# Modular arithmetic and trap door ciphers

## Joshua Holden

## Rose-Hulman Institute of Technology

Like other branches of mathematics, number theory has seen many surprising
developments in recent years. One of the most surprising is the fact that
number theory, long considered the most "useless" of any field of
mathematics, has become vital to the development of modern codes and
ciphers. As an example, the RSA cryptosystem, developed in the 1970's by
Rivest, Shamir, and Adleman, uses some ideas that are very easy to
understand. Yet, these ideas underlie large portions of both modern number
theory and modern cryptography. We will explore these ideas, and show how
they make RSA the first practical "trap door" cipher. This means that
anyone can encode a message but only the recipient can decode it!

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