Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

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

Algorithm 3 Fixed-Q precomputation.

Input:

s = 6 t + 2

m =

the bit length of s,

Q \in G_{2}

Output:

Q^{'} = (G_{D B L}, G_{A D D}, π_{Q}, {π^{2}}_{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:
$T \leftarrow Q, G_{D B L} \leftarrow {\emptyset}, G_{A D D} \leftarrow {\emptyset}$
3:
for $i \leftarrow m - 2$ down to 0 do
4:
Compute $λ and c$ , such that $y_{Q} + λ x_{Q} + c$ is the line tangent to T
5:
$T \leftarrow [2] T$
6:
Append $(λ, c)$ to $G_{D B L}$
7:
if $s [i] = 1$ then
8:
Compute $λ and c$ , such that $y_{Q} + λ x_{Q} + c$ is the line joining T and Q
9:
$T \leftarrow T + Q$
10:
Append $(λ, c)$ to $G_{A D D}$
11:
else if $s [i] = - 1$ then
12:
Compute $λ and c$ , such that $y_{- Q} + λ x_{- Q} + c$ is the line joining T and $- Q$
13:
$T \leftarrow T - Q$
14:
Append $(λ, c)$ to $G_{A D D}$
15:
end if
16:
end for
17:
$R \leftarrow π (Q)$
18:
Compute $λ and c$ , such that $y_{R} + λ x_{R} + c$ is the line joining T and R
19:
$π_{Q} \leftarrow (λ, c)$
20:
$R \leftarrow - π^{2} (Q)$
21:
Compute $λ and c$ , such that $y_{R} + λ x_{R} + c$ is the line joining T and R
22:
${π^{2}}_{Q} \leftarrow (λ, c)$
23:
$Q^{'} \leftarrow (G_{D B L}, G_{A D D}, π_{Q}, {π^{2}}_{Q})$
24:
return $Q^{'}$