Table 3 Computation and Communcitaion efficiency comparison

Huang and Li [8]

H_p+3E

P+H_P+E

2P+M

2\(\vert \mathbb {G}_{1} \vert \)

\(\vert \mathbb {G}_{2} \vert \)

\(\vert \mathbb {G}_{1} \vert \)

Li et al. [12]

2P+2H_p+3E+H

P+2H_p+2E+M

2P+2E+M

2\(\vert \mathbb {G}_{1} \vert \) + \(\vert \mathbb {G}_{2} \vert \)

2\(\mathbb {G}_{1}\)

*

Baek et al. [3]

2P+H_p+2E+M+H

H_p+E

P+M+E+H

\(\vert \mathbb {G}_{1} \vert \,+\,\vert q \vert \)

\(\vert \mathbb {G}_{2} \vert \)

2\(\vert \mathbb {G}_{1} \vert \)

Rhee et al. [4]

P+H_p+2E+H

H_p+3E+M

P+H_p+2E

\(\vert \mathbb {G}_{1} \vert \)

2\(\vert \mathbb {G}_{1} \vert \)

2\(\vert \mathbb {G}_{1} \vert \)

Pan and Li [10]

H_p+3E+H

P+H_p+3E

2P+M

2 \(\vert \mathbb {G}_{1} \vert \)

2\(\vert \mathbb {G}_{1} \vert \) + \(\vert \mathbb {G}_{2} \vert \)

\(\vert \mathbb {G}_{1} \vert \)

Our Scheme

P+4E+2H

4E+2H+M

2P+M+E

2\(\vert \mathbb {G}_{1} \vert \,+\,\vert \mathbb {G}_{2} \vert \)

2\(\vert \mathbb {G}_{1} \vert \)

\(\vert \mathbb {G}_{1} \vert \)