Introduction to Applied Algebraic Systems

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

The use of public-key cryptosystems (PKC) received considerable attention. They are beneficial in encryption as well as signing which play an essential role in electronic banking and financial transactions. Elliptic Curve Cryptography (ECC) is one of best public key techniques because of its small key size and high security. Secure applications in smart cards present implementation challenges particular to the platform's memory, bandwidth, and computation constraints. ECC's unique properties make it especially well suited to smart card applications. ECC systems provide the highest strength per bit of any cryptosystem known today. The paper presents a new method for smart card implementation of elliptic curves, explaining how ECC can not only significantly reduce the cost, but also accelerate the deployment of smart cards in new applications. ECC permits reductions in key and certificate size that translate to smaller memory requirements (especially for EEPROM), which represent significant cost savings. This added functionality can play an effective role in electronic payment and online banking technologies.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

Elliptic curves have been intensively studied in algebraic geometry and number theory. Recently, they have been used in devising efficient algorithms for factoring integers [8, 9], for primality proving [2, 11, 12] and for the construction of pseudorAndom bit generators [4]. In Chapter 8 we study how elliptic curves can be used in constructing efficient and secure cryptosystems, while in Chapter 10 we will discuss how good error-correcting codes may be obtained by using elliptic curves.

In modern days security issues play a vital and integral part of information technology and its related applications such as ATM and smart cards. Cryptography has evolved as a important element of security process. Elliptic curve cryptography is used to develop a variety of scheme for security purpose. Point addition and point doubling arithmetic used in elliptic curve is applicable for prime field. When same arithmetic performs over extension field it increases the complexity of the application but enhances the security in the field of communication technology. This paper communicates Weierstrass equation as the root of elliptic curve. Also described the group operation on elliptic curve that is point addition and point doubling used for encryption application based on prime field Fp. The proposed work is based on elliptic curve group operations solved in extension field F p 2. This new approach is verified using System for Algebra and Geometry and Experimentation (SAGE) open source software.

Elliptic curve cryptography is a public key cryptosystem based on the elliptic curve discrete logarithm problem. The reason for the attractiveness of ECC is that there is no sub-exponential algorithm known to solve the elliptic curve discrete logarithm problem. This leads to lower memory requirements, lower computation requirements and higher security than other public key cryptosystems. This project describes an implementation of a micro coded elliptic curve processor using FPGA technology. This processor implements optimal normal basis field operations. The design is generated by a parameterized module generator which can accommodate arbitrary n and also produce field multipliers with different speed/area tradeoffs. The control part of the processor is micro coded, enabling curve operations to be incorporated into the processor hence reducing the chip's I/O requirements. The micro coded approach also facilitates rapid development and algorithmic optimization. for example, projective and affine coordinates were supported using different sets of micro codes. The design was successfully tested on a Xilinx Spartan -II kit device. Elliptic curve processor is particularly suitable for recourse limited devices because of its lower requirements of memory and computation. It is applicable to be implanted into smart cards, cellular phones and other hand-held devices so that secure communication can be provided. For based on implement the FPGA Using Verilog HDL.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

Elliptic curve cryptography (ECC) is based in one of the hardest arithmetic problems, the elliptic curve discrete logarithm problem, making ECC a reliable cryptographic technique. In addition to this advantage, ECC requires shorter keys compared to other public key algorithms, which suggests its use in low-end systems such as smart cards because of its efficiency and limited hardware requirements. Currently the leading smart card technology is Java card. Though both Java 2 standard edition (J2SE) 5.0 and Java card (JC) 2.2.1 include classes and interfaces making ECC available to Java programs, they are not complete, making it necessary to use additional, non-standard packages if the programmer wants to use ECC for different operations (digital signature, data encryption and decryption, etc.). In the present work, we provide a complete review of the state of the art of ECC in J2SE and Java card, with a complete list of operations available in both standard implementations and proprietary packages.

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”: “He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.” In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology.In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms.

A public-key encryption algorithm and a practical hybrid scheme are proposed in this paper. This algorithm uses the advanced transform techniques, i.e. the Number Theoretic Transforms (NTTs) to diffuse data, and uses a non-linear technique based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) to confuse data. The main advantage of the proposed algorithm is its high efficiency and capability. The plaintext can be encrypted in blocks in parallel, thus saves computational time. The proposed algorithm also can be easily applied in a hybrid scheme for practical applications.

Although the construction industry does not yet have a broad base of experience with bar code technology, a wide range of applications has recently been developed by member firms of the Construction Industry Institute. The findings of a research project supported by the Construction Industry Institute are summarized. Specific applications in the areas of quantity takeoff, field material control, warehouse inventory and maintenance, tool and consumable material issue, timekeeping and cost engineering, purchasing and accounting, and document control and office operations are discussed. Research findings indicate that bar code use in construction can improve the speed and accuracy of computer data entry, and produce tangible cost savings similar to those that have been documented in other industries.

Рассматривается теорема о вырождении некоммутативного обобщения спектральной последовательности Ходжа-де Рама, доказанная ранее первым автором. Приводится улучшенная и упрощенная версия ее доказательства, которая в явном виде использует спектральную алгебраическую геометрию. Также объясняется, почему для доказательства существенно необходимы методы алгебраической топологии.

Изложена краткая биография Игоря Ростиславовича Шафаревича, и приведен обзор всех его основных опубликованных научных результатов.

В классической теории представлений локально компактных групп известны конструкции унитарно двойственного пространства неприводимых представлений, преобразования Фурье и теорема Планшереля. В настоящей работе приводятся аналоги этих конструкций для дискретной группы Гейзенберга и ее неприводимых бесконечномерных представлений в векторном пространстве без топологии.