module-lattice-security

name: module-lattice-security description: > Module lattice security methodology for post-quantum cryptography. Covers unconditional verification of Weber's conjecture, Principal Ideal Problem solvability, Ring-LWE and Module-LWE security reductions, and cyclotomic field arithmetic. Combines computational number theory (Fukuda-Komatsu sieve, Herbrand's theorem) with lattice-based cryptographic security analysis. Activation: module lattice security, Weber conjecture, Ring-LWE, Module-LWE, post-quantum cryptography, lattice cryptography, 模格安全.

Module Lattice Security Methodology

Overview

This methodology addresses the mathematical foundations of lattice-based post-quantum cryptography, specifically the security of Ring-LWE (R-LWE) and Module-LWE (MLWE) schemes through verification of Weber's conjecture (1886) for cyclotomic fields.

Source Paper: Ming-Xing Luo. "Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for k ≤ 12" (arXiv: 2604.15858)

Background: Weber's Conjecture

Weber's conjecture (1886) governs three critical aspects of lattice-based cryptography:

1. Principal Ideal Problem (PIP) solvability: Whether every ideal in the ring of integers is principal, affecting the security of ideal lattice cryptography
2. Module freeness: Whether modules over rings of integers are free, impacting the tightness of security reductions
3. Worst-case-to-average-case reductions: The tightness of reductions in R-LWE and MLWE, which form the security foundation of post-quantum cryptographic schemes

Prior work verified Weber's conjecture for k ≥ 9 only under the Generalized Riemann Hypothesis (GRH), a conditional assumption. This paper provides the first unconditional proof for k ≤ 12.

Mathematical Framework

Cyclotomic Fields and Z_2-towers

K_n = Q(ζ_{2^n})  — cyclotomic field of 2^n-th roots of unity
O_{K_n} — ring of integers of K_n
Z_2-tower: K_1 ⊂ K_2 ⊂ K_3 ⊂ ... ⊂ K_n

The inductive structure of the cyclotomic Z_2-tower allows propagation of class group properties through the tower.

Weber's Conjecture Statement

For cyclotomic fields K = Q(ζ_m), the class number h_K satisfies specific divisibility properties related to the structure of the ideal class group.

Key Components of the Proof

1. Fukuda-Komatsu Computational Sieve:

   • Algorithm for computing class groups of cyclotomic fields
   • Provides explicit bounds on class number divisibility
   • Enables computational verification for specific values of k

2. Inductive Structure of Z_2-tower:

   • Class group behavior propagates through the tower
   • Iwasawa theory provides asymptotic understanding
   • Finite-level computations connect to infinite tower behavior

3. Herbrand's Theorem:

   • Relates class group structure to Bernoulli numbers
   • Provides analytic control over class number divisibility
   • Enables unconditional bounds without GRH

Implementation Pattern

Step 1: Identify the Cyclotomic Field

def cyclotomic_field(m):
    """Return the cyclotomic field Q(ζ_m)."""
    # m determines the degree φ(m) of the field
    return Q.adjoin_primitive_root(m)

Step 2: Apply Fukuda-Komatsu Sieve

def fukuda_komatsu_sieve(K, bound):
    """Compute class group information using Fukuda-Komatsu sieve."""
    # Sieve primes up to bound
    # Compute p-rank of class group for each prime p
    # Track divisibility of class number
    return class_group_data

Step 3: Propagate through Z_2-tower

def z2_tower_propagation(base_field, max_level):
    """Propagate class group properties through Z_2-tower."""
    results = []
    current = base_field
    for n in range(max_level):
        results.append(analyze_class_group(current))
        current = extend_to_next_level(current)
    return results

Step 4: Apply Herbrand's Theorem

def herbrand_bound(K, prime_p):
    """Compute Herbrand bound for p-divisibility of class number."""
    # Relate to Bernoulli numbers B_{p-1-k}
    # Check p-divisibility conditions
    return is_divisible

Step 5: Verify Weber's Conjecture

def verify_weber(k):
    """Verify Weber's conjecture for given k."""
    # For k ≤ 12, use unconditional methods
    # Combine sieve results, tower propagation, and Herbrand bounds
    return verification_result

Cryptographic Implications

Ring-LWE Security

R-LWE Security Reduction:
    Worst-case SIVP_γ in ideal lattices
        → Average-case R-LWE_{q,χ} in R = Z[x]/(f(x))
    
Reduction tightness depends on:
    - Whether R has class number 1 (PIP solvability)
    - Module structure over R
    - Weber's conjecture affects the reduction factor γ

Module-LWE Security

MLWE Generalizes R-LWE:
    - R-LWE is MLWE with rank k = 1
    - MLWE with rank k > 1 over ring R
    - Security reduction depends on module freeness over R
    
Weber's conjecture → module freeness → tight security reduction

Post-Quantum Cryptographic Schemes

When to Use This Skill

• Analyzing security of lattice-based post-quantum cryptographic schemes
• Understanding the mathematical foundations of Ring-LWE and Module-LWE
• Evaluating worst-case-to-average-case reduction tightness
• Working with cyclotomic fields in cryptographic contexts
• Computational number theory for cryptography
• Post-quantum cryptography standardization analysis

Trigger Keywords

• Weber conjecture, lattice-based cryptography
• Ring-LWE, Module-LWE, R-LWE, MLWE
• Principal Ideal Problem, PIP
• Cyclotomic fields, class groups
• Post-quantum cryptography, PQC
• Fukuda-Komatsu sieve, Herbrand's theorem
• Z_2-tower, Iwasawa theory
• 模格安全, 韦伯猜想, 后量子密码

Pitfalls

1. GRH dependency: Prior results for k ≥ 9 required the Generalized Riemann Hypothesis; the unconditional proof only covers k ≤ 12. For k > 12, GRH-dependent results still apply.

2. Computational complexity: The Fukuda-Komatsu sieve has exponential complexity in the degree of the field; practical only for small-degree cyclotomic fields.

3. Reduction tightness: Even with Weber's conjecture verified, the worst-case-to-average-case reduction for R-LWE/MLWE still has polynomial loss factors.

4. Practical security: Mathematical security reductions do not directly translate to concrete bit-security levels; parameter selection requires additional analysis.

5. Part I limitation: This is Part I of the series; further parts may address k > 12 or additional aspects of module lattice security.

References

• Ming-Xing Luo. "Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for k ≤ 12" (arXiv: 2604.15858)
• Lyubashevsky, Peikert, Regev. "On Ideal Lattices and Learning with Errors over Rings"
• Langlois, Stehlé. "Worst-case to Average-case Reductions for Module Lattices"
• Fukuda. "Computation of the class number of cyclotomic fields"