What do you think?
Rate this book
Elliptic Curves: Number Theory and 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 proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students.
Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired.
By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.
Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired.
By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.
- GenresMathematicsNumber Theory
440 pages, Hardcover
First published January 1, 2003
10 people are currently reading
73 people want to read
About the author
Lawrence C. Washington
13 books3 followersLawrence Clinton Washington (born 1951, Vermont) is an American mathematician, who specializes in number theory.
Washington studied at Johns Hopkins University, where in 1971 he received his B.A. and masters degree. In 1974 he earned his PhD at Princeton University under Kenkichi Iwasawa with thesis Class numbers and Z_p extensions.[1] He then became an assistant professor at Stanford University and from 1977 at the University of Maryland, where he became in 1981 an associate professor and in 1986 a professor. He held visiting positions at several institutions, including IHES (1980/81), Max-Planck-Institut für Mathematik (1984), the Institute for Advanced Study (1996), and MSRI (1986/87), as well as at the University of Perugia, Nankai University and the State University of Campinas.
Washington wrote a standard work on cyclotomic fields. He also worked on p-adic L-functions. He wrote a treatise with Allan Adler on their discovery of a connection between higher-dimensional analogues of magic squares and p-adic L-functions.[2] Washington has done important work on Iwasawa theory, Cohen-Lenstra heuristics, and elliptic curves and their applications to cryptography.
In Iwasawa theory he proved with Bruce Ferrero in 1979 a conjecture of Kenkichi Iwasawa, that the \mu-invariant vanishes for cyclotomic Zp-extensions of abelian number fields (Theorem of Ferrero-Washington).[3]
In 1979–1981 he was a Sloan Fellow.
(from Wikipedia)
Washington studied at Johns Hopkins University, where in 1971 he received his B.A. and masters degree. In 1974 he earned his PhD at Princeton University under Kenkichi Iwasawa with thesis Class numbers and Z_p extensions.[1] He then became an assistant professor at Stanford University and from 1977 at the University of Maryland, where he became in 1981 an associate professor and in 1986 a professor. He held visiting positions at several institutions, including IHES (1980/81), Max-Planck-Institut für Mathematik (1984), the Institute for Advanced Study (1996), and MSRI (1986/87), as well as at the University of Perugia, Nankai University and the State University of Campinas.
Washington wrote a standard work on cyclotomic fields. He also worked on p-adic L-functions. He wrote a treatise with Allan Adler on their discovery of a connection between higher-dimensional analogues of magic squares and p-adic L-functions.[2] Washington has done important work on Iwasawa theory, Cohen-Lenstra heuristics, and elliptic curves and their applications to cryptography.
In Iwasawa theory he proved with Bruce Ferrero in 1979 a conjecture of Kenkichi Iwasawa, that the \mu-invariant vanishes for cyclotomic Zp-extensions of abelian number fields (Theorem of Ferrero-Washington).[3]
In 1979–1981 he was a Sloan Fellow.
(from Wikipedia)
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
February 17, 2024
I found this book to be very readable. I basically skimmed it and thought through half the proofs but didn't work any problems. You should have a decent background in algebra, but that's about the only prerequisite (indeed, this is basically all I came in with). He starts with stacking cannonballs and ends with Fermat's Last Theorem, though of course there is no way a single book is going to give you more than a glimpse into Wiles' proof.
It covers of course the group law on elliptic curves, and presents algorithms that can be used to compute things like the order of groups. It covers complex multiplication and from there explains $e^{\pi \sqrt{163}}$, though parts of that one are "it can be shown [ref]".
By the end I had at least a passing understanding of what the modularity theorem is, though sadly I still have no understanding of why it should be true.
It covers of course the group law on elliptic curves, and presents algorithms that can be used to compute things like the order of groups. It covers complex multiplication and from there explains $e^{\pi \sqrt{163}}$, though parts of that one are "it can be shown [ref]".
By the end I had at least a passing understanding of what the modularity theorem is, though sadly I still have no understanding of why it should be true.
Displaying 1 of 1 review