# ECDH Key Exchange

The [**ECDH**](https://en.wikipedia.org/wiki/Elliptic-curve_Diffie–Hellman) (Elliptic Curve Diffie–Hellman Key Exchange) is **anonymous key agreement scheme**, which allows two parties, each having an elliptic-curve public–private key pair, to establish a **shared secret** over an insecure channel. **ECDH** is very similar to the classical **DHKE** (Diffie–Hellman Key Exchange) algorithm, but it uses **ECC point multiplication** instead of **modular exponentiations**. ECDH is based on the following property of EC points:

* (***a*** \* **G**) \* ***b*** = (***b*** \* **G**) \* ***a***

If we have two **secret numbers** ***a*** and ***b*** (two **private keys**, belonging to Alice and Bob) and an ECC elliptic curve with generator point **G**, we can exchange over an insecure channel the values (***a*** \* **G**) and (***b*** \* **G**) (the **public keys** of Alice and Bob) and then we can derive a shared secret: ***secret*** = (***a*** \* **G**) \* ***b*** = (***b*** \* **G**) \* ***a***. Pretty simple. The above equation takes the following form:

* alicePubKey \* bobPrivKey = bobPubKey \* alicePrivKey = ***secret***

The **ECDH** algorithm (Elliptic Curve Diffie–Hellman Key Exchange) is trivial:

1. **Alice** generates a **random** ECC key pair: {**alicePrivKey**, **alicePubKey** = alicePrivKey \* G}
2. **Bob** generates a **random** ECC key pair: {**bobPrivKey**, **bobPubKey** = bobPrivKey \* G}
3. Alice and Bob **exchange their public keys** through the insecure channel (e.g. over Internet)
4. **Alice** calculates **sharedKey** = bobPubKey \* alicePrivKey
5. **Bob** calculates **sharedKey** = alicePubKey \* bobPrivKey
6. Now both **Alice** and **Bob** have the same **sharedKey** == bobPubKey \* alicePrivKey == alicePubKey \* bobPrivKey

In the next section, we shall implement the ECDH algorithm and demonstrate it with code example.


---

# 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://cryptobook.nakov.com/asymmetric-key-ciphers/ecdh-key-exchange.md?ask=<question>
```

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.