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.

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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

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard and in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard and is under consideration for inclusion in some other ISO standards. Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponential-time algorithm is known for the elliptic curve discrete logarithm problem. For this reason, the strength-per-key-bit is substantially greater in an algorithm that uses elliptic curves. This paper describes the ANSI X9.62 ECDSA, and discusses related security, implementation, and interoperability issues.

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The Elliptic Curve Digital Signature Algorithm (ECDSA)

This paper introduces and makes concrete the concept of certificateless public key cryptography (CL-PKC), a model for the use of public key cryptography which avoids the inherent escrow of identity-based cryptography and yet which does not require certificates to guarantee the authenticity of public keys The lack of certificates and the presence of an adversary who has access to a master key necessitates the careful development of a new security model We focus on certificateless public key encryption (CL-PKE), showing that a concrete pairing-based CL-PKE scheme is secure provided that an underlying problem closely related to the Bilinear Diffie-Hellman Problem is hard

/pdf/certificateless-public-key-cryptography-3qm3rm01ss.pdf

Certificateless Public Key Cryptography

This paper introduces and makes concrete the concept of certificateless public key cryptography (CL-PKC), a model for the use of public key cryptography which avoids the inherent escrow of identity-based cryptography and yet which does not require certificates to guarantee the authenticity of public keys. The lack of certificates and the presence of an adversary who has access to a master key necessitates the careful development of a new security model. We focus on certificateless public key encryption (CL-PKE), showing that a concrete pairing-based CL-PKE scheme is secure provided that an underlying problem closely related to the Bilinear Diffie-Hellman Problem is hard.

Certificateless public key cryptography

We study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods. By incorporating the detection of simplicial faces and a storing/reusing scheme for face volumes we propose practical and theoretical improvements for two of the algorithms. Finally we present a hybrid method combining advantages from the two algorithmic classes. The behaviour of the algorithms is theoretically analysed for hypercubes and practically tested on a wide range of polytopes, where the new hybrid method proves to be superior.

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Exact Volume Computation for Polytopes: A Practical Study

We describe a generic algorithm for computing discrete logarithms in groups of known order in which a smoothness concept is available. The running time of the algorithm can be proved without using any heuristics and leads to a subexponential complexity in particular for finite fields and class groups of number and function fields which were proposed for use in cryptography. In class groups, our algorithm is substantially faster than previously suggested ones. The subexponential complexity is obtained for cyclic groups in which a certain smoothness assumption is satisfied. We also show how to modify the algorithm for cyclic subgroups of arbitrary groups when the smoothness assumption can only be verified for the full group.

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A general framework for subexponential discrete logarithm algorithms

We present a fast algorithm for building ordinary elliptic curves over finite prime fields having arbitrary small MOV degree. The elliptic curves are obtained using complex multiplication by any desired discriminant.

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Building Curves with Arbitrary Small MOV Degree over Finite Prime Fields

List of Tables. List of Figures. Foreword. Preface. 1. Public Key Cryptography. 2. The Group Law on Elliptic Curves. 3. Elliptic Curves Over Finite Fields. 4. The Discrete Logarithm Problem. 5. Counting Points On Elliptic Curves. References. Index

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Elliptic Curves and Their Applications to Cryptography: An Introduction

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O ( h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials.

/pdf/the-complexity-of-class-polynomial-computation-via-floating-3uuu2das1v.pdf

Andreas Enge

Papers

Exact Volume Computation for Polytopes: A Practical Study

A general framework for subexponential discrete logarithm algorithms

Building Curves with Arbitrary Small MOV Degree over Finite Prime Fields

Elliptic Curves and Their Applications to Cryptography: An Introduction

The complexity of class polynomial computation via floating point approximations