Privacy-Preserving Identity-Based Undeniable Signatures from Lattices with Controlled Verifiability for IoT Networks

. 2026 Sep 30;26(19):6205. doi: 10.3390/s26196205

Algorithm 2 KeyExt.

Input:

p p

, and

I D_{i}

Output: Keys pair

(P k, S k)

1:
KGC calculates $a_{I D_{i}} = H_{1} (I D_{i}) \in Z_{q}^{n \times m}$ ;
2:
Samples $s_{I D_{i}} = S a m p l e M a t (A, T, σ, a_{I D_{i}}) \in Z_{q}^{m \times m}$ , where $A s_{I D_{i}} = a_{I D_{i}} m o d q$ , $| | s_{I D_{i}} | | \leq σ \sqrt{m}$ , and $σ \geq | | \tilde{T} | | w (\sqrt{l o g m})$ ;
3:
Randomly chooses $s d \in {0, 1}^{*}$ as a seed;
4:
Calculates $L = H_{2} (s d) \in Z_{q}^{n \times m}$ ;
5:
Randomly selects $x \in D_{σ}^{m}$ ;
6:
Calculates $X = L x m o d q$ ;
7:
Returns $(P k, S k) = {(a_{I D_{i}}, L, X), (s_{I D_{i}}, x)}$ back to user $I D_{i}$ .