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.

/pdf/the-elliptic-curve-digital-signature-algorithm-ecdsa-33tvnr4ena.pdf

The Elliptic Curve Digital Signature Algorithm (ECDSA)

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

This paper describes the Elliptic Curve Cryptography algorithm and its suitability for smart cards.

https://dl.acm.org/doi/pdf/10.1145/1386853.1378356

Elliptic curve cryptography

This paper describes elliptic curve cryptosystems (ECCs), which are expected to become the next-generation public key cryptosystems, and also describes Fujitsu Laboratories’ study of ECCs. ECCs require a shorter key length than RSA cryptosystems, which are the de facto standards of public key cryptosystems, but provide equivalent security levels. Because of the shorter key length, ECCs are fast and can be implemented with less hardware. First, we outline ECC and describe a typical digital signature algorithm. Then, we describe our technology for parameter generation of a secure ECC and the implementation of a fast ECC by software and by a digital signal processor. ECCs are expected to enter widespread use as a base technology of electronic information services.

Elliptic curve cryptosystem

The basic pattern matching problem is to find the locations where a pattern occurs in a text. Recently, secure pattern matching has been received much attention in various areas, including privacy-preserving DNA matching and secure biometric authentication. The aim of this paper is to give a practical solution for this problem using homomorphic encryption, which is public key encryption supporting some operations on encrypted data.In this paper, we make use of the somewhat homomorphic encryption scheme presented by Lauter, Naehrig and Vaikuntanathan (ACM CCSW 2011), which supports a limited number of both additions and multiplications on encrypted data. In their work, some message encoding techniques are also presented for enabling us to efficiently compute sums and products over the integers. Based on their techniques, we propose a new packing method suitable for an efficient computation of multiple Hamming distance values on encrypted data. Our main extension gives two types of packed ciphertexts, and a linear computation over packed ciphertexts gives our desired results. We implemented the scheme with our packing method.Our experiments ran in an Intel Xeon at 3.07 GHz with our software library using inline assembly language in C programs. Our optimized implementation shows that the packed encryption of a text or a pattern, the computation of multiple Hamming distance values over packed ciphertexts, and the decryption respectively take about 3.65 milliseconds (ms), 5.31 ms, and 3.47 ms for secure exact and approximate pattern matching of a binary text of length 2048. The total time is about 12.43 ms, which would give the practical performance in real life. Our method gives both faster performance and lower communication than the state-of-the-art work for a binary text of several thousand bits in length.

Secure pattern matching using somewhat homomorphic encryption

Among many approaches for privacy-preserving biometric authentication, we focus on the approach with homomorphic encryption, which is public key encryption supporting some operations on encrypted data. In biometric authentication, the Hamming distance is often used as a metric to compare two biometric feature vectors. In this paper, we propose an efficient method to compute the Hamming distance on encrypted data using the homomorphic encryption based on ideal lattices. In our implementation of secure Hamming distance of 2048-bit binary vectors with a lattice of 4096 dimension, encryption of a vector, secure Hamming distance, and decryption respectively take about 19.89, 18.10, and 9.08 milliseconds (ms) on an Intel Xeon X3480 at 3.07 GHz. We also propose a privacy-preserving biometric authentication protocol using our method, and compare it with related protocols. Our protocol has faster performance and shorter ciphertext size than the state-of-the-art prior work using homomorphic encryption.

/pdf/packed-homomorphic-encryption-based-on-ideal-lattices-and-1v8seh6e1o.pdf

Packed homomorphic encryption based on ideal lattices and its application to biometrics

Somewhat homomorphic encryption is public key encryption supporting a limited number of both additions and multiplications on encrypted data, which is useful for performing fundamental computations with protecting the data confidentiality. In this paper, we focus on the scheme proposed by Lauter, Naehrig and Vaikuntanathan (ACM CCSW 2011), and present two types of packed ciphertexts based on their packing technique. Combinations of two types of our packing method give practical size and performance for wider computations such as statistical analysis and distances. To demonstrate its efficiency, we implemented the scheme with our packing method for secure Hamming distance, which is often used in privacy-preserving biometrics. For secure Hamming distance between two binary vekoshiba@mail.saitama-u.ac.jpctors of 2048-bit, it takes 5.31ams on an Intel Xeon X3480 at 3.07aGHz. This gives the best performance in the state-of-the-art work using homomorphic encryption.

Practical Packing Method in Somewhat Homomorphic Encryption

On the Number of the Pairing-friendly Curves

Somewhat homomorphic encryption is public key encryption supporting a limited number of additions and multiplications on encrypted data. This encryption gives a powerful tool in performing meaningful computations with protecting data confidentiality, whose property is suitable mainly in cloud computing. In this paper, we focus on the scheme proposed by Brakerski and Vaikuntanathan, and present two types of packed ciphertexts in order to improve performance and reduce size of the encrypted data. One type of our packed ciphertexts is based on the message encoding technique proposed by Lauter, Naehrig and Vaikuntanathan. While their technique empowers efficient secure computation of sums and products over the integers, our second type of packed ciphertexts enables efficient secure computation of more complex functionalities such as multiple inner products and multiple Hamming distances. We apply our packing method to construct several protocols for secure biometric authentication and secure pattern matching computations. Our implementation shows that our method gives faster performance than the state-of-the-art work in such applications. Copyright © 2015 John Wiley & Sons, Ltd.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/sec.1164

Jun Kogure

Papers

Secure pattern matching using somewhat homomorphic encryption

Packed homomorphic encryption based on ideal lattices and its application to biometrics

Practical Packing Method in Somewhat Homomorphic Encryption

On the Number of the Pairing-friendly Curves

New packing method in somewhat homomorphic encryption and its applications