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# 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).
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