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## Elliptic Curve Cryptography and Finite Fields

Back around 2000, I found a copy of Neal Koblitz’s text A Course in Number Theory and Cryptography at the Borders bookstore in Bangor, Maine.  I only worked my way through the first chapter, but was fascinated with these ideas.  I found Professor Koblitz’s website which, at the time, had a tutorial section on finite fields and elliptic curve cryptography (this may have been on the Certicom website, I can’t remember now).  I moved on to other forms of digital cryptography, like the Diffie-Hellman Key Exchange and RSA Cryptosystem, but always appreciated Prof. Koblitz’s work.  Recently, we dressed up for Halloween as a number and I chose to be the number 4.  As part of my costume, I drew the addition table for the Galois Field of order 4$GF(4)=^{GF(2)[x]}/_{x^2+x+1}$, and did a lot of thinking that week about the element a, which was defined as the root of the equation $0=x^2+x+1$ in $GF(2)$

This past week, I decided to look at the mathematics behind Bitcoin and blockchain, and lo and behold, it is Finite Fields and Elliptic Curve Cryptography – I don’t know why it took me so long to find this out, but now I’m excited about these topics.  I am a little skeptical about the current “Bitcoin bubble.”  I’m not sure that these valuations are sustainable, but from everything I’ve read, the blockchain algorithm behind Bitcoin is revolutionary and the mathematics is “supercool.”

Here’s a graph of the equation $y=x+1$ in $GF(2)$.

## Light Posting

I’m on vacation until September, and my work priority this summer is to prepare to team-teach a class on Technology & Privacy for the Winter 2010 quarter.

If you’re interested in this topic and haven’t read Privacy On the Line by Whitfield Diffie and Susan Landau and The Eavesdroppers by Samuel Dash, they are both excellent.

I’ll probably post every few weeks, but, at the moment, my priorities are elsewhere.

## Digital Security

Someone once said to me that higher mathematics was useless.  I replied that since it controlled every computer on the planet, it seemed pretty useful to me.  One of my favorite areas of math is abstract algebra.  Abstract algebra grew out of the work of French mathematician Evariste Galois around 1830.

In the 1500’s Italian mathematicians had worked to find formulas to solve the cubic (x³) and quartic (x^4) equations.  Then, for almost 300 years, mathematicians worked to find a solution for the quintic, or fifth degree equation.  Finally, in the early 1800’s, the work of Galois and Abel proved that there is no general solution for equations with powers of x higher than 4.

From this work grew the field of abstract algebra.  Prior to Galois and Abel, between 1750 and 1820, many mathematicians contributed significant ideas to what would become known as abstract algebra.  Euler, Gauss, Legendre and Lagrange (1750-1820 or so) all worked with what is known as modular arithmetic.  Modular systems and discrete mathematics became very important in the mid-20th century with the development of the digital computer.

Here’s a link to a paper I wrote about some of these ideas

Mathematics, Communication and Secrecy