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An Undeniable Signature Scheme Utilizing Module Lattices

Kunal Dey, Mansi Goyal, Bupendra Singh, Aditi Kar Gangopadhyay

TL;DR

This work develops a module lattice-based post-quantum undeniable signature system based on the GPV framework utilizing module lattices, with the security assured by the hardness of the SIS and LWE problems.

Abstract

An undeniable signature scheme is type of digital signature where the signer retains control over the signature's verifiability. Therefore with the approval of the signer, only an authenticated verifier can verify the signature. In this work, we develop a module lattice-based post-quantum undeniable signature system. Our method is based on the GPV framework utilizing module lattices, with the security assured by the hardness of the SIS and LWE problems. We have thoroughly proved all the desired securities for the proposed scheme. Finally, we have implemented our protocol for different sets of parameters. The purpose of opting a module variant rather than a ring variant is to provide greater flexibility in selecting parameters.

An Undeniable Signature Scheme Utilizing Module Lattices

TL;DR

This work develops a module lattice-based post-quantum undeniable signature system based on the GPV framework utilizing module lattices, with the security assured by the hardness of the SIS and LWE problems.

Abstract

An undeniable signature scheme is type of digital signature where the signer retains control over the signature's verifiability. Therefore with the approval of the signer, only an authenticated verifier can verify the signature. In this work, we develop a module lattice-based post-quantum undeniable signature system. Our method is based on the GPV framework utilizing module lattices, with the security assured by the hardness of the SIS and LWE problems. We have thoroughly proved all the desired securities for the proposed scheme. Finally, we have implemented our protocol for different sets of parameters. The purpose of opting a module variant rather than a ring variant is to provide greater flexibility in selecting parameters.

Paper Structure

This paper contains 19 sections, 11 theorems, 13 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Organization of the paper
  4. Preliminaries
  5. Gaussian on Lattices gentry2008trapdoorschi2015lattice
  6. Some Hard Problems in Lattice
  7. Hard Random Lattices
  8. Trapdoor Functions
  9. The GPV Signature
  10. Undeniable Signature Scheme
  11. Module Lattice
  12. Building Trapdoor Function for Module Lattice
  13. Gaussian Preimage Sampling
  14. Proposed Undeniable Signature Scheme on Module Lattice
  15. Security
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $m \geq 2n \log q$, where $q$ is a prime number and $n$ is a positive integer. Then, for all $s \geq \omega(\sqrt{\log m})$ and for any but a $2 q^{-n}$ fraction of matrices $\mathbf{A} \in \mathbb{Z}_q^{n \times m}$, the following holds:

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 8 more