Common use of Bilinear Pairings Clause in Contracts

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 Xxxxxxx 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.

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) /= ƒ= 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 Xxxxxxx 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.