Elliptic Curve Cryptography

Elliptic curve cryptography (ECC) is a popular choice for the security of data online, including key exchanges, digital signatures, and encryption. ECC is often used in digital signatures, email and message security, and many credit cards. It’s important that ECC is secure! We’re going to look into the mathematical aspect of ECC to see how secure it is: the equation itself is expressed as y²=x³+ax+b, where a and b are some coefficients.

Elliptic curves are a trapdoor function, where it is easy to get from one value to the next, however given that final value, it would be complicated to work backwards and achieve the initial value. The curve itself is in a sideways bell curve, and is always symmetric across the x-axis.

You can connect two points on the elliptic curve with a line, and it will also touch the elliptic curve a third time. If you were to reflect this point across the x-axis, this point will be the sum of the values of the two initial points. Typically, in elliptic curve cryptography, you would repeatedly be doing these actions (often denoted by x•P).

For 256-bit whole numbers, the minimum amount of operations would be 0. To find the greatest amount of operations, we have to consider the greatest possible binary expansion, which would be 2²⁵⁵+2²⁵⁴+...+2¹+2⁰, meaning for a maximum amount of 255 point doublings. The point doublings will sum to a total of 510 point operations.

Using elliptic curve cryptography is often a lot more preferred in comparison to other public key cryptography options since there is often a lesser amount of storage used. Elliptic curve cryptography is often compared to RSA, which utilizes prime factorization to achieve certain values. Although RSA is faster than ECC, there is a difference in storage, making ECC more optimal today.

A large portion of popular cryptographic methods are done through the combination of algebra, number theory, and combinatorics, and shows how so much of cryptography is based on mathematics. If you’re interested, there's many different types of cryptographic systems that are really cool, such as RSA, AES, and the Caesar Cipher.