ML-GS: Module Lattice-Based Group Signature Scheme

Notations	Description
$Z_{q}^{n}$	n-dimensional vector over modulo q
$Z_{q}^{n \times m}$	n × m matrix over modulo q
R	polynomial Z[[x] / (x^d + 1)
R_q	quotient ring Z[_q[x] / (x^d + 1)
x ← D	x is sampled from the distribution D
∥a∥	$\sqrt{\sum a_{i}^{2}}$
∥a∥_∞	$\max_{j} \| a_{j} \|$
∥w∥	$\sqrt{\sum {‖ w_{i} ‖}^{2}}$ , where w = (w₀, …., w)∈ R^k
∥w∥_∞	$\max_{j} {‖ w_{j} ‖}_{\infty}$ , where w = (w₀, …., w_k) ∈ R^k
⊥	the algorithm outputs fail
negl(λ)	a non-negligible function about λ
B	the set of integers representable by a single byte, i.e., {0,1,…,255}

Table 2.

The Parameters for ML-GS Scheme

Parameter	Recommended Parameters
Parameter	PARM I	PARM II
q	1073738753	1073738753
Q	3329	3329
n	256	256
(k, l)	(4,4)	(6,5)
η	83832	83290
η₁	3	3
τ	39	49
$γ_{1} \geq 11 τ \sqrt{k n} η$	≈ 2³¹	≈ 2³¹
BKZ blocksize b to break SIS	364	583
Classical security	106	170
Quantum security	96	154

Table 3.

Algorithm Notations and Execution Time (ms)

Notation	Description	Execution time
T_M–TrapGen	Trapdoor generation algorithm on module lattices	248.09
T_M–Sample	Trapdoor sampling algorithm on module lattices	31.10

Table 4.

The Time Cost for ML-GS Scheme

	Setup	KeyGen	Sign	Verify
PARM I	248.09	56.54	89.04	57.24
PARM II	426.40	96.12	179.67	114.48

Table 5.

The comparison of time cost

	KeyGen	Signature	Verify	Security
[12]	T_Sample	3dT_Mul + T_S	dT_Mul + T_S	CPA-Anonymity
[23]	T_Sample	7dT_Mul + T_S	2dT_Mul + T_S	CPA-Anonymity
[ours]	T_M–Sample	5dT_Mul	3dT_Mul	CCA-Anonymity

Time Cost Comparison of GS Schemes (ms).

Table 6.

Comparison of Storage Cost

	Public Key Size	Secret Key Size	Signature Size
[12]	48KB	6KB	92KB
[23]	268.5KB	68.7KB	386.1KB
Ours	44.9KB	7KB	18.0KB

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DOI: https://doi.org/10.2478/ias-2025-0007 | Journal eISSN: 1554-1029 | Journal ISSN: 1554-1010

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Language: English

Page range: 106 - 124

Published on: Dec 31, 2025

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

group signature,

lattice,

post-quantum cryptography,

module learning with error,

module short integer solution

Related subjects:

Computer sciences,

Fundamentals of computer sciences,

Theoretical computer sciences,

IT-security and cryptology

© 2025 Deng Pan, Yatao Yang, Weitao Sun, Shuaibo Wang, Ke Wang, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

Volume 20 (2025): Issue 3 (December 2025)

ML-GS: Module Lattice-Based Group Signature Scheme

Figures & Tables

Figure 1.

Table 1.

Figure 2.

Figure 3.

Figure 4.

Table 2.

Table 3.

Table 4.

Table 5.

Figure 5.

Figure 6.

Table 6.

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