Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

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

Algorithm 6 Adaptive 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_{k}^{'}, Q_{k + 1}, \dots, Q_{d})

∣ Q_{j}^{'}

is the precomputation tuple for

Q_{j} \in G_{2}, 1 < k \leq 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 $j \leftarrow 1$ tokdo
4:
$c n t [j] \leftarrow 1$
5:
end for
6:
for $j \leftarrow k + 1$ toddo
7:
$T_{j} \leftarrow Q_{j}$
8:
end for
9:
for $i \leftarrow m - 2$ down to 0 do
10:
$f \leftarrow f^{2}$
11:
for $j \leftarrow 1$ to k do
12:
$G_{D B L}, G_{A D D} \leftarrow Q_{j}^{'}$
13:
$λ, c \leftarrow G_{D B L} [i]$
14:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
15:
$f \leftarrow f \cdot g$
16:
if $s [i] \neq 0$ then
17:
$λ, c \leftarrow G_{A D D} [c n t [j]]$
18:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
19:
$c n t [j] \leftarrow c n t [j] + 1$
20:
$f \leftarrow f \cdot g$
21:
end if
22:
end for
23:
for $j \leftarrow k + 1$ to d do
24:
$f \leftarrow f \cdot L_{T_{j}, T_{j}} (P_{j}), T_{j} \leftarrow [2] T_{j}$
25:
if $s [i] = 1$ then
26:
$f \leftarrow f \cdot L_{T_{j}, Q_{j}} (P_{j}), T_{j} \leftarrow T_{j} + Q_{j}$
27:
else if $s [i] = - 1$ then
28:
$f \leftarrow f \cdot L_{T_{j}, - Q_{j}} (P_{j}), T_{j} \leftarrow T_{j} - Q_{j}$
29:
end if
30:
end for
31:
end for
32:
for $j \leftarrow 1$ tokdo
33:
${π^{'}}_{Q}, {(π^{'})}_{Q}^{2} \leftarrow Q_{j}^{'}$
34:
$λ, c \leftarrow {π^{'}}_{Q}$
35:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
36:
$f \leftarrow f \cdot g$
37:
$λ, c \leftarrow {(π^{'})}^{2}_{Q}$
38:
Compute $g \leftarrow (y_{P} + λ x_{P} + c)$
39:
$f \leftarrow f \cdot g$
40:
end for
41:
for $j \leftarrow k + 1$ toddo
42:
$R \leftarrow π (Q_{j}), f \leftarrow f \cdot L_{T_{j}, R} (P_{j}), T_{j} \leftarrow T_{j} + R$
43:
$R \leftarrow π^{2} (Q_{j}), f \leftarrow f \cdot L_{T_{j}, - R} (P_{j}), T_{j} \leftarrow T_{j} - R$
44:
end for
45:
$f \leftarrow f^{(p^{12} - 1) / r}$
46:
returnf