# Discrete Logarithm Problem with Elliptic Curves
Consider an elliptic curve $$E$$, with generator $$G$$ and another element $$T$$. Then DLP is finding $$d$$, such that
> $$G+G+G+...+G$$ $$(d \ times)$$ $$= d\*G = T$$
To understand the practical implications, we visualize this geometrically on the curve $$E$$ :- we first find generator $$G$$ on $$E$$, then add $$G$$ to it to locate $$2G$$.
This is considered the first hop (computationally, it is executing the group operation). Then, we add $$G$$ again to mark the value of $$3G$$, which is the second hop. By repeating this process, $$d$$ times, we finally reach point $$T$$ on the curve $$E$$.
As it turns out, finding the number of hops is computationally hard.
So, the starting point is known for elliptic curves, i.e., the generator $$G$$, the final point is known, $$T$$, then DLP is essentially to find the number of hops taken to reach $$T$$ from $$G$$.
IMPORTANT Note - for elliptic curves, we use the discrete logarithm problem as:
> $$d$$ is private key and $$T$$ is the public key\
> $$d=K\_{pr}$$ - an integer, which is the number of hops in above explanation\
> $$T = dG = K\_{pub}$$ This is a point on curve $$E$$, i.e., group element, i.e., it has an $$x$$ and $$y$$ co-ordinate so $$T = ( x\_T, y\_T )$$
As we define $$d$$ as the private key and $$T$$ as the public key, the practical implication of the above is that given a known public key, it is very hard to compute the private key.
*Terminology note* - Generator is also commonly referred to as "base point", so $$G$$ is a base point for curve $$E$$.
**Credits**
Special thanks to Prof. Christof Paar for providing a fantastic [Introduction to Cryptography](https://www.youtube.com/channel/UC1usFRN4LCMcfIV7UjHNuQg).
---
# Agent Instructions: Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.
Perform an HTTP GET request on the current page URL with the `ask` query parameter:
```
GET https://hub.bsvblockchain.org/higher-learning/bsv-academy/digital-signatures/ecdsa-prerequisites/discrete-logarithm-problem-with-elliptic-curves.md?ask=
```
The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.
Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.