Semigroup-homomorphic Signature

2025-03-21

AI Summary

This work resolves a long-standing open problem in cryptography—constructing secure homomorphic signatures over semigroups—first posed in 2002. We present the first computationally efficient, compact, and tightly secure homomorphic signature scheme for arbitrary finite semigroups, with a rigorous security proof under the standard Short Integer Solution (SIS) assumption on lattices. The scheme supports arbitrary combinations of signatures under semigroup operations while guaranteeing functional correctness. Furthermore, we extend it to linear semigroup homomorphic signatures, incorporating privacy-preserving mechanisms such as signature randomization and unlinkability. Unlike prior homomorphic signature schemes restricted to groups—requiring invertibility—our construction operates over general semigroups, thus eliminating the need for inverses. This breakthrough establishes the first efficient, secure, and privacy-aware homomorphic signature framework for non-invertible algebraic structures, enabling authenticated computation over broader classes of algebraic objects and opening a new paradigm for cryptographic authentication in semigroup-based settings.

Technology Category

• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

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.

Problem

• Construct secure semigroup-homomorphic signature scheme
• Prove security based on SIS problem hardness
• Extend to privacy-preserving lattice-based linear scheme

Innovation

• First semigroup-homomorphic signature scheme
• Security based on SIS problem hardness
• Extends to lattice-based linear scheme