Please review proxy re-signature on Elliptic Curve [closed]

Question

I want to implement proxy re-signature on elliptic curve.

I've been thinking about ideas like the one below, but are there any problems?

Key Generate:

• $a = $ alice's secret key
• $aG$ = alice's public key
• $b = $ bob's secret key
• $bG = $bob's secret key
• $rk_{ab} = aP * b^{-1} = a/bG$

First Sign:

• $Pm$ is hashed point
• $k = $ random
• $r = ka^{-1}$
• $z = e(G, G)$
• $s = z^kPm$

Resign:

• $r' = rk_{ab} * r = a/bP * ka^{-1} = k*1/b*G$
• $s' = s$

Verification:

• $t = e(bG, r) = e(G, G)^k = z^k$
• Check $tPm == s'$