Many of the challenges and opportunities facing citizens in the twenty-first
century require some level of mathematical proficiency. Some obvious ones
are optimization problems in business, managing your household’s budget,
weighing the economic policies and proposals of political candidates, and
of course the ever-important quest to build the best fantasy sports team
possible and, if not winning your local NCAA basketball pool, at least doing
well enough to avoid embarrassment! As important as these are, there are
many other applications of mathematics going on quietly around us all the
time. In this book we concentrate on issues arising from cryptography, which
we’ll see is far more than soldiers and terrorists trying to communicate
in secret. We use this as the vehicle to introduce you to a lot of good,
applicable mathematics; for much of the book all you need is high school
algebra and some patience. These are not cookbook problems to help you
perfect your math skills, but rather the basis of modern commerce and
security! Equally important, you’ll gain valuable experience in how to think
about and approach diﬃcult problems. This is a highly transferable skill
and will serve you well in the years to come.
Cryptography is one of the oldest studies, and one of the most active
and important. The word cryptography comes from two Greek words:
κρυτττ ` oςς (kryptos), meaning secret, and γραϕω ` (grapho), meaning to
write. As these roots imply, it all began with the need for people to com-
municate securely. The basic setup is that there are two people, and they
must be able to quickly, easily, and securely exchange information, often in
the presence of an adversary who is actively attempting to intercept and
decipher the messages.
In the public mind, the most commonly associated images involve the
military. While war stories make for dramatic examples and are very im-
portant in both the development of the field and its applications, they are
only part of the picture. It’s not just a subject for soldiers on the battlefield.
Whenever you make an online purchase, you’re a player. This example has
many of the key features.