Schnorr signature

\[\begin{align*} &X = xG\\ \\ &R = rG\\ &c = H(X,R,m)\\ \\ &s = r + cx\\ &sig = (R,s)\\ \\ &sG = R + cX\\ \end{align*}\]

参考BIP340

MuSig

https://eprint.iacr.org/2018/068.pdf

n-of-n signature

rouge key attack

3-round, H(Ri) pre-commitment

\[\begin{align*} &X_{i} = x_{i}G \\ &L = H(X1, ..., Xn) \\ \\ &a_{i} = H(L, X_{i}) \\ &X = Σa_{i}X_{i} \\ \\ &R_{i} = r_{i}G \\ &R = ΣR_{i} \\ \\ &c = H(X, R, m)\\ &s_{i} = r_{i} + ca_{i}x_{i} \\ &s = Σs_{i} \\ \\ &round 1: H(R_{i}) \\ &round 2: R_{i} \\ &round 3: s_{i} \\ \end{align*}\]

参考BIP327

MuSig2

https://eprint.iacr.org/2020/1261.pdf

2-round, almost 1-round

omdl

\[\begin{align*} &R_{i}' = r_{i}'G \\ &R' = ΣR_{i}' \\ \\ &R_{i}'' = r_{i}''G \\ &R'' = ΣR_{i}'' \\ \\ &R_{i} = R'_{i} + bR''_{i}\\ &R = ΣR_{i}\\ \\ &b=H(X, R', R'', m)\\ &r_{i}=r'_{i}+br''_{i}\\ \\ &c = H(X, R, m)\\ &s_{i}=r_{i}+ca_{i}x_{i}\\ &s = Σs_{i}\\ \\ &round 1: R_{i}', R_{i}''\\ &round 2: s_{i}\\ \end{align*}\]

MuSig-DN

https://eprint.iacr.org/2020/1057.pdf

Deterministic Nonce

2-round, nizk proof

\[\begin{align*} &sk_{i} \rightarrow x_i, u_i, k_i \\ \\ &host~key: U_{i} = u_{i}G\\ &for~NIZK~proof: k_i \\ \\ &K = \{ (X_i, U_i) | 1 \leq i \leq n \} \\ &V = H(K, m)\\ \\ &r_i = f(u_{i}V)\\ &R_i = r_{i}G\\ &R = ΣR_i\\ \\ &c = H(X, R, m)\\ \\ &s_i = r_i + ca_{i}x_{i}\\ &s = Σs_i\\ \\ &NIZK~proof: (Bulletproofs)\\ &ρ_i = RandDer(k_i, (K, m))\\ &π_i = Π.Prv(crs, (U_i, V, R_i), u_i; ρ_i)\\ \\ &round 1: R_i, π_i\\ &round 2: s_i\\ \end{align*}\]