عنوان

Modern cryptography and elliptic curves :

1470435829

9781470435820

Modern cryptography and elliptic curves :

[Book]

a beginner's guide /

Thomas R. Shemanske.

Providence, Rhode Island :

American Mathematical Society,

[2017]

©2017

xii, 250 pages :

illustrations ;

22 cm.

Student mathematical library ;

volume 83

Includes bibliographical references and index.

Three Motivating Problems -- Back to the Beginning -- Some Elementary Number Theory -- A Second View of Modular Arithmetic -- Public-Key Cryptography and RSA -- A Little More Algebra -- Curves in Affine and Projective Space -- Applications of Elliptic Curves -- Appendix A: Deeper Results and Concluding Thoughts -- Appendix B: Answers to Selected Exercises

0

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. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry combine to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bézout's theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard's method of factorization, Diffie-Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra's elliptic curve factorization method and ECC. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises further guide the exploration. -- Provided by publisher.

Cryptography, Textbooks.

Curves, Elliptic, Textbooks.

Geometry, Algebraic, Textbooks.

Algebraic geometry-- Arithmetic problems. Diophantine geometry-- Applications to coding theory and cryptography.

Computer science-- Instructional exposition (textbooks, tutorial papers, etc.)

Computer science-- Theory of data-- Data encryption.

Computersicherheit

Cryptography.

Curves, Elliptic.

Elliptische Kurve

Geometry, Algebraic.

Information and communication, circuits-- Communication, information-- Cryptography.

Kryptologie

Number theory-- Arithmetic algebraic geometry (Diophantine geometry)-- Elliptic curves over global fields.

Number theory-- Computational number theory-- Factorization.

Number theory-- Elementary number theory-- Elementary number theory.

Number theory-- Finite fields and commutative rings (number-theoretic aspects)-- Algebraic coding theory; cryptography.

Number theory-- Instructional exposition (textbooks, tutorial papers, etc.)

Public-Key-Kryptosystem

Quantencomputer

Quantum theory-- Axiomatics, foundations, philosophy-- Quantum computation.

Shemanske, Thomas R.,1952-

American Mathematical Society,issuing body.

20200823025755.0

rda

[Book]

Y