We note the number of nodes that have a secret share, and the threshold of the network. Each node is identified by their long term public key and have a secret key .
We call the set of qualified node that have successfully ran the DKG, i.e. each node in have a partial share corresponding to the distributed secret key and public key where is the generator of the group (without indices means we refer to the "keys of the DKG").
We call the threshold network as Generalized Threshold.
We call the Lagrange basis polynomials the following:
such that and with
When not specified otherwise, we will use a pairing equipped elliptic curve of type III. Namely:
- There are three groups of order , with associated generators .
- There exists an efficiently computable bilinear map .
- We will place the key on the group: but that is not fixed.
Specifically for our current deployment, Medusa uses the bn254 family of curves.