Core Concepts

This paper proposes a method for constructing specific orthogonal bases in p-adic fields with large residue degrees to enhance the security of p-adic signature schemes against known attacks.

Abstract

Zhang, C., & Deng, Y. (2024). A Method of Constructing Orthogonal Basis in p-adic Fields. *arXiv preprint arXiv:2410.17982*.

This paper addresses the security vulnerability of a p-adic signature scheme proposed in 2021, stemming from the use of totally ramified extension fields. The authors aim to develop a method for constructing orthogonal bases in p-adic fields with large residue degrees to enhance the scheme's security.

The authors utilize the properties of p-adic fields, including residue degree, ramification index, and orthogonal bases, to develop a constructive method. They leverage the concept of cyclotomic cosets and properties of roots of unity to construct the orthogonal basis. They further provide an algorithm to find the minimal polynomial of a primitive element in the constructed field, enabling practical implementation.

- The paper proves a theorem providing an equivalent condition for an orthogonal basis in a p-adic field extension, linking it to the linear independence of basis elements over a finite field.
- It proposes a novel method to construct specific orthogonal bases in p-adic fields with large residue degrees, leveraging roots of unity and Eisenstein polynomials.
- The authors modify the existing p-adic signature scheme by incorporating the constructed orthogonal basis, enhancing its resistance against the identified attack.

The proposed method successfully constructs orthogonal bases in p-adic fields with desired properties, directly addressing the security flaw in the original p-adic signature scheme. This construction, based on well-established mathematical concepts, provides a concrete approach to improve the security of p-adic cryptographic schemes.

This research significantly contributes to the field of p-adic cryptography by providing a practical solution to a critical security concern. It paves the way for developing more secure and reliable cryptographic primitives based on p-adic lattices, potentially contributing to post-quantum cryptography.

While the paper provides a concrete construction, it lacks a formal security proof for the modified signature scheme. Further research is needed to analyze the computational hardness of underlying problems in p-adic fields and formally prove the security of the proposed scheme against various attack models. Additionally, exploring other applications of the orthogonal basis construction method in p-adic cryptography and beyond remains an open avenue.

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Stats

In 2021, the first p-adic signature scheme and public-key encryption cryptosystem were introduced.
These schemes have good efficiency but are shown to be not secure.
The attack succeeds because the extension fields used in these schemes are totally ramified.
In order to avoid this attack, the extension field should have a large residue degree.

Quotes

Key Insights Distilled From

by Chi Zhang, Y... at **arxiv.org** 10-24-2024

Deeper Inquiries

While the paper claims "good efficiency" for the modified p-adic signature scheme based on experimental results, a direct comparison to established post-quantum signature schemes (PQC) like Dilithium, Falcon, or SPHINCS+ is missing. The efficiency evaluation seems limited to internal comparisons with the previous, insecure version.
Here's a breakdown of potential efficiency factors and trade-offs:
Orthogonal Basis Construction: The algorithm relies on finding specific roots of unity, Eisenstein polynomials, and performing polynomial arithmetic over Qp. The computational complexity of these operations, especially for large prime numbers and extension fields, needs careful analysis and comparison with the underlying lattice operations in lattice-based PQC.
Signature Generation: Finding the closest vector (CVP) is generally a hard problem even in standard lattices. The paper leverages the constructed orthogonal basis to simplify this step. However, the efficiency gain compared to techniques like lattice basis reduction algorithms used in lattice-based PQC requires further investigation.
Signature Size and Verification: The paper doesn't provide details about the size of the signature (r, v). Comparing the signature size and the complexity of the verification step with other PQC schemes is crucial for a complete efficiency assessment.
Trade-offs:
Security Level: Increasing the security level in lattice-based PQC often involves increasing the lattice dimension or using larger moduli, directly impacting key and signature sizes. The p-adic scheme's security level is tied to the choice of primes (p, q) and the extension degree. Larger primes and degrees might lead to slower computations.
Implementation Complexity: Efficient implementations of lattice-based PQC often rely on optimized algorithms and data structures for lattice arithmetic. The p-adic scheme might require specialized algorithms for computations in p-adic fields, potentially leading to higher implementation complexity.
In conclusion, while the modified p-adic signature scheme presents an interesting approach, a thorough efficiency comparison with established PQC schemes, considering factors like key sizes, signature sizes, and computational complexity of all operations, is necessary. Furthermore, understanding the trade-offs between security level, efficiency, and implementation complexity is crucial for a comprehensive evaluation.

Yes, the reliance on specific mathematical structures like roots of unity and Eisenstein polynomials in constructing the orthogonal basis could potentially introduce vulnerabilities. Here's why:
Structure Exploitation: Attackers with specialized knowledge of these structures might be able to exploit their properties to develop attacks that are not apparent when considering the problem from a general lattice perspective. For example, they might discover relationships or weaknesses specific to the chosen roots of unity or the structure of Eisenstein polynomials that allow them to recover the private key or forge signatures more efficiently.
Hidden Subgroups: The use of roots of unity introduces a connection to the hidden subgroup problem (HSP), which has been extensively studied in the context of quantum algorithms. While the paper focuses on classical attacks, the presence of these structures might make the scheme susceptible to future quantum attacks, especially if efficient quantum algorithms for specific HSP instances are discovered.
Choice of Parameters: The security of the scheme relies heavily on the choice of the primes p and q, the Eisenstein polynomial G(x), and the random numbers ai. Poor choices or insufficient randomness in these parameters could create weaknesses exploitable by attackers. For instance, if the Eisenstein polynomial has a special form or the ai values are not truly random, it might allow an attacker to reduce the problem's complexity.
Mitigations:
Parameter Selection: Rigorous analysis and careful selection of parameters are crucial. Choosing parameters with provable security properties and ensuring sufficient randomness during key generation can mitigate some risks.
Security Proofs: Developing formal security proofs that consider the specific mathematical structures used in the scheme is essential. These proofs should demonstrate that breaking the scheme is at least as hard as solving well-studied hard problems, even when accounting for the additional structure.
Diversity of Approaches: Exploring alternative constructions of orthogonal bases that do not rely solely on these specific structures could increase confidence in the scheme's long-term security.
In summary, while the use of roots of unity and Eisenstein polynomials offers advantages in constructing the orthogonal basis, it also introduces potential vulnerabilities. Thorough security analysis, careful parameter selection, and the pursuit of diverse approaches are essential to mitigate these risks and ensure the scheme's robustness against attacks.

Yes, the connection between p-adic numbers and certain physical systems suggests intriguing possibilities for applying p-adic cryptography to secure communication protocols in areas like quantum information processing and cyber-physical systems:
Quantum Information Processing:
Error Correction: P-adic numbers have found applications in error correction codes, a fundamental aspect of quantum computing. Exploring p-adic cryptographic techniques in conjunction with p-adic error correction codes could lead to novel approaches for securing quantum information.
Quantum-Resistant Primitives: The search for quantum-resistant cryptographic primitives is crucial for securing future quantum communication networks. While the current state of p-adic cryptography doesn't guarantee quantum resistance, the unique properties of p-adic numbers might offer alternative avenues for developing such primitives.
Secure Control of Cyber-Physical Systems:
Signal Processing: P-adic analysis has shown promise in signal processing, particularly for systems with non-linear or chaotic behavior often found in cyber-physical systems. Integrating p-adic cryptographic techniques into these systems could enable secure communication and control, even in the presence of noise or adversarial disturbances.
Privacy-Preserving Control: P-adic encryption schemes, if proven secure and efficient, could be used to protect sensitive data in cyber-physical systems while still allowing for necessary computations. This is particularly relevant for applications like smart grids, healthcare systems, or autonomous vehicles, where privacy is paramount.
Challenges and Opportunities:
Bridging the Gap: A significant challenge lies in bridging the gap between the mathematical theory of p-adic numbers and the practical implementation of secure communication protocols in these domains.
Efficiency and Scalability: For real-world applications, p-adic cryptographic schemes need to be efficient and scalable to handle the demands of quantum information processing or complex cyber-physical systems.
Interoperability: Ensuring interoperability with existing communication protocols and standards is crucial for seamless integration.
In conclusion, while still in its early stages, the connection between p-adic numbers and physical systems presents exciting opportunities for applying p-adic cryptography to secure communication in quantum and cyber-physical domains. Further research is needed to overcome challenges and unlock the full potential of this approach for building secure and reliable systems of the future.

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