Predicate encryption against master-key tampering attacks

Cybersecurity

Table 3 Theoretical analysis of TRF

Reference	Description	Assumption	Size of PP	Cost of evaluation
(Faust et al. 2014)^**	-	$f: \mathbb {Z}_{p}^{t} \times \mathbb {Z}_{p}^{t} \rightarrow \mathbb {Z}_{N}$	$t\|\mathbb {Z}_{p}\|$	-
(Qin et al. 2015)^***	DDH	$f: \mathbb {H}^{n\times n} \times \mathbb {Z}_{p}^{n} \rightarrow \mathbb {H}^{n}$	$2\|\mathbb {Z}_{p}\| +(n^{2}+1)\|\mathbb {H}\|$	2n²Exp
(Bellare and Cash 2010)	DDH	$f: \mathbb {Z}_{p}^{n+1} \times \{0, 1\}^{n} \rightarrow \mathbb {H}$	$(n+1)\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|$	1Exp
	DLIN	$f: (\mathbb {Z}_{p}^{2\times 2})^{n+1} \times \{0, 1\}^{n} \rightarrow \mathbb {H}$	$4(n+1)\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|$	1Exp
(Goyal et al. 2011)	q-DHI	$f: \mathbb {Z}_{p} \times \mathbb {Z}_{p} \rightarrow \mathbb {H}$	$\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|$	1Exp

^*$\mathbb {H}$ is a group of prime order p over $\mathbb {Z}_{N}$. $|\mathbb {Z}_{p}|$ and $|\mathbb {H}|$ denote the size of an element in $\mathbb {Z}_{p}$ and $\mathbb {H}$, respectively. Exp denotes a modular exponentiation in $\mathbb {H}$
^**Here we consider the simplest t-wise independent hash function $f = \sum _{i=0}^{t-1}a_{i}x_{i} \bmod p \bmod N$ where $\mathbf {a} \xleftarrow {\$} \mathbb {Z}_{p}$
^***Only partial PP of OT-LF is considered here

Reference	Description	Assumption	Size of PP	Cost of evaluation
(Faust et al. 2014)^**	-	\(f: \mathbb {Z}_{p}^{t} \times \mathbb {Z}_{p}^{t} \rightarrow \mathbb {Z}_{N}\)	\(t\|\mathbb {Z}_{p}\|\)	-
(Qin et al. 2015)^***	DDH	\(f: \mathbb {H}^{n\times n} \times \mathbb {Z}_{p}^{n} \rightarrow \mathbb {H}^{n}\)	\(2\|\mathbb {Z}_{p}\| +(n^{2}+1)\|\mathbb {H}\|\)	2n²Exp
(Bellare and Cash 2010)	DDH	\(f: \mathbb {Z}_{p}^{n+1} \times \{0, 1\}^{n} \rightarrow \mathbb {H}\)	\((n+1)\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|\)	1Exp
	DLIN	\(f: (\mathbb {Z}_{p}^{2\times 2})^{n+1} \times \{0, 1\}^{n} \rightarrow \mathbb {H}\)	\(4(n+1)\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|\)	1Exp
(Goyal et al. 2011)	q-DHI	\(f: \mathbb {Z}_{p} \times \mathbb {Z}_{p} \rightarrow \mathbb {H}\)	\(\|{\mathbb {Z}_{p}}\| + \|\mathbb {H}\|\)	1Exp