Cryptography Reference
In-Depth Information
CHAPTER 1
Introduction
God created the integers. All the rest is the work of man.
—Leopold Kronecker
If you look at zero you see nothing; but look through it and you will see the
world.
—Robert Kaplan, The Nothing That Is: A Natural History of Zero
T O BE INVOLVED WITH MODERN cryptography is to dive willy-nilly into number
theory, that is, the study of the natural numbers, one of the most beautiful areas
of mathematics. However, we have no intention of becoming deep-sea divers who
raise sunken treasure from the mathematical ocean floor, which in any case is
unnecessary for cryptographic applications. Our goals are much more modest.
On the other hand, there is no limit to the depth of involvement of number theory
with cryptography, and many significant mathematicians have made important
contributions to this area.
The roots of number theory reach back to antiquity. The Pythagoreans—the
Greek mathematician and philosopher Pythagoras and his school—were already
deeply involved in the sixth century B . C . E . with relations among the integers,
and they achieved significant mathematical results, for example the famed
Pythagorean theorem, which is a part of every school child's education. With
religious zeal they took the position that all numbers should be commensurate
with the natural numbers, and they found themselves on the horns of a serious
dilemma when they discovered the existence of “irrational” numbers such as 2 ,
which cannot be expressed as the quotient of two integers. This discovery threw
the world view of the Pythagoreans into disarray, to the extent that they sought
to suppress knowledge of the irrational numbers, a futile form of behavior oft
repeated throughout human history.
Two of the oldest number-theoretic algorithms, which have been passed
down to us from the Greek mathematicians Euclid (third century B . C . E .) and
Eratosthenes (276-195 B . C . E .), are closely related to the most contemporary
encryption algorithms that we use every day to secure communication across
the Internet. The “Euclidean algorithm” and the “sieve of Eratosthenes” are both
quite up-to-date for our work, and we shall discuss their theory and application
in Sections 10.1 and 10.5 of this topic.
 
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