In this section, we describe the signing process of the protocol. For any set \(S \in \{1,\dots,n\}\) of \(t+1\) participants who participate to sign a message \(M\), let \(w_i=\lambda_{i,S}\cdot sk_i \pmod{p}\). Note that by Feldman's VSS, \(sk=\sum_{i \in S} w_i\). Note that since \(pk_i=g^{sk_i} \) is public after the key generation process, hence the value \(W_i=g^{w_i}=pk_i^{\lambda_{i,\mathcal{S}}}\) can also be publicly computed. The signing protocol follows a \(6\) steps process below:

Sign\((M)\langle \{P_i(sk_i)\}_{i=1}^n\rangle\):

  1. Each participant \(P_i\) chooses \(d_{i},e_{i} \in \mathbb{Z_p}\) and broadcasts \((D_{i},E_{i})=(g^{d_{i}},g^{e_{i}})\). Denote \(B=\{(i,D_i,E_i)\}_{i \in S}\).

  2. For each \(j \neq i\), each \(P_i\) uses Schnorr protocol (see Supporting Algorithms) to check the validity of \((D_i,E_i)\). If any check fails then the protocol aborts.

  3. Each \(P_i\) computes \(\rho_j=\mathsf{H}(j,M,B)\) for all \(j \in S\). Each \(P_i\) then computes the group commitment \(R=\prod_{j \in S} D_jE_j^{\rho_j}\) and the challenge \(c=\mathsf{H}(R,\mathsf{pk},M)\), then broadcasts \((\rho_i,R,c)\).

  4. Each \(P_i\) computes \(z_i=d_i+e_i\rho_i+\lambda_{i,S}\cdot \mathsf{sk_i} \cdot c\) and broadcasts \(z_i\).

  5. Each \(P_i\) computes \(R_i=D_iE_i^{\rho_i}\) and broadcasts \(R_i\).

  6. For each \(i\), participants check if \(R=\prod_{i\in S}R_i\) and \(g_i=R_i\mathsf{pk_i}^{c \lambda_{i,S}}\). If any check fails, report the misbehaving \(P_i\) and the protocol is aborted. Otherwise, compute \(z=\sum_{i \in S}z_i\) and returns \(\sigma=(R,z)\).