Identity-Based Proxy Signature with Message Recovery over NTRU Lattice

. 2023 Mar 4;25(3):454. doi: 10.3390/e25030454

Algorithm 4 Message recovery

1:
Choose $y_{1}, y_{2} \in D_{Z^{N}, σ}$
2:
Divide the message u into two parts $u = u_{1} ∥ u_{2}$ and make $| u_{1} | = l_{2}$ , if $| u | < l_{2}$ then let $u_{2} = ⊥$ .
3:
Compute $α = H_{2} (y_{1} + y_{2} h) .$
4:
Compute $u_{1}^{^{'}} = F_{1} (u_{1}) ‖ (F_{2} (F_{1} (u_{1})) ⨁ u_{1}) .$
5:
Compute $r = α ⨁ u_{1}^{^{'}} .$
6:
Compute $C = H_{3} (r, u_{2})$
7:
Compute $z_{i} = s_{i} C + y_{1}, z_{i + 1} = s_{i + 1} C + y_{2} .$
8:
if Nothing is outputted then
9:
$Restart$
10:
end if
11:
return $(u_{2}, z_{i}, z_{i + 1})$ on message m with probability $m i n (\frac{D_{Z^{N}, σ}}{M D_{Z^{N}, σ, s k u}}, 1)$ , where $M = O (1) .$