- Series: Discrete Mathematics and Its Applications
- Hardcover: 842 pages
- Publisher: Chapman and Hall/CRC; 1 edition (July 19, 2005)
- Language: English
- ISBN-10: 1584885181
- ISBN-13: 978-1584885184
- Product Dimensions: 7.2 x 2 x 10.2 inches
- Shipping Weight: 3.7 pounds (View shipping rates and policies)
- Average Customer Review: 6 customer reviews
- Amazon Best Sellers Rank: #1,661,484 in Books (See Top 100 in Books)
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Handbook of Elliptic and Hyperelliptic Curve Cryptography (Discrete Mathematics and Its Applications) 1st Edition
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… very comprehensive coverage of this vast subject area … a useful and essential treatise for anyone involved in elliptic curve algorithms … this book offers the opportunity to grasp the ECC technology with a diversified and comprehensive perspective. … This book will remain on my shelf for a long time and will land on my desk on many occasions, if only because the coverage of the issues common to factoring and discrete log cryptosystems is excellent.
―IACR Book Reviews, June 2011
… the book is designed for people who are working in the area and want to learn more about a specific issue. The chapters are written to be relatively independent so that readers can focus on the part of interest for them. Such readers will be grateful for the excellent index and extensive bibliography. … the handbook covers a wide range of topics and will be a valuable reference for researchers in curve-based cryptography.
―Steven D. Galbraith, Mathematical Reviews, Issue 2007f
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As expected, a substantial portion of the book is devoted to point counting methods. One of the methods discussed is the p-adic approach to counting the number of points on an elliptic curve over a field with a small characteristic, with the three most practical ones given the most attention. One of these, the Satoh algorithm, first computes the p-adic approximation of the canonical lift of an elliptic curve E over a finite field F(q), where q = p^d and p is a small prime. This involves lifting the j-invariants using a multivariate version of Newton's root finding algorithm. The trace of the Frobenius endomorphism must then be recovered, and this is done by using the action of the lift on a holomorphic differential on the lift. The resulting factoring problems are formidable, so instead the q-th Verschiebung, which is the dual isogeny to the Frobenius endomorphism is used. The Verschiebung is a separable morphism and the trace of an endomorphism is the trace of its dual. These facts are used to express the trace of the Frobenius endomorphism as a product (modulo q) of coefficients in Z(q). These coefficients are then calculated using certain polynomials.
Another algorithm using the p-adic approach to counting is the Arithmetic-Geometric-Mean (AGM) algorithm, which is discussed for the 2-adic case. As the name implies, this method is based on the AGM iteration, wherein a sequence of elliptic curves is constructed all of which are 2-isogenous to each other. This sequence is constructed so that it converges to the canonical lift of an ordinary elliptic curve, and then an explicit formula for the trace of the Frobenius map is derived. It is then shown how to extend the AGM algorithm to hyperelliptic curves by interpreting it as a special case of the Riemann duplication formula for theta functions.
The third p-adic algorithm discussed is called the Kedlaya algorithm and involves working with the affine curve associated to a hyperelliptic curve of genus g. Associated with this affine curve is its `dagger algebra,' the latter of which is discussed in the book and has its origins in the Monsky-Washnitzer cohomology for nonsingular affine curves over a finite field. This cohomology, which is currently listed under the classification of `rigid cohomology' is a cohomology for algebraic fields over fields of nonzero characteristic and can be considered to be a version of de Rham cohomology (in positive characteristic). In arises when one attempts to lift the Frobenius endomorphism on the coordinate ring of the curve to the coordinate ring of a lift of the curve. Taking the p-adic completion of the coordinate ring of the lift results in a de Rham cohomology which is even larger than the coordinate ring (the limit of exact differentials may not be exact), and so one works with a subring of the completion, which is called the `dagger ring.' The Frobenius endomorphism on the coordinate ring can then be lifted to a (Z(q)) endomorphism on the dagger ring. One can then define differentials of elements in the dagger ring, yielding a module over the dagger ring. The kernel and cokernel of this differential map can then be used to construct the zeroth and first Monsky-Washnitzer cohomology groups. The lift of the Frobenius endomorphism to the dagger ring induces an endomorphism on the cohomology groups, and this allows a Lefschetz fixed point formula to be proved, thus giving the number of rational points on the curve. The Kedlaya algorithm essentially follows this approach to do the point counting, but outputting the zeta function and working only for p greater than or equal to 3.
The book is not just a discussion on theoretical developments and computational algorithms, as an entire part of the book is devoted to applications. One of the applications discussed is that of `smart cards' which to date have been one of most widely used applications of cryptography. An entire chapter is spent on the hardware of smart cards, followed by one on how to attack the implementations of cryptosystems. One particular method for extracting the keys from inside a tamper-proof device involves the use of `power consumption analysis,' which is discussed in some detail in this chapter. The power consumption curve of the device or smart card is analyzed by the attacker, and this, coupled with an understanding of cryptographic algorithms, allows the keys to be compromised. Countermeasures against these attacks are discussed in the next chapter. The discussion is general enough in these chapters to give the motivated reader enough information to experiment with both attacking and with designing and testing effective countermeasures.
This is a reference book, not a textbook.
I gave it a 5-star rating because it would not be fair to down-grade the book because of my own limitations. However, in all honesty, I think most people interested in Elliptic Curve Cryptography will want to be introduced at "lower altitude".
It takes into discussion both theoretical and practical aspects of the domain.