There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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).

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Attribute-based encryption for fine-grained access control of encrypted data

We introduce a short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel.

/pdf/short-signatures-from-the-weil-pairing-5ajupg0cpi.pdf

Short Signatures from the Weil Pairing

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.

https://cdn.preterhuman.net/texts/cryptography/Hankerson,%20Menezes,%20Vanstone.%20Guide%20to%20elliptic%20curve%20cryptography%20(Springer,%202004)(ISBN%20038795273X)(332s)_CsCr_.pdf

Guide to Elliptic Curve Cryptography

We present the first efficient Identity-Based Encryption (IBE) scheme that is fully secure without random oracles We first present our IBE construction and reduce the security of our scheme to the decisional Bilinear Diffie-Hellman (BDH) problem Additionally, we show that our techniques can be used to build a new signature scheme that is secure under the computational Diffie-Hellman assumption without random oracles

/pdf/efficient-identity-based-encryption-without-random-oracles-54xabxckwn.pdf

Efficient identity-based encryption without random oracles

Galois representations attached to points on Shimura varieties

PROBLEM TO BE SOLVED: To secure binding updates in a wireless telecommunications system. SOLUTION: A public key is generated by using a home address of a mobile host. A home agent, such as a router, generates a private key by using public cryptographic parameters corresponding to the mobile host or the public key. A node of a communication destination uses the public key to encrypt a shared key and sends the encrypted shared key to the mobile host. The mobile host decrypts the shared key by using its original private key. The shared key is used for signing the binding update. Thereafter, the node of the communication destination utilizes the shared key to verify the authenticity of the binding update. COPYRIGHT: (C)2004,JPO

Method of securing binding update by using address based key

Emmy Noether in Erlangen

Cohomology of fiber systems and Mordell-Weil groups of abelian varieties

The generic fibre of a fibre system of polarized abelian varieties with level structure, and with endomorphism structure coming from a CMfield, is defined over the function field of the moduli space for the abelian varieties. We prove that the points on this generic abelian variety which are defined over the function field of the moduli space form a finite group. The methods of proof generalize those of MordellWeil groups of generic abelian varieties, Invent. Math. 81 (1985), 71-106, to which this paper is a sequel. Introduction. In this paper we will consider fibre systems of polarized abelian varieties over C characterized by having a fixed CM-field embedded in their endomorphism algebras. If V is the moduli space for such abelian varieties (with level structure), and W is the fibre variety constructed in [4], then the fibre over the generic point of V is an abelian variety defined over the function field of V. We consider the points of this abelian variety which are defined over C(V). Our result is THEOREM. If dimV > 0, then the Mordell-Weil group of the generic fibre is finite. Here, V is isomorphic to a product of domains of the form {r x s complex matrices Z1 ZtZ > 0} modulo the action of a discrete subgroup of a unitary group. The setting for which the theorem holds is described more precisely in ? 1. The analogous theorem was proved for V a noncompact quotient of the complex upper half plane by Shioda (Theorem 5.1 of [5]). When V is a quotient of a Hilbert-Siegel space Hr and V is the moduli space for abelian varieties whose endomorphism algebras contain a fixed totally real field, CM-field, totally indefinite quaterion algebra over a totally real field, or quaternion algebra over a CM-field, the analogous theorem was proved in [6] (see also [7]). The present paper makes use of new methods to remove the restriction on the base variety V in the CM-field case. I owe many thanks to Goro Shimura for suggesting the problem and giving guidance and advice, and to Robert Indik for the proof of Theorem 6. NOTATION. We will write diag(A, ... , A,) for the matrix with blocks A1,... , A, on the diagonal, I(r) for the identity matrix in GLr(Z), and t-y for the transpose of the matrix -y. Received by the editors March 26, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11GIO, llG15; Secondary 14K22. The author was partially supported by an NSF Postdoctoral Fellowship. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

https://www.ams.org/proc/1988-104-03/S0002-9939-1988-0929425-7/S0002-9939-1988-0929425-7.pdf

Alice Silverberg

Papers

Galois representations attached to points on Shimura varieties

Method of securing binding update by using address based key

Emmy Noether in Erlangen

Cohomology of fiber systems and Mordell-Weil groups of abelian varieties

Mordell-Weil groups of generic abelian varieties in the unitary case