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.

Handbook of Applied Cryptography

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF(2'). We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

https://www.ams.org/mcom/1987-48-177/S0025-5718-1987-0866109-5/S0025-5718-1987-0866109-5.pdf

Elliptic curve cryptosystems

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).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

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 discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

/pdf/use-of-elliptic-curves-in-cryptography-credwc2ehh.pdf

Use of Elliptic Curves in Cryptography

The Weil Pairing, first introduced by Andre Weil in 1940, plays an important role in the theoretical study of the arithmetic of elliptic curves and Abelian varieties. It has also recently become extremely useful in cryptologic constructions related to those objects. This paper gives the definition of the Weil Pairing, describes efficient algorithms to calculate it, gives two applications, and describes the motivation to considering it.

/pdf/the-weil-pairing-and-its-efficient-calculation-221mqsj27f.pdf

The Weil Pairing, and Its Efficient Calculation

Communications between a Host Computing System and a number of remote terminals is enhanced by a data compression method which modifies the data compression method of Lempel and Ziv by addition of new character and new string extensions to improve the compression ratio, and deletion of a least recently used routine to limit the encoding tables to a fixed size to significantly improve data transmission efficiency.

Data compression method

To encode data representative of correlated images, a sequence of arrays of source image data is provided in which each array is representative of an image. Each array of source image data is partitioned into a plurality of source image-data blocks. For each sequence of source image-data blocks, at least one prototypical image-data block is derived from at least a portion of the source image data of the sequence. For each source image-data block, a translation coefficient and a number of scaling coefficients equal to the number of prototypical image-data blocks associated with the sequence are determined. A difference image-data block is formed for each source image-data block from elements of the source image-data block minus the associated translation coefficient minus the sum in turn of each associated scaling coefficient multiplied by elements of the corresponding prototypical image-data block. Each difference image-data block is subjected to an image-data compression procedure to form a set of data-compression-encoded difference image-data elements. Each set of data-compression-encoded difference image-data elements may be stored in digital storage in association with the scaling coefficients and the translation coefficient associated with the corresponding source image-data block.

Image data processing of correlated images

The data compression methods of Ziv and Lempel are modified and augmented, in three ways in order to improve the compression ratio, and hold the size of the encoding tables to a fixed size. The improvements are in the area of dispensing with any uncompressed output, ability to use fixed size encoding tables by using a replacement strategy, and more rapid adaptation by widening the class of strings which may be added to the dictionary. Following Langdon, we show how these improvements also provide an adaptive probabilistic model for the input data. The issue of data structures for efficient implementation is also addressed.

Victor S. Miller

Papers

Use of Elliptic Curves in Cryptography

The Weil Pairing, and Its Efficient Calculation

Data compression method

Image data processing of correlated images

Variations on a theme by Ziv and Lempel