Elliptic curves number theory and cryptography

Elliptic curves are a very important new area of mathematics which has been greatly explored over the past few decades. An introduction to mathematical cryptography springerlink. Jul 20, 2015 elliptic curves are a very important new area of mathematics which has been greatly explored over the past few decades. With additional exercises, this edition offers more comprehensive coverage of the fund. The number theory chapters are equally detailed and quite extensive, using a treatment of elliptic curves over the rationals and over the complex numbers to develop all of the most uptodate number theoretic results pertaining to elliptic curves.

It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. Number theory and cryptography, second edition develops the theory of elliptic curves to provide a basis for. The objective of this course is to introduce students with basic concepts, facts and algorithms concerning elliptic curves over the rational numbers and finite fields and their applications in cryptography and algorithmic number theory. Menezesy march 23, 2005 elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous literature on the subject. The goal of these talks is to tell you something about the theory of elliptic curves, with an emphasis on those aspects that are of interest in cryptography.

There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and digital signature authentication see elliptic curve cryptography and elliptic curve dsa. Number theorists study prime numbers as well as the properties of. In 1994 andrew wiles, together with his former student richard taylor, solved one of the most famous maths problems of the last 400 years. Silverman and john tate, rational points on elliptic curves, undergraduate texts in mathematics, springerverlag, 1992. Just what are elliptic curves and why use a graph shape in cryptography. Washington introduction the basic theory weierstrass equations the group law projective space and the point at infinity proof of associativity. Annals of mathematics, mathematical sciences research institute 126 1986. Menezes elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. Moduli spaces and arithmetic dynamics, crm monograph series 30, ams, 2012. Oct 24, 20 another uncertainty about elliptic curve cryptography is related to patents. The table of contents for the book can be viewed here. Number theory and cryptography, second edition discrete mathematics and its applications on.

Elliptic curves in number theory and cryptography techylib. Elliptic curve cryptography and coding theory according to the lagranges theorem, h is always an integer. Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of fermats last theorem. This is how elliptic curve public key cryptography works. Ijca elliptic curve diffiehellman ecdh analogy for.

For example, in the 1980s, elliptic curves started being used in cryptography and elliptic curve techniques were developed for factorization and primality testing. This graduatelevel course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography. However, most books on the subject assume a rather high level of. The number theory chapters could be sampled, if not taken in their entirety, for a. Ecc requires smaller keys compared to nonec cryptography based on plain galois fields to provide equivalent security. For alice and bob to communicate securely over an insecure network they can exchange a private key over this network in the following way. Readings elliptic curves mathematics mit opencourseware. Curves, elliptic, number theory, cryptography publisher.

Number theory, elliptic curves, arithmetic and diophantine geometry, number theoretic aspects of dynamical systems, cryptography. The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. Syllabus elliptic curves mathematics mit opencourseware. Are there cases of the serre modularity conjecture which are relatively easy to prove for some elliptic representations, other than by langlandstunnell, perhaps for those on frey curves. They also find applications in elliptic curve cryptography ecc and integer factorization. Elliptic curves are especially important in number theory, and constitute a major area of current research. Ijca solicits original research papers for the may 2020 edition. In the 1980s and 1990s, elliptic curves played an impor tant role in the proof of fermats last theorem. If youd like a little bit more context, check out my primer on cryptography for some basic information. Home archives volume 176 number 10 elliptic curve diffiehellman ecdh analogy for secured wireless sensor networks. For a number of years, i have been moving within and between the overlapping mathematics and cryptography communities. It is possible to write endlessly on elliptic curves.

There are over patents that cover specific uses of elliptic curves owned by blackberry through their 2009 acquisition of certicom. Many of these patents were licensed for use by private organizations and even the nsa. Elliptic curves and cryptography aleksandar jurisic alfred j. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. How elliptic curve cryptography works technical articles. I should add that elliptic curve cryptography is actually the use of certain elliptic curves and rational points over a finite field. From number theory to network communication series. Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and. Migration of legacy information system based on business process theory. Context is probably ellipticcurve cryptography but im not sure, the math is a bit over my head. This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. This course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography.

Number theory and cryptography, second edition discrete mathematics and its applications at. Advanced topics in the arithmetic of elliptic curves, springerverlag, gtm 151, 1995. The public key is created by agreeing on a standard generator point in an elliptic curve group elliptic curve mathematics is a branch of number theory and multiplying that point by a random. Ellipses, elliptic function s, and elliptic curves. The reason that we use elliptic curves for the key exchange is because they allow longer keys to be generated with fewer bits of data exchanged between computers. Assuming solely a modest background in elementary amount idea, groups, and fields, elliptic curves.

Nov 21, 20 elliptic curves in number theory and cryptography 1 a historical overview. Number theory and cryptography, second edition develops the theory of elliptic curves to provide a basis for both number theoretic and. We will therefore take a stroll through the history of mathematics, encountering first the ellipse, moving on to elliptic integrals, then to elliptic functions, jumping back to elliptic curves, and eventually making the connection between elliptic functions and elliptic curves. Elliptic curves, modular forms and cryptography springerlink. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Click here for a cv and complete list of publications books. Number theory and cryptography discrete mathematics and its applications on. An introduction, with marc hindry, springerverlag, gtm 201, 2000. Number theory and cryptography, second edition develops the theory of elliptic curves to. A relatively easy to understand primer on elliptic curve. Number theory and cryptography, second edition discrete mathematics and its applications 2nd edition by washington, lawrence c.

May 28, 2003 elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of fermats last theorem. To kick things off, here is a very brief summary provided by wikipedia and myself with the help of my friend ut00random suggested in the 1980s, elliptic curve cryptography is now a very succesful cryptographic approach which uses very deep results about algebraic geometry and algebraic number theory into its theory and implementation. In elliptic curve cryptography, the group used is the group of rational points on a given elliptic curve. Elliptic curves number theory and cryptography second edition by lawrence c. Number theory and cryptography, second edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. Elliptic curves number theory and cryptography second. A private key is a number priv, and a public key is the public point dotted with itself priv times. Number theory and cryptography, 2nd edition by lawrence c. Elliptic curves number theory and cryptography request pdf.

This book is an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. Join researchgate to discover and stay uptodate with the latest research from leading experts in elliptic curves and. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves. Washington introduction the basic theory weierstrass equations the group law projective space and the point at.

The chapters on elliptic curve cryptography could be approached similarly, and readers interested only in elliptic curve cryptography might be able to skip or skim some of the more technical material in chapters 3 and 4 in order to get right to the cryptography. Download it once and read it on your kindle device, pc, phones or tablets. In 1994 andrew wiles, together with his former student richard taylor, solved one of the most famous maths problems of the last 400 years, fermats last. An elliptic curve cryptosystem can be defined by picking a prime number as a maximum, a curve equation and a public point on the curve. Ellipticcurve cryptography ecc is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields. Number theory and cryptography, second edition discrete mathematics and its applications 2nd edition by washington. This course note aims to give a basic overview of some of the main lines of study of elliptic curves, building on the students knowledge of undergraduate algebra and complex analysis, and filling in background material where required especially in number theory and geometry. Elliptic curve cryptography elliptic curve cryptography ecc was discovered in 1985 by victor miller ibm and neil koblitz university of washington as an alternative mecha. While this is an introductory course, we will gently work our way up to some fairly advanced material, including an overview of the proof of fermats last theorem. Journal of number theory elliptic curve cryptography. Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the.

To understand ecc, ask the company that owns the patents. Elliptic curves in number theory and cryptography 1 a historical overview. An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. Number theory and cryptography introduces every the cryptographic and amount theoretic sides of elliptic curves, interweaving the thought of elliptic curves with their functions. My background is in number theory, and i became intrigued with cryptography after elliptic curves were introduced to the eld. Number theory and cryptography, second edition discrete mathematics and its applications by lawrence c. A friendly introduction to number theory, pearson, 1997. Silverman brown university and ntru cryptosystems, inc. Elliptic curve cryptography ecc is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields. Summer school on computational number theory and applications to cryptography university of wyoming june 19 july 7, 2006 0. Larry washington department of mathematics university of maryland.