Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes

Chi Zhang

TL;DR

The paper addresses the security of $p$-adic lattice based cryptosystems for signatures and public-key encryption, which rely on the hardness of the Longest Vector Problem and Closest Vector Problem in local fields. It introduces a deterministic polynomial-time orthogonalization approach in totally ramified local fields and uses it to recover a uniformizer, build an orthogonal basis, and solve $LVP$/$CVP$, thereby forgoing signatures and decrypting ciphertexts. The contributions include a concrete attack with complexity advantages in the totally ramified case, a more efficient variant when $\gcd(n,p)=1$, a toy example illustrating the steps, and a discussion of countermeasures such as enforcing non-totally-ramified extensions (adding residue degree). The findings highlight a critical vulnerability in these schemes under the tested conditions and guide future design toward more robust constructions that resist orthogonalization-based attacks.

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.

An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes

TL;DR

The paper addresses the security of -adic lattice based cryptosystems for signatures and public-key encryption, which rely on the hardness of the Longest Vector Problem and Closest Vector Problem in local fields. It introduces a deterministic polynomial-time orthogonalization approach in totally ramified local fields and uses it to recover a uniformizer, build an orthogonal basis, and solve /, thereby forgoing signatures and decrypting ciphertexts. The contributions include a concrete attack with complexity advantages in the totally ramified case, a more efficient variant when , a toy example illustrating the steps, and a discussion of countermeasures such as enforcing non-totally-ramified extensions (adding residue degree). The findings highlight a critical vulnerability in these schemes under the tested conditions and guide future design toward more robust constructions that resist orthogonalization-based attacks.

Abstract

Lattices have many significant applications in cryptography. In 2021, the -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 -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 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 -adic lattices are not suitable in constructing cryptographic primitives. We propose some possible modifications to avoid our attack at the end of this paper.
Paper Structure (20 sections, 15 theorems, 92 equations, 1 table, 3 algorithms)

This paper contains 20 sections, 15 theorems, 92 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Facts about Local Fields
  4. Norm and Orthogonal Basis
  5. $p$-adic Lattice
  6. LVP and CVP
  7. Successive Maxima
  8. Signature Scheme and Public-key Encryption Cryptosystem
  9. The $p$-adic Signature Scheme
  10. The $p$-adic Public-key Encryption Cryptosystem
  11. Orthogonalization in Local Fields
  12. Orthogonalization in Totally Ramified Fields
  13. LVP in General Local Fields
  14. Break the Schemes
  15. The Attack
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $V$ be a vector space over $\mathbb{Q}_p$ of finite dimension $n>0$, and let $\left|\cdot\right|$ be a norm on $V$. Then there is a decomposition $V=V_1+V_2+\cdots+V_n$ of $V$ into a direct sum of subspaces $V_i$ of dimension $1$, such that for any $v_i\in V_i$, $i=1,2,\dots,n$.

Theorems & Definitions (32)

  • Proposition 2.1: ref-5
  • Definition 1: orthogonal basis ref-5
  • Proposition 2.2: ref-2
  • Lemma 2.3: ref-3
  • Definition 2: $p$-adic lattice ref-3
  • Definition 3: orthogonal basis of a $p$-adic lattice ref-3
  • Definition 4: ref-1'ref-1
  • Definition 5: ref-1'ref-1
  • Definition 6: ref-1'ref-1
  • Definition 7: successive maxima ref-3.5
  • ...and 22 more