Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

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

Algorithm 5 Fixed-Q Product method.

Input:

s = 6 t + 2

m =

the bit length of s,

P = {(P_{1}, \dots, P_{d}) ∣ P_{j} \in G_{1}}

Q^{'} = {(Q_{1}^{'}, \dots, Q_{d}^{'}) ∣ Q_{j}^{'}

is the precomputation tuple for

Q_{j} \in G_{2}}

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 $j \leftarrow 1$ toddo
4:
$c n t [j] \leftarrow 1$
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:
$G_{D B L}, G_{A D D} \leftarrow Q_{j}^{'}$
10:
$λ, c \leftarrow G_{D B L} [i]$
11:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
12:
$f \leftarrow f \cdot g$
13:
if $s [i] \neq 0$ then
14:
$λ, c \leftarrow G_{A D D} [c n t [j]]$
15:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
16:
$c n t [j] \leftarrow c n t [j] + 1$
17:
$f \leftarrow f \cdot g$
18:
end if
19:
end for
20:
end for
21:
for $j \leftarrow 1$ toddo
22:
${π^{'}}_{Q}, {(π^{'})}_{Q}^{2} \leftarrow Q_{j}^{'}$
23:
$λ, c \leftarrow {π^{'}}_{Q}$
24:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
25:
$f \leftarrow f \cdot g$
26:
$λ, c \leftarrow {(π^{'})}^{2}_{Q}$
27:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
28:
$f \leftarrow f \cdot g$
29:
end for
30:
$f \leftarrow f^{(p^{12} - 1) / r}$
31:
returnf