MultiSig

n~n

GIVEN: a account $(p,P)$, 3 user $(x_1,X_1)$, $(x_2,X_2)$, $(x_3,X_3)$. Where $X=(A,B)$ and $x=(a,b)$.

Compute $X'=X_1+X_2+X_3$ and $x'=x_1+x_2+x_3$, $a'$ is shared to the 3 user.
Send $P$ to $(X')$, the new account is $P'=H_s(rA')G+B'$, where $r$ is a random scalar and $R$ is published to all. This new account can be spent iff $p'$, which is $p'=H_s(rA')+b'$ or $p'=H_s(a'R)+b'$.
Every user computes a partial key image $J_1=b_1H_p(P')$, $J_2$, $J_3$, the key image is $J=H_s(a'R)H_p(P')+J_1+J_2+J_3$.
Set $P'$ the $s$-th account of $P_N$, such that $P'=P_s$.
Every user picks a random scalar $u_1$, $u_2$, $u_3$, compute $u=u_1+u_2+u_3$.
Randomly choose scalar $s_i$ for $i\neq s$, compute:
1. $L_s=uG, R_s=uH_p(P_s), c_{s+1}=H_s(m,L_s,R_s)$
2. $L_{s+1}=s_{s+1}G+c_{s+1}P_{s+1}, R_{s+1}=s_{s+1}H_p(P_{s+1}), c_{s+2}=H_s(m,L_{s+1},R_{s+1})$
3. ...
4. $L_{s-1},R_{s-1},c_s$
Every user computes $s_{s,1}=u_1-c_sb_1$, $s_{s,2}$ and $s_{s,3}$, which are shared.
Compute $s_s=s_{s,1}+s_{s,2}+s_{s,3}-c_sH_s(a'R)=u-c_s(b'+H_s(a'R))$.