Computability Clause Samples

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 ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (CDH) problem and the k-Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ 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, ..., ▇, ▇ + 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 .

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 ▇▇▇▇▇▇-▇▇▇▇▇▇▇ problem. For instance, symmet- ric pairings lead to a strict separation between the intractability of the Computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ problem and the hardness of the corresponding de- cision problem. The security of our proposal is based on the hardness of the computational ▇▇▇▇▇▇- ▇▇▇▇▇▇▇ (CDH) problem, Divisible computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ and K-Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ expo- nent, which are described below: • Computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (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]. 1. The Discrete Logarithm Problem (DLP): given P, Q ∈ G1, the DLP in G1 is to find an integer n, such that Q = n ╳ P, whenever such an integer exists.

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 ▇▇▇▇ 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 ▇. ▇▇▇▇ on the other hand, always reaches its optimal value.

Computability. There is an efficient algorithm to compute  for all  ∈ .

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 ▇▇▇▇ [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 ▇.