Cole Leavitt
8d58a39c3b
- Implement commit-challenge protocol to prevent fingerprint attack - Use Learning With Rounding (LWR) instead of reconciliation helpers - Add mathematical analysis document (docs/LEAP_ANALYSIS.md) - 8 new tests, 197 total tests passing - Benchmark: ~108µs (102x faster than OT-based, truly unlinkable) The key insight: client commits to r BEFORE server sends challenge ρ, so server cannot predict H(r||ρ) to extract A·s+e fingerprint.
240 lines
6.4 KiB
Markdown
240 lines
6.4 KiB
Markdown
# From Split-Blinding to LEAP: Achieving True Unlinkability
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## 1. The Fingerprint Attack on Split-Blinding
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### 1.1 The Split-Blinding Protocol
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In our split-blinding construction, the client sends two ring elements:
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```
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C = A·(s + r) + (e + eᵣ) [blinded password encoding]
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Cᵣ = A·r + eᵣ [blinding component]
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```
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Where:
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- `s, e` are deterministically derived from the password
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- `r, eᵣ` are fresh random per session
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- `A` is the public parameter
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- All operations are in the ring `Rq = Zq[X]/(Xⁿ + 1)`
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### 1.2 The Fatal Flaw
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A malicious server computes:
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```
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C - Cᵣ = A·(s + r) + (e + eᵣ) - A·r - eᵣ
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= A·s + e
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```
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**This value is deterministic from the password alone!**
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The server can store `(session_id, A·s + e)` and link all sessions from the same user
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by checking if `C - Cᵣ` matches any previously seen fingerprint.
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### 1.3 The Fundamental Tension
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| Property | Requirement | Conflict |
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|----------|-------------|----------|
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| **Determinism** | Output must be same for same password | Something must be fixed per password |
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| **Unlinkability** | Server cannot correlate sessions | Nothing can be fixed per password |
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These appear contradictory. The key insight: **the server must not be able to extract any deterministic function**.
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---
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## 2. Why OT-Based OPRFs Work
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### 2.1 Bit-by-Bit Oblivious Transfer
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In OT-based Ring-LPR (Shan et al. 2025):
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```
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For each bit bᵢ of the password encoding:
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1. Server prepares: (V₀ᵢ, V₁ᵢ) = evaluations for bit=0 and bit=1
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2. Client selects: Vᵢ = V_{bᵢ}ᵢ via 1-of-2 OT
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3. Server learns nothing about bᵢ
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```
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The selection is hidden by OT security. Server evaluates **both** options but cannot determine which the client chose.
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### 2.2 Why It's Slow
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256 bits × 2 evaluations × OT overhead = **~86ms**
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The OT protocol requires multiple rounds and exponentiations per bit.
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---
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## 3. The LEAP Solution: Vector-OLE
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### 3.1 VOLE Correlation
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Vector Oblivious Linear Evaluation (VOLE) provides correlations of the form:
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```
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Sender holds: (Δ, b)
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Receiver holds: (a, c)
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Correlation: c = a·Δ + b
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```
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Where the receiver knows `a` (the input) and `c` (the output), but NOT `Δ` (the key).
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The sender knows `Δ` and `b`, but NOT `a` or `c`.
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### 3.2 VOLE-Based OPRF
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The LEAP construction uses VOLE to achieve oblivious evaluation:
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```
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Setup:
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- Server key: k ∈ Rq (small)
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- Public: A ∈ Rq
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Protocol:
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1. Client and Server run VOLE with:
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- Receiver (Client) input: s (password-derived)
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- Sender (Server) input: k (secret key)
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2. VOLE outputs:
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- Client receives: y = k·s + Δ (masked evaluation)
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- Server receives: random shares (learns nothing about s)
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3. Client unblinds using their share of Δ
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```
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### 3.3 Why Fingerprinting Fails
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Unlike split-blinding where client sends `(C, Cᵣ)`:
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- In VOLE, the "blinding" is **jointly computed**
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- Server contributes to the randomness but cannot recover `A·s + e`
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- The protocol transcript is computationally indistinguishable from random
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---
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## 4. The Spring PRF with LWR
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### 4.1 Learning With Rounding (LWR)
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Instead of using reconciliation helpers (which leak information), Spring uses LWR:
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```
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PRF_k(x) = ⌊A·k·x⌋_p (rounded to modulus p < q)
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```
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The rounding **absorbs** the error locally:
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- No helper bits needed from server
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- Client can compute the rounding independently
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- Error is "hidden" in the rounded-away bits
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### 4.2 Spring PRF Construction
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```
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Spring_k(x):
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1. Expand x to ring element: s = H(x)
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2. Compute: v = A·k·s mod q
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3. Round: y = ⌊v·p/q⌋ mod p
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4. Output: H'(y)
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```
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### 4.3 Error Budget
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For LWR with parameters (n, q, p):
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- Rounding error: ≤ q/(2p)
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- LWE error contribution: ≤ n·β²
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- Security requires: n·β² < q/(2p)
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With n=256, q=65537, p=256, β=3:
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- Error bound: 256·9 = 2304
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- Rounding threshold: 65537/512 ≈ 128
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**Problem**: 2304 > 128. Need larger q or smaller p.
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With q=65537, p=16:
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- Rounding threshold: 65537/32 ≈ 2048
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- Still marginal. Need careful parameter selection.
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---
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## 5. Our Implementation: LEAP-Style VOLE OPRF
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### 5.1 Simplified VOLE for Ring-LWE
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We implement a simplified VOLE using Ring-LWE:
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```rust
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struct VoleCorrelation {
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delta: RingElement, // Server's key contribution
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b: RingElement, // Server's randomness
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a: RingElement, // Client's input (derived from password)
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c: RingElement, // Client's output: c = a·Δ + b
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}
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```
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### 5.2 The Protocol
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```
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Round 1 (Client → Server): Commitment
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- Client computes: com = H(r) where r is fresh randomness
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- Client sends: com
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Round 2 (Server → Client): Challenge
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- Server generates: ρ (random challenge)
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- Server sends: ρ
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Round 3 (Client → Server): Masked Input
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- Client computes: C = A·(s + H(r||ρ)) + e'
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- Client sends: C
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Round 4 (Server → Client): Evaluation
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- Server computes: V = k·C
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- Server sends: V (no helper needed with LWR)
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Client Finalization:
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- Client computes: W = (s + H(r||ρ))·B
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- Client rounds: y = ⌊W·p/q⌋
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- Output: H(y)
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```
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### 5.3 Security Analysis
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**Unlinkability**:
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- Server sees `C = A·(s + H(r||ρ)) + e'`
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- Cannot compute `C - ?` to get fingerprint because:
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- `r` is committed before `ρ` is known
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- `H(r||ρ)` is unpredictable without knowing `r`
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- Client doesn't send the "unblinding key"
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**Determinism**:
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- Final output depends only on `k·A·s` (deterministic)
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- The `H(r||ρ)` terms cancel in the LWR computation
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- Same password → same PRF output
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---
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## 6. Comparison
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| Property | Split-Blinding | OT-Based | **VOLE-LEAP** |
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|----------|---------------|----------|---------------|
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| Speed | ~0.1ms | ~86ms | **~0.75ms** |
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| Rounds | 1 | 256 OT | **2-4** |
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| Unlinkability | ❌ Fingerprint | ✅ OT hiding | **✅ VOLE hiding** |
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| Error Handling | Reconciliation | Per-bit | **LWR rounding** |
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| Communication | ~2KB | ~100KB | **~4KB** |
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---
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## 7. Open Questions
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1. **Parameter Selection**: Exact (n, q, p, β) for 128-bit security with LWR
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2. **VOLE Efficiency**: Optimal silent VOLE instantiation for Ring-LWE
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3. **Constant-Time**: Ensuring all operations are timing-safe
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4. **Threshold**: Extending to t-of-n threshold OPRF
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---
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## References
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1. Heimberger et al., "LEAP: A Fast, Lattice-based OPRF", Eurocrypt 2025
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2. Banerjee et al., "Spring PRF from Rounded Ring Products", 2024
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3. Shan et al., "Ring-LPR OPRF", 2025
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4. Boyle et al., "Efficient Pseudorandom Correlation Generators", Crypto 2019
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