The primary goal of cryptography is to encrypt data so that it can be transmitted over insecure communication channels. Cryptography has existed for over 4,000 years and different forms have been developed over time, but all with the same goal. Cryptography can be sorted into the following categories: symmetric and asymmetric cryptography. Elliptic curves, which are the main focus of this paper, are part of asymmetric encryption. They are determined by the equation: 𝑦^2 + 𝑎1𝑥𝑦 + 𝑎3𝑦 = 𝑥^3 + 𝑎2𝑥^2 + 𝑎4𝑥 + 𝑎6 although the shorter Weierstrass form is more often used: 𝑦^2 = 𝑥^3 + 𝑎𝑥 + 𝑏. To understand cryptography that uses elliptic curves, it is important to know the addition of points on elliptic curves, which can be divided into adding a point to itself (point doubling) or adding two different points (point addition). The cryptography of elliptic curves is based on the discrete logarithm problem, that is, how to find 𝑛 so that 𝑄 = 𝑛𝐺, where 𝑛𝐺 represents the addition of point 𝐺 to itself n times (point doubling). With the advancement of quantum computers, elliptic curve cryptography will no longer be secure due to the easy solution of the discrete logarithm problem. There are organizations involved in standardizing new cryptographic systems that would be resistant to quantum computers.