Table 9.

Complexity analysis in terms of cumulative circuit depth2 (CCD) and the number of homomorphic multiplications (HMs) for secure approximation division (Algorithm 3).

Algorithm 3	CCD	# of HMs
1: Compute $\widehat{X}$	$⌊\mathsf{\text{log}}\left(d-1\right)⌋+1$	d − 1
2: Compute ${\widehat{X}}^{\prime }$	$⌊\mathsf{\text{log}}\left(C-2\right)⌋+⌊\mathsf{\text{log}}\left(d-1\right)⌋+2$	C − 2
3:For c = 0,1, ⋯, C − 1
4: For i = 0,1, ⋯, d − 1
5: Calculate ${\widehat{h}}_{cd+i}{\widehat{x}}^i$	−	−
6: end for
7: end for
8: $\widehat{a}=\widehat{0}$
9: For c = 0,1, ⋯, C − 1
10: ${\widehat{a}}^{\prime }=\widehat{0}$	−	−
11: For i = 0,1, ⋯, d − 1
12: ${\widehat{a}}^{\prime }={\widehat{a}}^{\prime }+{ĥ}_{cd+i}{\widehat{x}}^i$	−	−
13: end for
14: $\widehat{a}=\widehat{a}+{\widehat{a}}^{\prime }{\widehat{x}}^{cd}$	$⌊\mathsf{\text{log}}\left(C-2\right)⌋+⌊\mathsf{\text{log}}\left(d-1\right)⌋+3$	1
15: end for
Total:	$⌊\mathsf{\text{log}}\left(C-2\right)⌋+⌊\mathsf{\text{log}}\left(d-1\right)⌋+3$	2C + d − 3