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Ethereum: In Elliptic Curve Addition
As we dive deeper into the world of blockchain and cryptocurrency, it’s essential to understand the underlying technology that makes it all work. One of the critical components of Ethereum is elliptic curve addition, which plays a vital role in its security and scalability.
What is Elliptic Curve?
Before we proceed, let’s take a brief detour into the world of elliptic curves. An elliptic curve is a type of mathematical curve that’s used to represent points on a two-dimensional plane with coordinates (x, y). These coordinates are defined by a pair of numbers (a, b) called the “y-intercept” and “x-intercept,” respectively.
Elliptic Curve Operations
In the context of cryptography, elliptic curves provide a secure way to perform operations such as addition, subtraction, multiplication, and exponentiation. These operations are performed using the curve’s built-in algorithms, which ensure that the results are secure and resistant to tampering.
EC Multiplication
When it comes to Elliptic Curve (EC) multiplication, there are two main methods: point addition and point doubling. Point addition involves adding two points on the elliptic curve, while point doubling is used for faster computations when multiple points need to be multiplied together.
Point addition is performed using a specific formula that takes into account the coordinates of the two points being added. This formula ensures that the result lies within the elliptic curve, ensuring its security.
EC Addition
In Ethereum, EC addition refers to the process of adding two points on an elliptic curve. To perform this operation, we need to use a specific algorithm that involves:
- Point addition: We add two points, P and Q, using the formula
P + Qin point addition.
- Point doubling: If multiple points are being multiplied together, we can use point doubling to reduce the number of computations required.
Where Does the Second Point Come From?
In EC addition, the second point is generated using a specific algorithm that ensures it lies within the elliptic curve. The exact formula for generating the second point depends on the type of elliptic curve being used (e.g., Sei-Kun-Mukai (SKM) or J-de Brion-Soule-Henry-Whorl-Elliot (J-DEH-HE)).
For SKM, the second point is generated as Q = P + y A, where A is an element of the elliptic curve. For J-DEH-HE, the second point is generated as Q = P - x B, where B is another element of the elliptic curve.
In both cases, the algorithm ensures that the second point lies within the elliptic curve and follows a specific set of rules to ensure its security.
Conclusion
Ethereum’s EC addition is a critical component of its cryptographic infrastructure. By understanding how EC multiplication and additional work, we can appreciate the complexity and sophistication behind Ethereum’s underlying technology. While this article has only scratched the surface of elliptic curves in Ethereum, I hope it provides a solid foundation for further exploration and learning.
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