The Numeroscope

An Interactive Laboratory of Numbers and Cryptography

by Paul Burchard

What is the Numeroscope?

Mathematically-based cryptography will form the basis for our emerging digital economy. The Numeroscope is a multi-media, interactive laboratory for exploring the the number systems which form the basis for today's cryptography, Along the way, we will use the exploration of these somewhat exotic number systems to illuminate the meaning of familiar algebraic operations (like +, -, ×, ÷, , Log).

Technical Summary

Acknowledgments

Awards

Modules

Addition and Multiples

Introducing the "clock numbers", which work like ordinary integers, except that they wrap around after a certain number of "hours".

Multiplication, Powers, and Logarithms

From the right point of view, either addition or multiplication can be made to look simple. But because the relationship between these two operations is profound, no single point of view can make them both look simple at the same time.

Factoring Number Systems

When the number of hours on the clock dial is not a prime and so can be factored, the additive and multiplicative views of the number system can be clarified by geometrically "factoring out" the influence of each of the primes involved.

Applications to Cryptography (RSA)

The RSA public key cryptosystem is explained in terms of the multiplicative structure of clocks with pq hours (with p, q prime).

The Square Root

Clock numbers modulo an odd prime p always have square roots, but only if we are willing to admit "imaginary" roots from an larger number system. This is analogous to the relationship between the real and complex number systems.

Solving Polynomial Equations

More generally, just as the "imaginary" number i = is the solution of the equation x² + 1 = 0, we can create even larger number systems by imagining that we have solutions to more complicated polynomial equations.

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