By Hu Xiong, Zhen Qin, Athanasios V. Vasilakos

As an intermediate version among traditional PKC and ID-PKC, CL-PKC can stay away from the heavy overhead of certificates administration in conventional PKC in addition to the main escrow challenge in ID-PKC altogether. because the creation of CL-PKC, many concrete structures, protection types, and functions were proposed over the last decade. Differing from the opposite books out there, this one presents rigorous therapy of CL-PKC.

Definitions, particular assumptions, and rigorous proofs of protection are supplied in a way that makes them effortless to appreciate.

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If these properties hold, then we say the group (G, ◦G ) and the group (H, ◦H ) are isomorphic and write this as G ∼ = H. An isomorphism from a group to another group provides an alternate and equivalent approach to think about the structure of groups. For example, if the group (G, ◦G ) is finite and G ∼ = H, then the group (H, ◦H ) must be finite and have the same order as G. Also, if there exists an isomorphism f from a group (G, ◦G ) to another group (H, ◦H ), then f −1 is an isomorphism from (H, ◦H ) to (G, ◦G ).

C/a is defined as c × a−1 mod N ). We stress that division by a is only defined when a is invertible. If c × a = b × a mod N and a is invertible, then we may divide each side of the equation by a (or, equivalently, multiply each side by a−1 ) to obtain (c × a) × a−1 = (b × a) × a−1 mod N ⇒ c = b mod N We see that in this case, division works “as expected” by adopting the idea of invertible integers. The natural question is that which integers are invertible modulo a given modulus N ? 5 Let a, N be integers with N > 1.

M − 1}. 9 Let (G, ◦) be a finite group with order m, and the element g ∈ G features with the order i. Then i|m. 1, g m = 1. 7, we have g m = g [m mod i] when element g features with the order i. If def i m, then i = [m mod i] is a nonzero integer smaller than i such that g i = 1. This contradicts the fact that i is the order of the element g. 3 If (G, ◦) is a group with prime order p, then this group is cyclic. Furthermore, all elements of G other than the identity element are generators of (G, ◦).