MPC 2-n

The 2-n protocol is based on the Lindell-17 2-2 protocol, and modified by OKX to support more than 2 parties.

Lagrange interpolation

The Lagrange interpolation formula is a method to find a unique polynomial that passes through a given set of points. The formula is as follows:

Given n+1 points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$ , the Lagrange interpolation polynomial $P(x)$ is expressed as:

$$P(x) = \sum_{i=0}^{n} y_i \cdot L_i(x)$$

where $L_i(x)$ is the Lagrange basis polynomial, defined by:

$$L_i(x) = \prod_{\substack{0 \leq j \leq n \\ j \ne i}} \frac{x - x_j}{x_i - x_j}$$

Each term $L_i(x)$ is constructed by taking the product of all other points, excluding the current point $(x_i, y_i)$ .

Try moving the points around to see how the polynomial changes:

SSS

SSS, or Shamir's Secret Sharing, is a cryptographic protocol that allows a secret to be split into multiple shares, such that the secret can only be reconstructed when a sufficient number of shares are combined. The protocol is designed to be secure even if some of the shares are compromised.

It was invented by Adi Shamir in 1979 in his paper How to share a secret.

SSS is based on polynomial interpolation, where the secret is the y-intercept of the polynomial. The shares are the y-values of the polynomial at different x-values.

VSS

VSS, or Verifiable Secret Sharing, is a cryptographic protocol that allows a secret to be split into multiple shares, such that the secret can only be reconstructed when a sufficient number of shares are combined. The protocol is designed to be secure even if some of the shares are compromised.

It was invented by Feldman in 1987.

VSS is similar to SSS, but with the additional property that the shares can be verified to be correct. This is done by attaching a proof to each share, which can be used to verify that the share is consistent with the other shares.

DKG

Distributed Key Generation (DKG) is a cryptographic protocol that allows a group of parties to generate a shared public key without revealing their private keys. The protocol is designed to be secure even if some of the parties are malicious.

Recovery

Escape

TSS