Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

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

Algorithm 2 Products of optimal Ate pairings on the BN curve (Product method).

Input:

s = 6 t + 2

m =

the bit length of s,

P_{j} \in G_{1}

Q_{j} \in G_{2}

, where j is

1, \dots, d

Output:

e_{p r o d} (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:
$f \leftarrow 1$
3:
for $j \leftarrow 1$ tondo
4:
$T_{j} \leftarrow Q_{j}$
5:
end for
6:
for $i \leftarrow m - 2$ down to 0 do
7:
$f \leftarrow f^{2}$
8:
for $j \leftarrow 1$ to d do
9:
$f \leftarrow f \cdot L_{T_{j}, T_{j}} (P_{j}), T_{j} \leftarrow [2] T_{j}$
10:
if $s [i] = 1$ then
11:
$f \leftarrow f \cdot L_{T_{j}, Q_{j}} (P_{j}), T_{j} \leftarrow T_{j} + Q_{j}$
12:
else if $s [i] = - 1$ then
13:
$f \leftarrow f \cdot L_{T_{j}, - Q_{j}} (P_{j}), T_{j} \leftarrow T_{j} - Q_{j}$
14:
end if
15:
end for
16:
end for
17:
for $j \leftarrow 1$ toddo
18:
$R \leftarrow π (Q_{j}), f \leftarrow f \cdot L_{T_{j}, R} (P_{j}), T_{j} \leftarrow T_{j} + R$
19:
$R \leftarrow π^{2} (Q_{j}), f \leftarrow f \cdot L_{T_{j}, - R} (P_{j}), T_{j} \leftarrow T_{j} - R$
20:
end for
21:
$f \leftarrow f^{(p^{12} - 1) / r}$
22:
returnf