Bilinear Pairing. × −→ Let G1 be a group of the order of a large prime number q and G2 be a multiplicative subgroup of a finite field F of the same order and P be a generator of G1. A map e : G1 G1 G2 is called a bilinear map if it has the following properties [21]: Bilinearity: e(aP, bQ) = e(P, Q)ab where P, Q ∈ G1 and a, b ∈ Zq∗; Non-degeneracy: P, Q ∈ G1, such that e(P, Q) 1; ∈
Bilinear Pairing. Let G1 and G2 be cyclic additive and multiplicative groups of prime order q, respectively. The generator of G1 is g1. Let e : G1 × G1 → G2 be a bilinear pairing, which satisfies the following properties: • Bilinearity: ∀P, Q ∈ G1 and ∀a, b ∈ Zq∗, e(aP, bQ) = e(P, bQ)a = e(aP, Q)b = e(P, Q)ab are satisfied. • Non-degenerate: ∀P, Q ∈ G1 such that e(P, Q) ƒ= 1. • Computable: for all P, Q ∈ G1, there is always an effective algorithm to compute e(P, Q). The security of our protocol is based on the following computationally infeasible problems. ∗ • Elliptic Curve Discrete Logarithm problem (ECDL): Let a ∈ Zq∗, given P, aP ∈ G1, and compute a. • Computational Diffie–Xxxxxxx problem (CDH): Let a, b ∈ Zq , given g1, ag1, and bg1, and find abg1. • Decisional Diffie–Xxxxxxx problem (DDH): Let a, b, c ∈ Zq∗, given g1, ag1, bg1, and cg1, and decide if e(ag1, bg1) = e(g1, cg1).
Bilinear Pairing. This subsection gives some preliminaries of bilinear pairing and its properties. Let G and Gτ be two groups of prime order q and let P be a generator of G, where G is additively represented and Gτ is multiplicatively. A map e: G × G → Gτ is said to be a bilinear pairing and the group G is called a bilinear group, if the following three properties hold:
Bilinear Pairing. × → G1 is an cyclic additive group and G2 is a cyclic multiplicative group with same order q. Assume that discrete logarithm problem (DLP) is hard in both G1 and G2. A mapping e : G1 G1 G2 which satisfies the following properties is called a bilinear pairing from a cryptographic point of view:
