Table 5 Comparison of parameters’ size.

Scheme	Public key	Secret key	Ciphertext
Duong et al.²⁴	\(((l + 3)mn + nk)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((4m + 2k)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Roy et al.²⁶	\(4mn + nk\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((n^{2} + 4m + 2k)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Xiao et al.²⁷	\(2mn\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2(m - nk^{\prime})nk^{\prime}\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2(m + k^{\prime})\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Ours	\((3mn + nk)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((2m + 2k + 1)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)

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Scheme	Public key	Secret key	Ciphertext
Duong et al.²⁴	\(((l + 3)mn + nk)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((4m + 2k)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Roy et al.²⁶	\(4mn + nk\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((n^{2} + 4m + 2k)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Xiao et al.²⁷	\(2mn\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2(m - nk^{\prime})nk^{\prime}\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(2(m + k^{\prime})\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)
Ours	\((3mn + nk)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\(m^{2} \left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)	\((2m + 2k + 1)\left\| {{\mathbf{\mathbb{Z}}}_{q} } \right\|\)

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