Matrix Subcode Equivalence Problem Enables New Signature Scheme Construction

Equivalence problems underpin many modern cryptographic systems, forming the basis of digital signatures and other security protocols, and researchers continually seek new mathematical challenges to build even more robust encryption. Magali Bardet and Charles Brion, from LITIS at the University of Rouen Normandie, alongside Philippe Gaborit, Mercedes Haiech, and Romaric Neveu from XLIM at the University of Limoges, introduce two novel problems, the Matrix Subcode Equivalence Problem and the Matrix Code Permuted Kernel Problem, to advance this field. This work establishes that solving the Matrix Subcode Equivalence problem is computationally as difficult as a known, complex problem, and applies this to create a new digital signature scheme. The resulting signature scheme offers a significant improvement over existing methods like SPHINCS+ and CROSS, achieving a smaller signature and public key size, and thus contributing valuable diversity to the landscape of post-quantum cryptography

Shorter Signatures Through Dual Support Decomposition

This research investigates methods for constructing more efficient digital signature schemes, with a particular emphasis on resilience against attacks from quantum computers. The advent of quantum computing poses a significant threat to currently deployed public-key cryptography, necessitating the development of post-quantum cryptographic algorithms. A central objective within this field is to minimise the size of digital signatures while simultaneously maintaining acceptable computational performance; these two goals frequently involve inherent trade-offs. Dual Support Decomposition emerges as a promising technique for reducing signature size, complementing other approaches such as threshold computation and multi-party computation, all of which aim to enhance both security and practicality. Crucially, these methods leverage the underlying mathematical structure of hard problems, including Pinocchio Key-Private (PKP), Syndrome Decoding (SD), and Ring-SD, to create more compact signatures, and careful parameter selection plays a vital role in optimising performance.

Computational cost is inextricably linked to the efficiency of fundamental operations, including syndrome decoding, matrix manipulations, and polynomial arithmetic, all of which significantly impact overall performance. Syndrome decoding, a core component of code-based cryptography, involves recovering the original message from a corrupted codeword, and its complexity directly affects signature generation and verification times. Proof generation and verification, essential for establishing the authenticity and integrity of signatures, contribute substantially to the computational burden; while proofs of knowledge can reduce signature size by providing succinct evidence of key ownership, they often introduce additional computational complexity. Threshold schemes distribute the computational load across multiple parties, enhancing security and fault tolerance, but introduce communication overhead, requiring careful optimisation to minimise latency and bandwidth requirements. Key size, a critical factor in both storage and communication costs, is heavily influenced by the parameters chosen for the underlying code or mathematical structure, with larger parameters generally leading to larger keys, and the dimensions of the code directly impacting key size and computational complexity.

Balancing signature size and computational cost remains a significant challenge in designing efficient post-quantum signature schemes. The inherent trade-offs between these two objectives necessitate a holistic approach, considering the specific requirements of the application and the available resources. Improving the efficiency of syndrome decoding algorithms, such as utilising fast Fourier transforms or specialised hardware accelerators, is a critical area for optimisation. Reducing communication overhead in threshold schemes, through techniques like proactive secret sharing or efficient broadcast protocols, is equally important. Exploiting the structure of underlying cryptographic problems, for example, by utilising structured codes or carefully chosen parameters, offers substantial performance gains. Zero-knowledge proofs, while computationally intensive, can enhance both efficiency and security by allowing verification without revealing sensitive information, although their implementation requires careful consideration of computational costs. The choice of parameters, including code dimensions, field sizes, and error correction capabilities, is crucial for achieving optimal performance, and requires a thorough understanding of the underlying mathematical properties.

Dual Support Decomposition, the central technique explored in this research, builds upon the principles of lattice-based cryptography, a promising candidate for post-quantum security. This technique involves representing cryptographic keys and signatures as short vectors in a high-dimensional lattice, and exploiting the inherent difficulty of finding such vectors. By carefully decomposing the lattice into smaller, more manageable sublattices, the computational complexity of signature generation and verification can be significantly reduced. This decomposition allows for the use of more efficient algorithms for lattice operations, such as basis reduction and shortest vector problem solving. Furthermore, Dual Support Decomposition can be combined with other techniques, such as zero-knowledge proofs and threshold schemes, to further enhance security and efficiency. The effectiveness of this technique depends on the choice of lattice parameters, including the dimension, basis, and error distribution, and requires careful analysis to ensure both security and performance.

The implications of this research extend beyond theoretical cryptography, impacting a wide range of applications that rely on secure communication and data integrity. These include secure email, digital signatures for legal documents, secure software updates, and blockchain technology. As quantum computers become increasingly powerful, the need for post-quantum cryptography will become more urgent, and efficient signature schemes will be essential for maintaining the security of these applications. The development of compact and efficient post-quantum signature schemes will not only enhance security but also reduce bandwidth requirements and storage costs, making these technologies more accessible and practical. Ongoing research focuses on optimising the performance of these schemes, exploring new cryptographic techniques, and developing standardised protocols for their deployment. In summary, this research presents ongoing efforts to develop compact and efficient post-quantum signature schemes, actively addressing inherent trade-offs to achieve the best possible performance for secure communication in the future, and contributing to the broader field of post-quantum cryptography.