Table 2 Summary of the Notations used in the Algorithm 2

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

Multiplicative groups of prime order p

g

Generator polynomial of \(\mathbb {G}_{1}\)

H(·)

Secure map-to-point hash function

h(·)

hash function maps cluster element of \(\mathbb {G}_{1}\) consistently to Z_p

tag_F

Tag of file F

Pk

Public key

Sk

Secret key

τ₁

Signature on block b₁

n

Total number of chunks in shared data

S

Shared information

c

Total number of customers in a cluster

d_1,i

Information proprietor of 1^st cluster

CL

Customer List

b_k

k^th block

id_k

k^th block identifier

E

Subset of q random blocks

τ_Re−key

Re-sign key

ρ_k\(\phantom {\dot {i}\!}{}^{(\tau _{Re-key})}\)

Re-Signature on revoked customer’s k^th block

Public parameters

(v_d, w_d, g_d, J_d)

\(F_{d_{1},i}\)

File owned by information proprietor (d₁) of i^th cluster

ϕ

Set of signatures on entire chunks in distributed information.

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

One exponentiation in \(\mathbb {G}_{1}\)

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

One multiplication in \(\mathbb {G}_{1}\)

Pair

Pairing operation on e: \(\mathbb {G}_{1}\) * \(\mathbb {G}_{2}\) →\(\mathbb {G}_{\mathrm {T}}\)

\({m}-{MulExp}^{t}_{\mathbb {G}}\)

t m term exponentiations \(\phantom {\dot {i}\!}\sum _{i=1}^{m} g^{a_{i}}\)