> For the complete documentation index, see [llms.txt](https://whitepaper.opz.io/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://whitepaper.opz.io/keyfusion/key-concepts-and-techniques/key-generation.md). # Key Generation A two-party method can be used to generate a public key for elliptic-curve signing (such as ECDSA or EdDSA). This two-party approach can be used to generate a public key $$Q$$ without either party knowing the private key $$x$$. Here's how it works: * **Concept of Elliptic-Curve Keys**: The private key $$x$$ corresponds to a public key $$Q$$, defined as $$Q=x⋅G$$, where $$G$$ is the generator of the elliptic-curve group. * **Objective**: The goal is for party $$1P1$$ to hold a share $$1x1$$, and party $$2P2$$ to hold a share $$2x2$$, with the condition $$2x=x1+x2$$. **The Process of Two-Party Key Generation** * **Initial Step**: Each party independently selects a random value: $$1P1$$ chooses $$1x1$$ and computes $$Q1=x1⋅G$$, and $$2P2$$ chooses $$2x2$$ and computes $$Q2=x2⋅G$$. * **Exchange and Calculation**: $$1P1$$ and $$2P2$$ exchange $$1Q1$$ and $$2Q2$$ respectively, and each then defines $$2Q=Q1+Q2$$. By elliptic-curve properties, $$Q=(x1+x2)⋅G=x⋅G$$, thus generating the public key without exposing $$x$$. **Addressing Security Concerns** * **Problem with Initial Approach**: If $$2P2$$ is corrupt, it could bias $$Q$$ by waiting for $$1Q1$$ from $$1P1$$ and then choosing $$2Q2$$ to manipulate $$Q$$. * **Solution - Commitment Scheme**: To prevent this, $$1P1$$ sends a commitment to $$1Q1$$, essentially a cryptographic "envelope" that hides $$1Q1$$ but binds $$1P1$$ to it. After $$2P2$$ sends $$2Q2$$, $$1P1$$ reveals $$1Q1$$. * **Ensuring Randomness and Security**: This method ensures that $$1Q1$$ and $$2Q2$$ are chosen independently. If one party is honest, the result is random and secure. Neither party knows $$x$$, as it's only additively shared between them. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## 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, and the optional `goal` query parameter: ``` GET https://whitepaper.opz.io/keyfusion/key-concepts-and-techniques/key-generation.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. 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.