# PVUGC-011: Collusive Randomness Cancellation - Stage 2 Standards Validation

**Issue:** PVUGC-011
**Stage:** Stage 2 (Standards Validation)
**Date:** 2025-10-29
**Decision Method:** Solo
**Verdict:** PERSISTS

---

## Evidence from v2.7

**Spec Location:**
- PVUGC-2025-10-27.md §6, line 200 (ρ_i selection: "Choose ρ_i ∈ Z_r* (non-zero)")
- PVUGC-2025-10-27.md §8, line 278 (PoCE-A constraint: "ρ_i ≠ 0 (via auxiliary ρ_i · u_i = 1)")
- PVUGC-2025-10-27.md §12, line 306 (Mandatory hygiene: "fresh ρ_i")

**v3.0 Status:**
- Novel (Medium severity) - Discovered during peer review

**v3.0 Baseline (report-peer_review-2025-10-26/PVUGC-011):**

The v3.0 peer review identified a second-order vulnerability: coalitions of k ≥ 2 armers can coordinate KEM randomness values {ρ_i} to achieve ∏ρ_i ≡ 1 (mod r), enabling:
- Limited grinding attacks on coalition's own contribution T_coalition
- Griefing via selective participation and protocol abortion
- Fairness violation (coordinated control vs. independent randomness assumption)

**Recommended Mitigation (v3.0):**
Three-phase commit-reveal protocol:
1. Phase 1 (Commitment): Each armer publishes c_i = SHA256("PVUGC_RHO_COMMIT/v1" || ρ_i || salt_i)
2. Phase 2 (Revelation): After all commitments collected, reveal (ρ_i, salt_i)
3. Phase 3 (Verification): Verify c_i == SHA256("..." || ρ_i || salt_i), abort on mismatch

---

## Assessment

**Search Results from v2.7:**

Searched for evidence of commit-reveal protocol implementation:
- Pattern: `commit.*reveal|PVUGC_RHO_COMMIT|salt_i|revelation|randomness.*independence`
- Result: No matches

**What's Specified in v2.7:**

Per-share validation only:
- Line 200: ρ_i ∈ Z_r* (individual non-zero check)
- Line 278: PoCE-A proves ρ_i ≠ 0 (cryptographic enforcement)
- Line 306: "fresh ρ_i" (hygiene requirement, no enforcement mechanism)

These checks prevent individual degeneracy (ρ_i = 0) but do not enforce independence across armers.

**Remaining Gaps:**

1. No commitment phase: Armers can coordinate ρ_i values in real-time
2. No temporal isolation: Coalition members can choose ρ_2 = ρ_1^{-1} after observing ρ_1
3. No domain tag "PVUGC_RHO_COMMIT/v1": Commit-reveal protocol absent
4. No salt mechanism: No cryptographic binding to prevent adaptive choice

**Evidence-Based Example:**

Coalition attack (per v3.0 analysis, lines 62-71):
```
Armer 1: ρ_1 = x (random)
Armer 2: ρ_2 = x^{-1} mod r (coordinated)
```

v2.7 validation:
- ρ_1 ≠ 0? YES (passes line 278 check)
- ρ_2 ≠ 0? YES (x ≠ 0 → x^{-1} ≠ 0)
- ∏ρ_i = x · x^{-1} = 1 mod r? YES (collusion succeeds)

Current spec has no mechanism to detect or prevent this multiplicative relationship.

**Impact Analysis (per v3.0):**

Not immediately exploitable for secret leakage (additive α = Σs_i structure), but enables:
- Pre-commitment grinding on T_coalition (coalition's combined adaptor share)
- Griefing attacks (delay/abort arming repeatedly at low cost)
- Future protocol brittleness (if extensions depend on randomness independence)

---

## Verdict: PERSISTS

**Justification:**

The v2.7 specification maintains only per-share validation (ρ_i ≠ 0) without cryptographic enforcement of randomness independence across armers. The v3.0-recommended commit-reveal protocol is absent from the specification. While first-order checks (individual degeneracy) are cryptographically enforced via PoCE-A, second-order collusive attacks remain unmitigated. This represents a deliberate architectural gap rather than an implementation detail.

**Severity Alignment:** Medium (per v3.0) - No direct fund theft, but enables griefing and violates fairness assumptions.

**Flag for Gap-Remediation:** Yes

**Gap Classification:** Type 5 (Implementation Guidance Gap)
- **What is specified:** Individual ρ_i ≠ 0 requirement
- **What is missing:** HOW to enforce randomness independence across armers
- **Security implication:** Enables coalition grinding and griefing attacks

---

## Cross-References

- v3.0 Peer Review: `report-peer_review-2025-10-26/PVUGC-011.md`
- Related Issue: PVUGC-006 (second-order degeneracy analysis, PARTIAL verdict)
- Gap Remediation: (To be created if gap-remediation workflow triggered)

---

**Date:** 2025-10-29
**Auditor:** Standards Compliance Auditor (Lead)
**Verdict:** PERSISTS