Computability. P, Q G1 there is an efficient algo- rithm to compute e(P, Q).
Computability. There is an efficient algorithm to compute e(P, Q) for all P, Q ∈ G1 . In this section, we generate the pair-wise key first which is used to compute the group session key afterward, then present our ID-based one round authenticated group key agreement protocol enlightened by the modified ID-based public key infrastructure described in section 3. Let U1 ,U 2 ,…,Un be the users who are going We note that the Weil [13] and Xxxx [23] pairings associated with supersingular elliptic curves or abelian varieties can be modified to create such bilinear maps.
Computability this means that there exists an efficient algorithm to compute e (P, P) ∀ P ∈ G. by X.
Computability. There is an efficient algorithm to compute for all ∈ .
Computability. There exists a polynomial time algorithm to compute e(P, Q), ∀P, Q ∈ G1. A bilinear map is defined as a probabilistic polyno- mial time algorithm (E) that takes a security pa- rameter k and returns a uniformly random tuple (G1, GT , e, g, q) of bilinear parameters, where g is the generator of G1 and e is the bilinear map. Consequences of Pairings. Pairings have important consequences on the hardness of certain variants of the Xxxxxx-Xxxxxxx problem. For instance, symmet- ric pairings lead to a strict separation between the intractability of the Computational Xxxxxx-Xxxxxxx problem and the hardness of the corresponding de- cision problem. The security of our proposal is based on the hardness of the computational Xxxxxx- Xxxxxxx (CDH) problem, Divisible computational Xxxxxx-Xxxxxxx and K-Bilinear Xxxxxx-Xxxxxxx expo- nent, which are described below: • Computational Xxxxxx-Xxxxxxx (CDH): Given g, gα, gβ for unknown α, β ∈ Zq, compute gαβ. •
Computability. There exists a polynomial time algorithm which can compute the value of e(P, Q) efficiently for all P, Q ∈ G1. For the details of the construction of such bilinear maps in a secure and efficient manner, please refer to [2, 3, 9, 12].
Computability. There is an efficient algorithm to compute ê(P, Q) for all P, Q ∈ G1. Note that the bilinearity of pairings also implies that ê : G1 × G1 G2 is symmetric. Thus, for any Q, R ∈ G1, the equality ê(Q, R) = ê(R, Q) holds. Both Q, R ∈ G1 can be represented by some generator P such that Q = aP and R = bP where a, b ∈ Z. Then it’s followed that ê(Q, R) = ê(aP, bP) = ê(P, P)ab = ê(bP, aP) = ê(R, Q). The map ê may be computed using a Weil pairing [41] or a Xxxx pairing [25] on an elliptic curve over Fq. In principle, the antisymmetry of the Weil pairing forces the two subgroups to be distinct. However, given a supersingular curve1 one may define a modified Weil pairing on a single subgroup of order q using distortion maps introduced by Verheul [59]. Distortion maps (also called endomorphisms) makes it possible to send points from one subgroup of the l-torsion to another. Of the two, the Weil pairing has simpler mathematic properties. However, it does not always reach the optimal value for x. Xxxx on the other hand, always reaches its optimal value.
Computability. There exists an efficient algorithm to compute eˆ(u, v) for any u, v ∈ G1. The security of our protocol is based on the hardness of the computational Xxxxxx-Xxxxxxx (CDH) problem and the k-Bilinear Xxxxxx-Xxxxxxx Exponent (BDHE) problem [5], which are as follows: CDH Problem: Given g, gα, gβ for unknown α, β ∈ Zq, compute gαβ. CDH Assumption: Let B be an algorithm which has advantage Adv(B) = Pr ΣB(g, gα, gβ) = gαβΣ in solving the CDH problem. The CDH assumption is that Adv(B) is negligible for any polynomial- time algorithm B. k-BDHE Problem: Given g, h, and yi = gα in G for i = 1, 2, ..., x, x + 2, ..., 2k as input, compute eˆ(g, h)αk . Since the input vector is missing the term gαk+1 , the bilinear map does not seem to help computing e(g, h)αk+1 .
