The private key $k^n_i$ is generated as:

$k^n_i = \text{Hash}{\mathbf{KeyGen}}(\text{'ed25519'},K{\text{role}}, R_e, i) \mod q$

Where:

• $K_{\text{role}}$ is the secret role key obtained from the SDP protocol.
• $R_e$ is the epoch randomness, a predefined randomness source for epoch $e$.
• $i$ is the $iterator$ ensuring uniqueness when multiple keys are generated.
• $q$ is the order of the Ed25519 curve, ensuring a valid scalar for Ed25519.