Algorithm 1: Proposed digital signature algorithm.

Input: n,${\mathcal{V}}_{ID}$, ${RSU}_{ID}$

Output: Vehicles key pairs, i.e., public and private, the master secret key of RSU (${RSU}_{msk}$)

1.Global public key component $p:\mathrm{prime}\mathrm{number}{2}^{L-1}<p<2^L$   2.Deployed public parameters $g,p,a$ and b   3.Choose random number $x\in (1,2,3\cdots q-1)$   4.Computes vehicle private key $V_{Prk}=(x·g)$   5.Again choose random number $y\in (1,2,3\cdots q-1)$   6.Computes vehicle public key $V_{Puk}=(y·g)$   7.Computes SDN controller master secret key ${SDN}_{msk}=k·g$ mod p   8.Where k: any integer number ($0<k<q$)

Signature:

For all registered vehicles and RSU do

1.Assign unique ${ID}_i$ to each registered vehicles such as $V_{ID}$, RSU as ${RSU}_{ID}$   2.Computes pseudo ID of vehicle as H (${\mathcal{V}}_{ID}$ || K|| nj), $j=1,2,3\cdots n$   3.Computes $C:y^2+h\left(x\right)y=f\left(x\right)$   4.Computes keys pairs of RSU such as ${RSU}_{prk}=(x·g⨁\mathrm{Nonce})$   5.Computes $r_{sx}=r_{{sx}_{i-1}}⨁H\left(V_{r_{i-1}}\right)$   6.Applied the master key of SDN controller for signature   7.Computes public key of SDN controller such as ${SDN}_{puk}=({SDN}_{msk}·g⨁\mathrm{Nonce})$   8.Computes $R=K·g\mathrm{mod}q$   9.Computes $S=\left[{\mathrm{K}}^{-1}(H\left(M\right)+x·R)\right]\mathrm{mod}q$

End for

Verifying:

1.$V=[(g^{u1}·y^{u2})modp]\mathrm{mod}q$   2.$U_1=\left[H\left(M^{\prime }\right)\omega \right]\mathrm{mod}q$   3.$S=K^{-1}(H\left(M\right)+x·R)$   4.$R=K·g\mathrm{mod}q$   5.$\omega ={\left(S^{\prime }\right)}^{-1}$   6.$U2=\left[\left(R^{\prime }\right)\omega \right]\mathrm{mod}q$   7.$V=R^{\prime }$