Table 2 Symbols and execution times(ms)

E

The time cost of one modular exponentiation

0.784

H_p

The time cost of one Hash-to-point operation

5.679

P

The time cost of one pairing operation

5.429

M

The time cost of one multiplication operation

2.108

H

The time cost of one hash function operation

0.008

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

the bit length of an element in \(\mathbb {G}_{1}\)

-

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

the bit length of an element in \(\mathbb {G}_{2}\)

-

|q|

the bit length of an element in \(\mathbb {Z}_{q}\)

-