Provably secure attribute-based encryption with attribute revocation and grant function using proxy re-encryption and attribute key for updating

Naruse, Takeru; Mohri, Masami; Shiraishi, Yoshiaki

doi:10.1186/s13673-015-0027-0

Table 1 Key Size, Ciphertext Size and Computation Amount

From: Provably secure attribute-based encryption with attribute revocation and grant function using proxy re-encryption and attribute key for updating

	Yu etal’s scheme [ 3 ]	Hur et al’s scheme [ 4 ]	The proposed scheme
PK	\( \begin{array}{l}\left(3\left\|U\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	2 × \|G\| + \|G _T\|	\( \begin{array}{l}\left(2\left\|U\right\|+2\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)
SK	(2\|U\| + 1) × \|G\|	\( \begin{array}{l}\left(2\left\|S\right\|\kern0.5em +1\right)\times \left\|G\right\|\\ {}+ \log \kern0.5em \left\|N\right\|\times \left\|K\right\|\end{array} \)	(\|S\| + 2) × \|G\|
CT	\( \begin{array}{l}\left(\left\|U\right\|+1\right)\\ {}+\left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	\( \begin{array}{l}\left(2\left\|I\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	\( \begin{array}{l}\left(2\left\|I\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)
RK	r\|U\| × \|Z _p\|	(2\|N\| − 1) × \|K\|	r\|U\| × \|X _p\|
Enc	(\|U\| + 2) × exp	(2\|I\| + 2) × exp	(2\|I\| + 2) × exp
Ext	(2\|U\| + 1) × exp	(2\|S\| + 2) × exp	(\|S\| + 2) × exp
Re-enc	\|R _CT\| × exp	\|R _CT\| × exp	\|R _CT\| × exp
Re-key	\|R _SK\| × exp	\|R _SK\| × exp	\|R _SK\| × exp
Dec	\( \begin{array}{l}\left(\left\|U\right\|+1\right)\times \widehat{e}\\ {}+\left(\left\|U\right\|1\right)\kern0.5em \times \kern0.5em exp\end{array} \)	\( \begin{array}{l}\left(2\left\|R\right\|+1\right)\times \widehat{e}\\ {}+\left(2\left\|R\right\|+2\right)\times exp\end{array} \)	\( \begin{array}{l}\left(2\left\|R\right\|+1\right)\times \kern0.5em \widehat{e}\\ {}+\left(2\left\|R\right\|+2\right)\times \kern0.5em exp\end{array} \)

Exp:ex ponentiation in G, ê: bilinear pairing,
|U|: the number of attributes defined in the system,
|S|: the number of attributes in user’s key,
|R| the number of user’s attributes satisfying an acces structure,
r: the number of times the attribute revocation event occurs,
|R _SK|: the number of updated attributes (secret key),
|R _CT|: the numbet of updated attributes (ciphertext), |N|: the number of total user,
|I|: the number of attributes am acces structure, |K|: size of the common key.

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	Yu etal’s scheme [ 3 ]	Hur et al’s scheme [ 4 ]	The proposed scheme
PK	\( \begin{array}{l}\left(3\left\|U\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	2 × \|G\| + \|G _T\|	\( \begin{array}{l}\left(2\left\|U\right\|+2\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)
SK	(2\|U\| + 1) × \|G\|	\( \begin{array}{l}\left(2\left\|S\right\|\kern0.5em +1\right)\times \left\|G\right\|\\ {}+ \log \kern0.5em \left\|N\right\|\times \left\|K\right\|\end{array} \)	(\|S\| + 2) × \|G\|
CT	\( \begin{array}{l}\left(\left\|U\right\|+1\right)\\ {}+\left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	\( \begin{array}{l}\left(2\left\|I\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)	\( \begin{array}{l}\left(2\left\|I\right\|+1\right)\\ {}\times \left\|G\right\|+\left\|{G}_T\right\|\end{array} \)
RK	r\|U\| × \|Z _p\|	(2\|N\| − 1) × \|K\|	r\|U\| × \|X _p\|
Enc	(\|U\| + 2) × exp	(2\|I\| + 2) × exp	(2\|I\| + 2) × exp
Ext	(2\|U\| + 1) × exp	(2\|S\| + 2) × exp	(\|S\| + 2) × exp
Re-enc	\|R _CT\| × exp	\|R _CT\| × exp	\|R _CT\| × exp
Re-key	\|R _SK\| × exp	\|R _SK\| × exp	\|R _SK\| × exp
Dec	\( \begin{array}{l}\left(\left\|U\right\|+1\right)\times \widehat{e}\\ {}+\left(\left\|U\right\|1\right)\kern0.5em \times \kern0.5em exp\end{array} \)	\( \begin{array}{l}\left(2\left\|R\right\|+1\right)\times \widehat{e}\\ {}+\left(2\left\|R\right\|+2\right)\times exp\end{array} \)	\( \begin{array}{l}\left(2\left\|R\right\|+1\right)\times \kern0.5em \widehat{e}\\ {}+\left(2\left\|R\right\|+2\right)\times \kern0.5em exp\end{array} \)