Bilinear Pairings Clause Samples

Bilinear Pairings. Denote the additive cyclic group by G1 and the multiplicative group by G2, with both having high prime order q. Let P be a generator of G1. Then, the bilinear pairing e : G1 × G1 → G2 should satisfy the followings: • Bilinear: Given P1, P2, Q, Q2 ∈ G1, then e(P1 + P2, Q1) = e(P1, Q1)e(P2, Q1), e(P1, Q1 + Q2) = e(P1, Q1)e(P1, Q2) and e(aP1, bQ1) = e(abP1, Q1) = e(P1, abQ1) = e(bP1, aQ1) = e(P1, Q1)ab for any a, b ∈ Zq∗. • Nondegenerate: There exist P, Q ∈ G1, such that e(P, Q) /= 1, with 1 the identity element of G2. • Computable: For any P, Q ∈ G1, the value e(P, Q) is efficiently computed. The following related mathematical problems are considered. • The Elliptic Curve Discrete Logarithm Problem (ECDLP). This problem states that given two points R and Q of an additive group G, generated by an elliptic curve (EC) of order q, it is computationally hard for any polynomial-time bounded algorithm to determine a parameter x ∈ Zq∗, such that Q = xR. • The Elliptic Curve Diffie ▇▇▇▇▇▇▇ Problem (ECDHP). Given two points R = xP, Q = yP of an additive group G, generated by an EC of order q with two unknown parameters x, y ∈ Zq∗, it is computationally hard for any polynomial-time bounded algorithm to determine the EC point xyP.
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Bilinear Pairings. In recent years, the bilinear pairings have been found various applications in cryptography and have been used to construct some new cryptographic primi- tives [1, 4]. Let G1 be a cyclic additive group and G2 be a cyclic multiplicative group of the same prime order q. We assume that the discrete logarithm problems in both G1 and G2 are hard. A bilinear pairing is a map e : G1 G1 G2 which satisfies the following properties:
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Bilinear Pairings. In recent years, the bilinear pairings have been found various applications in cryptography and have been used to construct some new cryptographic primi- tives [1, 4]. Let G1 be a cyclic additive group and G2 be a cyclic multiplicative group of the same prime order q. We assume that the discrete logarithm problems in both G1 and G2 are hard. A bilinear pairing is a map e : G1 × G1 → G2 which satisfies the following properties: q 1. Bilinear: For any P, Q ∈ G1 and a, b ∈ Z∗, we have e(aP, bQ) = e(P, Q)ab. 2. Non-degenerate: There exists P ∈ G1 and Q ∈ G1 such that e(P, Q) /= 1.
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Related to Bilinear Pairings

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