The Wonderful World of Elliptic Curve Cryptography

Introduction

So what protects your privacy and security probably more than anything else on the Internet? That will be Elliptic Curve , and especially:

y² = x³+ax+b

where 4a³+27b² ≠ 0 (and which is need to avoid singular points). The most popular curve is a Secp256k1 (or Curve 25519), and is defined with a=0 and b=7:

y² = x³+7 [Link]

In this, with ECC (Elliptic Curve Cryptography), we take a random number (n), and a point on the elliptic curve (G), and then multiply them together to produce P:

P = n G

G will be an (x,y) point on the curve that both Bob and Alice will agree to. n will then be Bob’s private key, and P will be his public key. The challenge is that if n is a 256-bit random value, it will be extremely difficult to find the value, even though we know G and P.

So let’s look at a bit of Python code in getting an elliptic curve setup:

In this case we see that _a is 0 and _b is 7 (y² = x³+7), and that we have a _Gx and a _Gy value. We also have _p which is a prime number in…