Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

. 2021 Apr 19;21(8):2859. doi: 10.3390/s21082859

Algorithm 1 Optimal Ate pairing on the BN curve.

Input:

s = 6 t + 2

m =

the bit length of s,

P \in G_{1}

Q \in G_{2}

Output:

e (P, Q)

1:
Write s in signed binary form, $s = \sum_{i = 0}^{m - 1} s [i] 2^{i}$ with $s [i] \in {- 1, 0, 1}$
2:
$T \leftarrow Q, f \leftarrow 1$
3:
for $i \leftarrow m - 2$ down to 0 do
4:
$f \leftarrow f^{2} \cdot L_{T, T} (P), T \leftarrow 2 T$
5:
if $s [i] = 1$ then
6:
$f \leftarrow f \cdot L_{T, Q} (P), T \leftarrow T + Q$
7:
else if $s [i] = - 1$ then
8:
$f \leftarrow f \cdot L_{T, - Q} (P), T \leftarrow T - Q$
9:
end if
10:
end for
11:
$R \leftarrow π (Q), f \leftarrow f \cdot L_{T, R} (P), T \leftarrow T + R$
12:
$R \leftarrow π^{2} (Q), f \leftarrow f \cdot L_{T, - R} (P), T \leftarrow T - R$
13:
$f \leftarrow f^{(p^{12} - 1) / r}$
14:
returnf