# ECDSA Signing Process

The Elliptic Curve Digital Signature Algorithm (ECDSA) involves a two-party process where neither party knows the private key $$x$$, yet they can jointly create a signature on a message $$m$$. Here's an overview of the ECDSA signing function:

1. **Nonce Generation and Computation**:
   * A random nonce $$k$$ is chosen, and $$R=k⋅G$$ is computed.
   * The "x-coordinate" of the elliptic-curve point $$R$$ is denoted as $$r$$.
2. **Signature Computation**:
   * The signature component $$s$$ is calculated as $$s=k−1⋅(H(m)+r⋅x)modq$$.
3. **Signature Output**:
   * The signature is output as a pair $$(r,s)$$.

**Additively Homomorphic Encryption in ECDSA**

* **Encryption Type**: The process uses additively homomorphic encryption, which allows for encrypted values to be efficiently added or multiplied by a scalar without decryption. Paillier encryption, a common form of this encryption, is often used.

**Two-Party Computation Steps**

* **Key Generation**:
  * Parties choose random $$1x1$$ and $$2x2$$ and exchange $$Q1=x1⋅G$$ and $$Q2=x2⋅G$$. Commitment schemes are used for security.
  * $$1P1$$ generates a Paillier key-pair $$(pk,sk)$$ and sends $$ckey​=Encpk​(x1)$$ to $$2P2$$, along with a zero-knowledge proof.
* **Signature Generation on Message** $$m$$:
  * Parties independently generate $$1k1$$ and $$2k2$$, and exchange $$R1=k1⋅G$$ and $$R2=k2⋅G$$.
  * The nonce $$k$$ and $$R$$ are set as $$k=k1⋅k2$$ and $$R=k1⋅R2=k2⋅R1$$.
  * $$2P2$$ computes a "partial signature" using the homomorphic properties of the encryption, performing a series of encrypted operations.
  * $$1P1$$ decrypts the received ciphertext, multiplies it by $$1−1k1−1​$$ to obtain $$s$$, and verifies the signature $$(r,s)$$.

**Security Considerations**

* **Protection Against Private Key Exposure**: The protocol ensures neither party knows $$k$$ or the private key $$x$$.
* **Handling Corrupt Parties**: If $$2P2$$ is corrupt, it can provide incorrect encryption, but since $$1P1$$ verifies the signature first, no advantage is gained by $$2P2$$ from this.


---

# 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://whitepaper.opz.io/keyfusion/key-concepts-and-techniques/ecdsa-signing-process.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.