Table 1. Notations and symbols.

Notation	Definition
$i{d}_i$	The identifier of the file $F_i$
M	The number of files
N	The number of keywords
$\mathcal{L}(\cdot )$	A bit-length of $\cdot$
$|\cdot |$	Number of elements in set $\cdot$
$S{L}_{w_1}^i{|}_{|A|}$	Get the first |A| bits of $S{L}_{w_1}^i$
$S{L}_{w_1}^i{|}_{-(|A|-|B|)}$	Get the last $(|A|-|B|)$ bits of $S{L}_{w_1}^i$
$\tilde{w}$	The query phrase
R	A set of ciphertext satisfying phrase search
$proof$	Verification result, $1$: valid, $0$: invalid
||	Concatenation symbol, $a||b$ denotes the concatenation of message a and b.
$r_{i,j}$	Number of positions of keyword $w_i$ in file $F_j$