# Discrete Logarithm Problem with Elliptic Curves

### The Problem

Consider an **elliptic curve E**, with a **generator point G** and another point **T** on the curve.

The **Elliptic Curve Discrete Logarithm Problem (ECDLP)** is:

* Find **d** such that:

**d × G = T**

Here, **d** is an integer, and **T** is the result of adding **G** to itself **d times**.

***

### Visualizing the Process

We can picture this geometrically:

1. Start with **G** on curve **E**.
2. Add **G + G = 2G** → the **first hop**.
3. Add **G again** → **3G** (the second hop).
4. Continue this process **d times** until reaching **T**.

👉 The challenge is that while **G** (the starting point) and **T** (the endpoint) are known, finding the exact number of hops (**d**) is **computationally hard**.

<figure><img src="/files/75bQ029bJ7RYMyTt3W4p" alt=""><figcaption></figcaption></figure>

### Practical Implications

* In elliptic curve cryptography:
  * **Private key (Kₚᵣ)** = **d** (the number of hops).
  * **Public key (Kₚᵤᵦ)** = **T = d × G** (the point reached on the curve).
* Since **T** is a point on the curve, it has both **x** and **y** coordinates:\
  **T = (xₜ, yₜ)**
* Given **T (public key)**, it is extremely hard to compute **d (private key)**.

***

### Terminology

* The **generator G** is also called the **base point** of the elliptic curve.
* In practice:
  * Choose a secure elliptic curve **E**.
  * Select a base point **G**.
  * Private key = **d**.
  * Public key = **T = d × G**.

***

### Key Takeaway

The **Elliptic Curve Discrete Logarithm Problem (ECDLP)** underpins the security of elliptic curve cryptography:

* **Easy direction**: Compute **T = d × G**.
* **Hard direction**: Given **T** and **G**, recover **d**.

This one-way property is what makes elliptic curves secure for cryptographic applications like **ECDSA**.


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