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9

I try to provide a brief intro. ABE Attribute-based encryption (ABE) is a relatively recent approach that reconsiders the concept of public-key cryptography. In traditional public-key cryptography, a message is encrypted for a specific receiver using the receiver’s public-key. Identity-based cryptography and in particular identity-based encryption (IBE) ...


3

This is a bilinear pairing used in cryptography. More precisely it's an evaluation of an appropriate pairing friendly elliptic curve equation.(1,2,3,4,5)


2

Surely bilinearity helps and and you have to be aware of the fact that $e(g^a,g^b)\cdot e(g^a,g^{-b})=e(g^a,g^b)\cdot e(g^a,g^b)^{-1}=1$. You see that from your last equation $$=e(g^{a\lambda_i}\cdot H(x)^{-v_ir_i},g^{u})\cdot e(g^{r_i}, g^{u't}\cdot H(x)^{v_i(u+\gamma)})\cdot e(g^{r_i}, g^{-tu'})\cdot e(g, H(x)^{v_i\gamma r_i})^{-1}$$ you get ...


2

There really isn't a difference. It is just author preference in notation. Some authors prefer to write the pairing operations multiplicatively $e(P^a, Q^b)=e(P,Q)^{ab}$ while others prefer to write it additively $e(aP,bQ)=e(P,Q)^{ab}$. This comes from the fact that in $e : \mathbb{G}_1\times \mathbb{G}_2\to\mathbb{G}_T$, $\mathbb{G}_1$ and $\mathbb{G}_2$ ...


2

If you need something that already exists, have a look at the advanced Crypto software collection and specifically cpabe — which implements ciphertext-policy attribute-based encryption scheme that uses C and PBC library for pairings.


2

I think you have a lack of knowledge on pairings and finite fields. Your definition of the pairing $e(X,Y)=g^{XY} \bmod p$ is not correct. A pairing is defined as a map $e : \mathbb{G}_1 \times \mathbb{G}_2 \to \mathbb{G}_T$ with the property \begin{align}\text{for all }g_1 \in \mathbb{G}_1 \text{ and } g_2 \in \mathbb{G}_2: e(g_1^a,g_2^b) = ...


2

CP-ABE fits naturally with RBAC, whereas KP-ABE not so much. Better analogies can be made if you think of attributes as "tags" of the encrypted object/document, instead of the users. For instance, imagine a confidential document about nuclear weapons which is encrypted under the attributes NUCLEAR and TOPSECRET. Then, only a user with a key for attributes ...


1

My understanding of this is as follows: Monotonic access structure: if $\mathbb{A}$ is a set of attributes satisfying an access structure $T$, then any $\mathbb{A}'$ such that $\mathbb{A} \subset \mathbb{A}'$ also satisfies $T$. For example, consider $T = A \cap B$, then both $\mathbb{A}=\{A,B\}$ and $\mathbb{A}'=\{A,B,C\}$ satisfy $T$. Non-monotonic ...


1

Simply speaking, if any superset of the set satisfying the access structure satisfies the access structure, we call the structure monotonic. Let $\{1,2,...,n\}$ be a set of indices. An access structure is a collection $\mathbb{A}$ of non-empty subsets of $\{1,2,3,...,n\}$. We say a collection (or an access structure) $\mathbb{A} \subseteq 2^{\{1,2,...,n\}}$ ...



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