In July of 2005, a summer school organized by the Real Sociedad Matematica Espanola was held at the Universidad Internacional Menendez Pelayo on "Recent Trends in Cryptography" with the goal of giving the attendees an introduction to some of the new ways that mathematics is used by cryptographers and cryptanalysts. The school consisted of four short courses and a handful of lectures on a variety of topics related to stream ciphers and public-key cryptography. Notes from some of these courses and talks have been compiled and published in a new volume in the American Mathematical Society's *Contemporary Mathematics* series. While the five papers collected vary quite a bit in terms of the background they assume and the quality of their exposition, they all serve as good introductions to the topics at hand.

Two of the articles deal with stream ciphers, which are symmetric methods of encryption in which the plaintext is combined (typically with an xor relation) with a pseudorandom sequence which are traditionally generated by using a linear feedback shift-registers (LFSRs). The first of these two articles, Amparo Fuster-Sabater on "Cellular Automata in Stream Ciphers," discusses how cellular automata can be used to reproduce the outputs of various LFSRs, such as clock-controlled shrinking generators. The next article is "Linear and Nonlinear sequences and Applications to Stream Ciphers" by Tor Helleseth, and in this paper Helleseth gives an introduction to stream ciphers LFSRs in terms of what they are and how they can be used. This highlights one of my few complaints with the volume — the articles are published in alphabetical order by author and not necessarily in the order which one would want to read them to get the most out of them. In fact, I struggled through Fuster-Sabater's article only to find some things much easier to understand after I read the paper by Helleseth.

These articles are followed by "An Introduction to Pairing-Based Cryptography" by Alfred Menezes, in which the author begins by introducing the very notion of bilinear pairings. He then discusses some of the exciting ways that they can be used in cryptography both for variations on the standard Diffie-Hellman problem and for Identity-Based Encryption, an idea which has taken off in cryptographic circles in recent years. After a whirlwind introduction to some results about elliptic curves, Menezes then goes on to discuss the Tate Pairing on elliptic curves, and how it can be used to realize some of the formal constructions discussed in the earlier sections, as well as what conditions one might find desirable (or not) when choosing elliptic curves to use in implementations.

An article by Phong Nguyen entitled "Public-Key Cryptanalysis" is almost as ambitious as the title sounds, as the author gives very quick summaries of the very idea of public key encryption as well as a number of actual cryptosystems (RSA, Elgamal, Knapsack, etc) and some elementary attacks on them. The later sections of this article discuss more sophisticated attacks including a family of attacks Nguyen calls 'Square Root attacks' such as Pollard's Rho Method and Baby-Step/Giant-Step. There is also significant room dedicated to lattices and their properties, including how they can be used to help with factoring and attacking various cryptosystems. A final article in the volume is by Igor Shparlinski and deals with using elliptic curves to generate pseudorandom numbers.

As one would expect from articles based on summer school courses, the articles tend to emphasize readability over detail. Most of the authors are ambitious in the scope of their articles but give overviews rather than the full stories and often the authors only sketch proofs or omit proofs of the theorems entirely. On the flip side, the bibliographies of all of the articles are quite extensive and the authors give plenty of guidance if you are left wanting to learn more. While you will not become an expert in any of the fields after reading these articles, you will get a good sense of some of the modern approaches used in cryptography, and I imagine many readers will have their interest piqued. And for a graduate student or researcher interested in learning about recent trends in cryptography, this book gives as good a starting point as I have seen.

Darren Glass is an Assistant Professor of Mathematics at Gettysburg College whose research interests include Number Theory, Algebraic Geometry, and Cryptography. He can be reached at

dglass@gettysburg.edu.