This article was first published on Loopring Protocol - Medium
Elliptic Curve Cryptography (ECC) was introduced in 1985 and has been one of the biggest advances in the field since then. It took 25 years of trial and testing before it was used in production by OpenSSL. Delays like this aren’t uncommon since the bridge between theoretical and practical cryptography can only be proven through the test of time.
ECC’s biggest downside lies in the inherent fact that it’s complex. Its upside lies in the fact that its 256-bit key is stronger and more efficient than RSA’s 4096-bit key. Rather than relying on large numbers alone, elliptic curves obtain their security by combining points on mathematical curves.
So what are elliptic curves?
We define elliptic curves as a group of x and y coordinates represented on a graph via an equation such as y^2=x^3–7x+10 represented below. Wherever there exists a valid x-value which corresponds to a y-value, we call that a pair on the curve that satisfies the equation. Example points for our example equation are represented below.
Real-world elliptic curves aren’t too different from this, although this is just used as an example.
You can try calculating a point yourself by plugging in the numbers:
The point (2, 2) exists on the graph above and is considered a valid pair. An example of an invalid pair would be where the x-value is less than ~ -3.1, since no y-value can be determined when substituted into the equation.
Elliptic Curve Integers
Our elliptic curve depicted above can be represented as a group of integers represented by each y-value modulo a prime number.
Below is the group of integers represented by the equation y^2=x^3–7x+10 mod 19....
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Loopring Protocol - Medium