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 field), but 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 1934. 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:
- The reminder that the fastest known algorithm to solve the elliptic curve discrete logarithm problem takes p^1/2 steps for a finite field Fp (i.e. the algorithms therefore are not really better than "black box" algorithms).
- The brief historical discussion on public key cryptography.
- The motivational discussion for the Lenstra algorithm using simple calculations that leads to a failed attempt to find the reciprocal of an integer modulo p. 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 discussion on the Frobenius map in the context of elliptic curves over F2 and its use in finding the number of points of an elliptic curve over a finite field.
- The motivational discussion for the use of distortion maps, due to the degeneracy of the Weil pairing. The distortion maps are used to define a modified Weyl pairing, which is proved to be non-degenerate.
Some omissions:
- Algorithms used to calculate the number of points of an elliptic curve over a finite field that are more efficient than brute-force counting or estimation using Hasse's formula.
- The proof that the torsion points of order m can be written as the product of two cyclic groups of order m. The authors apparently do not want to get into the notions of unramified and separable "isogenies" between elliptic curves and Galois extensions, both of which are used in the proof that they reference. Isogenies are mentioned in a footnote to the discussion on distortion maps, since the latter are isogenies.
- The proof verifying certain properties of divisors, namely that they are equal if the corresponding rational functions are constant multiples of each other, and that the degree of a divisor is zero if its sum is the zero element of the elliptic curve. The proofs were no doubt omitted due to their dependence on techniques from algebraic geometry.
- Quantum cryptography. This is discussed very briefly in the last chapter, but the subject is mature enough to be presented at the undergraduate level.
- Cryptography based on non-Abelian groups. One good example would be cryptography based on the mathematical theory of knots and braids (the braid group is non-Abelian), even 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.
- Hyperelliptic curves are discussed very briefly in the last chapter, but a full-fledged presentation could be done in the book without missing the targeted audience. Hyperelliptic curves are also mentioned after the discussion of the MOV algorithm, wherein the authors allude to the use of Weil descent to transfer the elliptic curve discrete logarithm problem to a discrete logarithm problem in a finite field F2^m when m is composite. 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.
Note: This review is based on a reading of chapters 5 and 8 of the book.
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- Reviewed in the United States on May 18, 2010
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- José Luis Castro GarcíaReviewed in Mexico on August 7, 2020
Good book
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