Table 1 Notations

(Pk_O,Sk_O)

Data owner’ keys

(Pk_R,Sk_R)

Receiver’ keys

(Pk_S,Sk_S)

Server’keys

κ

Security parameter

C_w

Keyword ciphertext

\(T_{w^{\prime }}\)

Search trapdoor

\(\mathcal {C},\mathcal {A}\)

Challenger and adversary

\(\mathcal {O}\)

Oracles

g,h,g₁

Generators of group G₁

G₁,G₂

Multiplicative cyclic groups with order q

q

A large prime number

H₁

A hash function \(H_{1}:G_{1} \rightarrow Z_{q}^{*}\)

H₂

A hash function \(H_{2}:\{0,1\}^{*} \rightarrow Z_{q}^{*}\)

pp

A global parameter

E

The time cost of one modular exponentiation

H_p

The time cost of one Hash-to-point operation

P

The time cost of one pairing operation

M

The time cost of one multiplication operation

H

The time cost of one hash function operation

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

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 \(\mathbb {Z}_{q}\)