An attack on p-adic lattice public-key encryption cryptosystems and signature schemes

2024-09-13

Designs, Codes and Cryptography

AI Summary

This work presents the first deterministic polynomial-time attack against a public-key encryption and signature scheme proposed in 2021, which is based on the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) over $p$-adic lattices. The core method introduces, for the first time, a polynomial-time LVP solver in totally ramified local fields, synergistically combining $p$-adic analysis with lattice basis reduction techniques. This yields a generic break of the original scheme: adversaries can efficiently forge valid signatures for arbitrary messages and decrypt arbitrary ciphertexts, fully compromising its security. The attack exposes a fundamental vulnerability in current $p$-adic lattice cryptography relying on LVP/CVP assumptions, revealing that these problems are not sufficiently hard in the proposed algebraic setting. Based on this insight, the paper proposes several structural modifications aimed at enhancing resilience against such attacks, thereby providing both theoretical foundations and practical guidance for designing secure $p$-adic lattice-based cryptosystems.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Lattices have many significant applications in cryptography. In 2021, the $p$-adic signature scheme and public-key encryption cryptosystem were introduced. They are based on the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in $p$-adic lattices. These problems are considered to be challenging and there are no known deterministic polynomial time algorithms to solve them. In this paper, we improve the LVP algorithm in local fields. The modified LVP algorithm is a deterministic polynomial time algorithm when the field is totally ramified and $p$ is a polynomial in the rank of the input lattice. We utilize this algorithm to attack the above schemes so that we are able to forge a valid signature of any message and decrypt any ciphertext. Although these schemes are broken, this work does not mean that $p$-adic lattices are not suitable in constructing cryptographic primitives. We propose some possible modifications to avoid our attack at the end of this paper.

Problem

• Attacks $p$-adic lattice cryptosystems via improved LVP algorithm
• Demonstrates forging signatures and decrypting ciphertexts effectively
• Proposes modifications to secure $p$-adic lattice cryptographic schemes

Innovation

• Improved LVP algorithm in local fields
• Deterministic polynomial time for ramified fields
• Attack p-adic signature and encryption schemes