xii PREFACE The first issue is the most obvious. You need to authorize your credit card company or bank to transfer funds to the merchant however, you’re not face-to-face with the seller, and you have to send your information through a probably very insecure channel. It’s imperative that no one is able to obtain your personal information and pretend to be you in future transactions! There are, however, two other very important items. The process must be fast people aren’t willing to wait minutes to make sure an order has been confirmed. Also, there’s always the problem of a message being corrupted. What if some of the message is mistransmitted or misread by the party on the other end? These questions lead us to the study of efficient algorithms and error detection and correction codes. These have found a wealth of ap- plications not just in cryptography, but also in areas where the information is not secret. Two great examples are streaming video and Universal Product Codes (UPC). In streaming video the information (everything from sports high- lights to CSPAN debates) is often unprotected and deliberately meant to be freely available to all what matters is being able to transmit it quickly and play it correctly on the other end. Fruits and vegetables are some of the few remaining items to resist getting a UPC barcode these black and white patterns are on almost all products. It may shock you to realize how these are used. It’s far more than helping the cashier charge you the proper amount they’re also used to help stores update their inventory in real time as well as correlate and analyze your purchases to better target you in the future! These are both wonderful examples of the need to detect and correct errors. These examples illustrate that problems and solutions arising from cryp- tography often have applications in other disciplines. That’s why we didn’t title this book as an introduction to cryptography, but rather to encryption. Cryptography is of course important in the development of the field, but it’s not the entire story. The purpose of this book is to introduce just enough mathematics to explore these topics and to familiarize you with the issues and challenges of the field. Fortunately, basic algebra and some elementary number the- ory is enough to describe the systems and methods. This means you can read this book without knowing calculus or linear algebra however, it’s im- portant to understand what “elementary” means. While we don’t need to use powerful theorems from advanced mathematics, we do need to be very clever in combining our tools from algebra. Fortunately we’re following the paths of giants, who have had numerous “aha moments” and have seen sub- tle connections between seemingly disparate subjects. We leisurely explore these paths, emphasizing the thought processes that led to these remarkable advances. Below is a quick summary of what is covered in this book, which we follow with outlines for semester-long courses. Each chapter ends with a collection of problems. Some problems are straightforward applications of
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