Hoffstein, Pipher, and Silverman provide a thorough treatment of the topics while keeping the material accessible. The book uses examples throughout the. Jeffrey Hoffstein (Author), Jill Pipher (Contributor), . Mathematical Cryptography (Undergraduate Texts in Mathematics) by Jeffrey Hoffstein Paperback $ An Introduction to Mathematical Cryptography. Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman. Springer-Verlag – Undergraduate Texts in Mathematics.

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This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature silver,an.

The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography.

The book includes an extensive bibliography and index; supplementary materials are available ;ipher. The book covers a variety of topics that are considered central to mathematical cryptography. Topics are well motivated, and there are a good number of examples and nicely chosen exercises. eilverman

An Introduction to Mathematical Cryptography

To me, this book is still the first-choice introduction to public-key cryptography. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling.

Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Read more Read less.

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An Introduction to Mathematical Cryptography

A Textbook for Students and Practitioners. Topics hoffsein well-motivated, hfofstein there are a good number of examples and nicely chosen exercises. Undergraduate Texts in Mathematics Paperback: Springer; Softcover reprint of the original 2nd ed.

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Read reviews that mention elliptic curves cryptography text math mathematical level class theory examples mathematics background introduction solid topics. Showing of 18 reviews. Top Reviews Most recent Top Reviews. There was a problem filtering reviews right now. Please try again later. Good start to this topic.

The only hiccup was that I tried to work through the text example myself and came up with different answers. This is because the text was wrong. There is an extensive errata file that you can get online. May sure you get it before starting to work through text examples and end-of-chapter exercises. I hope that a revised version is issued that corrects these errata.


Good introduction to intermediate level coverage of math-based crypto, however, I found the text hard to follow because the cross references were hard to look up. It really interferes with the reading process. Including a page number would help greatly.

For more purely math books, this isn’t as large of an issue bec there are so many propositions and theorems that it’s easy to isolate the back reference — but for this text, it is quite difficult. At least for the chapters that were studied by this reviewer, the authors of this book give an effective introduction to the mathematical theory used in cryptography at a level that can be approached by an undergraduate senior in mathematics. The field of cryptography is vast of course, and a book of this size could not capture it effectively.

The topics of primary importance are represented however, and the authors do a fine job of motivating and explaining the needed concepts. The authors give an elementary overview of elliptic curves over the complex numbers, and most importantly over finite fields whose characteristic is greater than 3.

The case where the characteristic is equal to 2 is delegated to its own section. In discussing the arithmetic of elliptic curves over finite fields, the authors give a good motivation for Hasse’s formula, which gives a bound for the number of points of the elliptic curve over a finite fieldbut they do not go into the details of the proof.

The Hasse formula is viewed in some texts as a “Riemann Hypothesis” for elliptic curves over finite fields, and was proven by Hasse in This reviewer has not studied Hasse’s proof, but a contemporary proof relies on the Frobenius map and its separability, two notions that the authors do not apparently want to introduce at this level of book however they do introduce the Frobenius map when discussing elliptic curves over F2.

Separability is viewed in some texts in elliptic curves as more of a technical issue, which can be ignored at an elementary level.

It arises when studying endomorphisms of elliptic curves of fields of non-zero characteristic, and involves defining rational functions. The Frobenius map is not separable, and this fact allows one to show that its degree is strictly greater than the number of points in its kernel. Taking the nth power of the Frobenius map and adding to it the endomorphism which simply multiplies elements by -1, one can show that the number of points of the elliptic curve is equal to the degree of this endomorphism.

Just a few more arithmetical calculations establishes Hasse’s estimate. Some more of the highlights of this part of the book: This failure is used to explain the workings of the Lenstra elliptic curve factorization algorithm in a way that it is better appreciated by the reader.

The distortion maps are used to define a modified Weyl pairing, which is proved to be non-degenerate.

The authors apparently do not want to get into the notions of unramified and separable “isogenies” between elliptic curves and Galois extensions, both of pkpher are used in the proof that they reference. Isogenies are mentioned in a footnote to the discussion on distortion maps, since the latter silgerman isogenies. The proofs were no doubt omitted due to their dependence on techniques from algebraic geometry.


This is discussed very briefly in the last chapter, but the subject is mature enough to be presented at the undergraduate level. One good example would be cryptography based on the mathematical theory of knots and braids oipher braid group is non-Abelianeven though this approach is in its infancy at the present time, and in almost all cases shown to be highly vulnerable to attacks.

It could have been included in the last chapter or possibly as a long exercise. The authors correctly don’t want to elaborate on Weil descent in any more detail, since it requires a solid knowledge of field extensions and theory of algebraic varieties at a level that one obtains in a graduate course in algebraic geometry.

Suffice it to say that the strategy of Weil descent involves finding a cover of the elliptic curve by a hyperelliptic curve that is defined over the extension of the ground field.

This approach has been shown to be problematic for Koblitz curves, the latter of which are discussed in the book. This review is based on a reading of chapters 5 and 8 of piipher book. I used this book to teach a class of math and computer science majors. The book is great. Elliptic Curves, Lattice Based Cryptography.

The systems are described and important attacks on the mathematical underpinnings are given in detail. The necessary mathematics is developed. Many homework problems on which the reader can practice are another strength.

The students have expressed how much they get from the book. I highly recommend it. I’m doing my honor’s thesis on theoretical Cryptography as an undergrad at Colby College, and this book has been the perfect resource. It is so clear, and many time teaches by using easy to understand concrete examples. This book is the perfect place to start if you want to learn about Crypto.

Herstein LC to fully grasp all the related Math fundamentals. It is for undergrads, but useful at grad level to any student who didn’t take the courses as an undergrad i.

See all 18 reviews. Customers who viewed this item also viewed. Protocols, Algorithms and Source Code in C. Introduction to Cryptography with Coding Theory 2nd Edition. Pages with related products.

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