Bilinear Pairing Sample Clauses

Bilinear Pairing. × → 1. Bilinearity: e(aP, bQ) = e(P, Q)ab for all P, Q ∈ G1 and a, b ∈ Zq∗. 2. Non-degeneracy: If a generator P ∈ G1 then e(P, P ) is a generator of G2, that is, e(P, P ) ƒ= 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) /= 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: 2.2.2.1 Bilinearity: e(aP, bP) = e(P, P)ab = e(P, abP) = e(abP, P) for all a, b ∈ Zq*;