Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semigroup-homomorphic Signature

Heng Guo, Kun Tian, Fengxia Liu, Zhiyong Zheng

TL;DR

The paper resolves the open problem of constructing secure semigroup-homomorphic signatures by introducing the first such scheme and a lattice-based linear extension. Security is established through tight reductions to the Short Integer Solution problem $SIS_{q,\\beta}$, supported by trapdoor lattices and discrete Gaussian sampling. The work also provides a privacy-preserving linear variant and discusses open challenges, including fixed-length signatures, alternative semigroups, standard EUF-CMA, and fully homomorphic signatures. Overall, it lays foundational work for semigroup-homomorphic cryptography and suggests several avenues for future research.

Abstract

In 2002, Johnson et al. posed an open problem at the Cryptographers' Track of the RSA Conference: how to construct a secure homomorphic signature on a semigroup, rather than on a group. In this paper, we introduce, for the first time, a semigroup-homomorphic signature scheme. Under certain conditions, we prove that the security of this scheme is based on the hardness of the Short Integer Solution (SIS) problem and is tightly secure. Furthermore, we extend it to a linear semigroup-homomorphic signature scheme over lattices, and this scheme can also ensure privacy.

Semigroup-homomorphic Signature

TL;DR

The paper resolves the open problem of constructing secure semigroup-homomorphic signatures by introducing the first such scheme and a lattice-based linear extension. Security is established through tight reductions to the Short Integer Solution problem , supported by trapdoor lattices and discrete Gaussian sampling. The work also provides a privacy-preserving linear variant and discusses open challenges, including fixed-length signatures, alternative semigroups, standard EUF-CMA, and fully homomorphic signatures. Overall, it lays foundational work for semigroup-homomorphic cryptography and suggests several avenues for future research.

Abstract

In 2002, Johnson et al. posed an open problem at the Cryptographers' Track of the RSA Conference: how to construct a secure homomorphic signature on a semigroup, rather than on a group. In this paper, we introduce, for the first time, a semigroup-homomorphic signature scheme. Under certain conditions, we prove that the security of this scheme is based on the hardness of the Short Integer Solution (SIS) problem and is tightly secure. Furthermore, we extend it to a linear semigroup-homomorphic signature scheme over lattices, and this scheme can also ensure privacy.

Paper Structure

This paper contains 17 sections, 20 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Semigroup-Homomorphic Signature: Definition and Security model
  3. Semigroup-Homomorphic Signature
  4. Linear Semigroup-Homomorphic Signature
  5. Tight security
  6. Primitives
  7. Semigroup homomorphic signature
  8. Construction
  9. Correctness
  10. Homomorphism
  11. Security
  12. A Linear Semigroup-Homomorphic Signature Scheme
  13. Construction
  14. Correctness
  15. Security
  16. ...and 2 more sections

Key Result

Theorem 1

(Worst-case to average-case reduction 23) For any polynomial bounded $m = \text{poly}(n)$, and any $\beta > 0$, if $q \geq \beta \cdot \omega(\sqrt{n\log n})$, then solving the average-case problem $\textbf{SIS}_{q,\beta}$ is at least as hard as solving the worst-case problem $SIVP_{\gamma}$ on any

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 31 more