Saying Goodbye to Napier’s Logs, and Hello To Elliptic Curves

A Bluffer’s Guide to converting discrete log problems into elliptic curve ones — it’s all about point adding and point multiplying

Before we start … let’s do some basics. John Napier found:

a^b a^d = a^{b+d}

and that:

(a^b)^c = a^{bc}

And, as every schoolchild should know:

5³ x 5⁷ = 5¹⁰

and that:

(6⁴)⁶=6²⁴

The Problem With Discrete Log Public Key

So, where is public-key encryption just now? Well, it all started with discrete logarithms (g^x mod p) and exponential ciphers (M^e mod p), and where people like Whitfield Diffie and Ron Rivest showed how a hard problem can be derived from exponentials. But that hard problem has become a bit easier as computing power has increased, so the prime number (p) has inflated itself to 2,048 bits and more. On the horizon, too, are quantum computers, and which don’t see our existing hard problems as hard anymore.

So, we are in-between, and the solution is to find something that works for all, and, especially allows limited processing devices…