. 2025 Aug 4;11:e3068. doi: 10.7717/peerj-cs.3068

Table 4. Efficiency analysis.

Scheme	Sig.comp	Ver.comp	AS
Boneh et al. (2009)	$(m + n + 1) T_{E_{1}} + (m + n - 1) T_{M_{1}} + m T_{H_{1}}$	$2 c T_{P} + (m + n c) T_{E_{1}} + (m + n c - 1) T_{M_{1}} + m T_{H_{1}}$	✓
Lin et al. (2018)	$(m + 4) T_{E_{1}} + (m + 1) T_{M_{1}} + (m + 1) T_{H_{1}}$	$3 c T_{P} + (c m + c n + 1) T_{E_{1}} + (c m + c n) T_{M_{1}} + (m + n + 1) T_{H_{1}} + c T_{E_{3}} + c T_{M_{3}}$	$\times$
Chang et al. (2019)	$(m + n + 3) T_{E_{1}} + (m + n + 1) T_{M_{1}} + (m + n) T_{H_{1}}$	$3 c T_{P} + (c m + c n + 1) T_{E_{1}} + (c m + c n) T_{M_{1}} + (m + n + 1) T_{H_{1}} + c T_{E_{3}} + c T_{M_{3}}$	$\times$
Lin, Xue & Huang (2021)	$T_{P} + (m + n + 1) T_{E_{1}} + (m + n) T_{M_{1}} + (m + 2) T_{H_{1}}$	$4 c T_{P} + (m + n c) T_{E_{1}} + (m + n c - 1) T_{M_{1}} + (m + 2 c) T_{H_{1}} + c T_{E_{3}} + c T_{M_{3}}$	✓
Li, Zhang & Liu (2022)	$(m + 2) T_{E_{1}} + m T_{M_{1}} + m T_{H_{1}}$	$2 c T_{P} + (m + c) T_{E_{1}} + (m + c - 1) T_{M_{1}} + m T_{H_{1}} + T_{E_{2}} + T_{M_{2}}$	✓
Our scheme	$(m + 3) T_{E_{1}} + (m + 1) T_{M_{1}} + (m + 1) T_{H_{1}}$	$2 c T_{P} + (m + 2 c) T_{E_{1}} + (m + 2 c - 1) T_{M_{1}} + T_{E_{2}} + T_{M_{2}}$	✓

Note:

AS, using asymmetric bilinear pairing operations.