Mathematical Background Sample Clauses
Mathematical Background. Let G1 be a cyclic additive group of prime order q , and G2 be a cyclic multiplicative group of the same order q . We let P denote the generator of G1 . A bilinear pairing is a map properties:
(1) Bilinearity e : G1 × G1 → G2 which satisfies the following e(aQ, bR) = e(Q, R)ab , where Q, R ∈ G1 , a, b ∈ Z * .
(2) Non-degeneracy e(P, P) ≠ 1 .
(3) Computability There is an efficient algorithm to compute e(Q, R) for all Q, R ∈ G1 . The Weil and ▇▇▇▇ pairings associated with supersingular elliptic curves or abelian varieties can be modified to create such admissible pairings, as in [9]. The following problems are assumed to be intractable within polynomial time. Definition 1 (Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (BDH) problem). Let G1 , G2 , P and e be as above. The BDH problem in < G1 , G2 , e > is as follows: Given < P , aP , bP , cP > with uniformly random choices of a, b, c ∈ Z * , compute 2 e(P, P)abc ∈ G . Definition 2 (Computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (CDH) problem). Let G1 and P be as above. The CDH problem in G1 is as follows: Given < P , aP , a, b ∈ Z * , compute abP ∈ G1 .
Mathematical Background. Definition (Decisional ▇▇▇▇▇▇-▇▇▇▇▇▇▇ Problem (DDHP)): As stated in [13], p and q are large prime numbers, where p = 2q + 1. Let g 2 Z⇤p be a generator of some cyclic group of quadratic residues Gp in Z⇤p. The two statements below are computationally indifferent from each other. • hga, gb, gabi, where a and b are integers randomly and independently selected from Zp. • hga, gb, gci, where a, b and c are integers randomly and independently selected from
Mathematical Background. Definition (Decisional ▇▇▇▇▇▇-▇▇▇▇▇▇▇ Problem (DDHP)): As stated in [13], p and q are large prime numbers, where p = 2q + 1. Let g ∈ Z∗p be a generator of some cyclic group of quadratic residues Gp in Z∗p. The two statements below are computationally indifferent from each other. • ⟨ga, gb, gab⟩, where a and b are integers randomly and independently selected from Zp. • ⟨ga, gb, gc⟩, where a, b and c are integers randomly and independently selected from Zp.
Mathematical Background