Standards and Technology * Provides full exposition on techniques for efficiently implementing finite-field and elliptic curve arithmetic* Distills complex mathematics and algorithms for easy understanding* Includes useful literature references, a list of algorithms, and appendices on sample parameters, ECC standards, and software toolsThis comprehensive, highly focused reference is a useful and indispensable resource for practitioners, professionals, or researchers in computer science, computer engineering, network design, and network data security.

2,893 citations

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10 Feb 2005TL;DR: A homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn and can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set.

Abstract: Let ψ be a 2-DNF formula on boolean variables x1,...,xn ∈ {0,1}. We present a homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn. In other words, given the encryption of the bits x1,...,xn, anyone can create the encryption of ψ(x1,...,xn). More generally, we can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set. We present a number of applications of the system:
In a database of size n, the total communication in the basic step of the Kushilevitz-Ostrovsky PIR protocol is reduced from $\sqrt{n}$ to $\sqrt[3]{n}$.
An efficient election system based on homomorphic encryption where voters do not need to include non-interactive zero knowledge proofs that their ballots are valid. The election system is proved secure without random oracles but still efficient.
A protocol for universally verifiable computation.

1,754 citations

### Cites background from "Identity-Based Encryption from the ..."

...The modified Weil pairing on the curve [25, 22, 4, 26] gives a bilinear map e : G×G→ G1 with the required properties....

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##### References

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01 Jan 1996TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.

Abstract: From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

13,597 citations

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23 Aug 1985TL;DR: In this article, the authors introduce a novel type of cryptographic scheme, which enables any pair of users to communicate securely and to verify each other's signatures without exchanging private or public keys, without keeping key directories, and without using the services of a third party.

Abstract: In this paper we introduce a novel type of cryptographic scheme, which enables any pair of users to communicate securely and to verify each other’s signatures without exchanging private or public keys, without keeping key directories, and without using the services of a third party. The scheme assumes the existence of trusted key generation centers, whose sole purpose is to give each user a personalized smart card when he first joins the network. The information embedded in this card enables the user to sign and encrypt the messages he sends and to decrypt and verify the messages he receives in a totally independent way, regardless of the identity of the other party. Previously issued cards do not have to be updated when new users join the network, and the various centers do not have to coordinate their activities or even to keep a user list. The centers can be closed after all the cards are issued, and the network can continue to function in a completely decentralized way for an indefinite period.

6,902 citations

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IBM

^{1}TL;DR: It is argued that the random oracles model—where all parties have access to a public random oracle—provides a bridge between cryptographic theory and cryptographic practice, and yields protocols much more efficient than standard ones while retaining many of the advantages of provable security.

Abstract: We argue that the random oracle model—where all parties have access to a public random oracle—provides a bridge between cryptographic theory and cryptographic practice. In the paradigm we suggest, a practical protocol P is produced by first devising and proving correct a protocol PR for the random oracle model, and then replacing oracle accesses by the computation of an “appropriately chosen” function h. This paradigm yields protocols much more efficient than standard ones while retaining many of the advantages of provable security. We illustrate these gains for problems including encryption, signatures, and zero-knowledge proofs.

5,313 citations

### "Identity-Based Encryption from the ..." refers background in this paper

...To analyze the security of certain natural cryptographic constructions Bellare and Rogaway introduced an idealized security model called the random oracle model [ 3 ]....

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01 Jan 1986TL;DR: It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.

Abstract: Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

4,680 citations

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01 Jan 1987TL;DR: Simple identification and signature schemes which enable any user to prove his identity and the authenticity of his messages to any other user without shared or public keys are described.

Abstract: In this paper we describe simple identification and signature schemes which enable any user to prove his identity and the authenticity of his messages to any other user without shared or public keys. The schemes are provably secure against any known or chosen message attack if factoring is difficult, and typical implementations require only 1% to 4% of the number of modular multiplications required by the RSA scheme. Due to their simplicity, security and speed, these schemes are ideally suited for microprocessor-based devices such as smart cards, personal computers, and remote control systems.

4,193 citations

### "Identity-Based Encryption from the ..." refers background in this paper

...Interestingly, the related notions of identity-based signature and authentication schemes, also introduced by Shamir [41], do have satisfactory solutions [ 15 , 14]....

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