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 5 Confirmation/Disavowal

Input:

p p

, and e
Output: Confirmation or Disavowal

1:
Signer and verifier calculate L and X;
2:
Chooses a permutation r over $(1, 2, \dots, m)$ ;
3:
Chooses a vector $π \in D_{σ}^{m}$ ;
4:
Calculates $F_{1} = H_{5} (r | | L π m o d 2 q)$ ;
5:
Calculates $F_{2} = H_{5} (r (π))$ ;
6:
Calculates $F_{3} = H_{5} (r (π + x))$ ;
7:
Calculates $F_{4} = H_{5} (r | | U π m o d 2 q)$ ;
8:
Sets the commitments $(F_{1}, F_{2}, F_{3}, F_{4})$ ;
9:
Verifier sends a challenge $f \in {1, 2, 3}$ to signer;
10:
Signer answers $(r, π)$ for $f = 1$ challenge;
11:
Answers $(r, π + x)$ for $f = 2$ challenge;
12:
Answers $(r (π), r (x))$ for $f = 3$ challenge;
13:
Verifier checks the validity of $F_{1}$ , $F_{2}$ , and $F_{4}$ for $f = 1$ challenge;
14:
Checks the validity of $F_{1}$ , $F_{2}$ , and $F_{4}$ for $f = 2$ challenge;
15:
If $F_{4} = H_{5} (r | | U (π + x) - e_{3})$ , outputs confirmation;
16:
else $F_{4} \neq H_{5} (r | | U (π + x) - e_{3})$ , outputs disavowal;
17:
Checks the validity of $F_{2}$ , $F_{3}$ , and $| | r (e) | | \leq β \sqrt{m}$ for $f = 3$ challenge;
18:
Outputs “Confirmation” when verifications are all successful;
19:
Otherwise, outputs “Disavowal”.