Suggestions for Further Reading

Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it.

—Samuel Johnson

In this chapter, we have summarized the basic mathematical facts that cryptologists are expected to know in order to have a decent understanding of the present-day public-key technology. Our discussion has been often more intuitive than mathematically complete. A reader willing to gain further insight in these areas should look at materials written specifically to deal with the specialized topics. Here are our (biased) suggestions.

There are numerous textbooks on introductory algebra. The books by Herstein [125], Fraleigh [96], Dummit and Foote [81], Hungerford [133] and Adkins and Weintraub [1] are some of our favourites. The algebra of commutative rings with identity (rings by our definition) is called commutative algebra and is the basic for learning advanced areas of mathematics like algebraic geometry and algebraic number theory. A serious study of these disciplines demands more in-depth knowledge of commutative algebra than we have presented in Section 2.13.1. Atiyah and MacDonald’s book [14] is a de facto standard on commutative algebra. Hoffman and Kunze’s book [127] is a good reference for linear algebra and matrix algebra.

Elementary number theory deals with the theory of (natural) numbers without using sophisticated techniques from complex analysis and algebra. Zuckerman et al. [316] can be consulted for a lucid introduction to this subject. The books by Burton [42] and Mollin [207] are good alternatives.

Thorough mathematical treatise on finite fields can be found in the books by Lidl and Niederreiter [179, 180] of which the second also deals with computational issues. Other books of computational flavour include those by Menezes [191] and by Shparlinski [274]. Also see the paper [273] by Shparlinski.

The use of elliptic curves in cryptography has been proposed by Koblitz [150] and Miller [205], and that of hyperelliptic curves by Koblitz [151]. A fair mathematical understanding of elliptic curves banks on the knowledge of commutative algebra (see above) and algebraic geometry. Hartshorne’s book [124] is a detailed introduction to algebraic geometry. Fulton’s book [99] on algebraic curves is another good reference. Rigorous mathematical treatment on elliptic curves can be found in Silverman’s books [275, 276]. The book by Koblitz [152] is elementary, but has a somewhat different focus than needed in cryptology. By far, the best short-cut is the recent textbook from Washington [298]. Some other books by Koblitz [150, 153, 154], Blake et al. [24], Menezes [192] and Hankerson et al. [123] are written for non-experts in algebraic geometry (and hence lack mathematical details), but are good from computational viewpoint. The expository reports [46, 47] by Charlap et al. provide nice elementary introduction to elliptic curves. For hyperelliptic curves, on the other hand, no such books are available. Koblitz’s book [154] includes a chapter on hyperelliptic curves. In addition, an appendix in the same book, written by Menezes et al. much in the style of Charlap et al. [46, 47], provides an introductory and elementary coverage.

In an oversimplified sense, algebraic number theory deals with the study of number fields. The books by Janusz [140], Lang [160], Mollin [208] and Ribenboim [251] go well beyond what we cover in Section 2.13. Also see [89]. For a more modern and sophisticated treatment, look at Neukirch’s book [216]. A book dedicated to p-adic numbers is due to Koblitz [149]. Course notes from one of the authors of this book can also be useful in this regard. The notes are freely downloadable from:

http://www.facweb.iitkgp.ernet.in/~adas/IITK/course/MTH617/SS02/

Analytic number theory deals with the application of complex analytic techniques to solve problems in number theory. Although we do not explicitly need this branch of mathematics (apart from a few theorems that we mention without proofs), it is rather important for the study of numbers. Consult the books by Apostol [12] and by Ireland and Rosen [136] for this. Also see [249]. For complex analysis, we recommend the book by Ahlfors [6]

Feller’s celebrated book [92] is a classical reference on probability theory. Grinstead and Snell’s book [121] is available in the Internet.