In a previous post I described mathematicians’ ongoing search for key properties of prime numbers. That effort may seem to belong entirely within the realm of pure mathematics; but surprisingly, the importance of primes goes far beyond the abstruse obsessions of ivory-tower mathematicians. In fact, the use of prime numbers underlies some of the most dramatic events in the news these past weeks: the story behind Edward Snowden’s revelations that the National Security Agency (NSA) is snooping on the communications of both American citizens and European diplomats.
While the Europeans have protested about their internal communications being intercepted by the NSA—ironically—the tools that one can use for protection from spying by anyone are readily accessible online, in the professional literature, and in publicly-available manuals and textbooks. These methods all rely on clever uses of prime numbers.
The essentials of these techniques are far from new. The foundations of a program to create codes so powerful that they could not be broken even if an eavesdropper were to use the entire available worldwide computing power were laid more than 35 years ago. The year 1976 saw the development of the Diffie-Hellman key exchange method (named after Whitfield Diffie and Martin Hellman; the names Ralph Merkle, James Ellis, Clifford Cocks, and Malcolm Williamson are often also associated with it); and the following, 1977, witnessed the appearance of the RSA algorithm. Both methods have advanced over the past three and a half decades, but information about their extensions is also readily available to anyone.
How do these techniques work? I will explain both methods here—necessarily in a simplified way. (Those interested in learning more can read some of the articles in the links that appear throughout this post.)
