opaque-lattice/docs/LEAP_ANALYSIS.md

From Split-Blinding to LEAP: Achieving True Unlinkability

1. The Fingerprint Attack on Split-Blinding

1.1 The Split-Blinding Protocol

In our split-blinding construction, the client sends two ring elements:

C   = A·(s + r) + (e + eᵣ)     [blinded password encoding]
Cᵣ  = A·r + eᵣ                 [blinding component]

Where:

• s, e are deterministically derived from the password
• r, eᵣ are fresh random per session
• A is the public parameter
• All operations are in the ring Rq = Zq[X]/(Xⁿ + 1)

1.2 The Fatal Flaw

A malicious server computes:

C - Cᵣ = A·(s + r) + (e + eᵣ) - A·r - eᵣ
       = A·s + e

This value is deterministic from the password alone!

The server can store (session_id, A·s + e) and link all sessions from the same user by checking if C - Cᵣ matches any previously seen fingerprint.

1.3 The Fundamental Tension

Property	Requirement	Conflict
Determinism	Output must be same for same password	Something must be fixed per password
Unlinkability	Server cannot correlate sessions	Nothing can be fixed per password

These appear contradictory. The key insight: the server must not be able to extract any deterministic function.

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2. Why OT-Based OPRFs Work

2.1 Bit-by-Bit Oblivious Transfer

In OT-based Ring-LPR (Shan et al. 2025):

For each bit bᵢ of the password encoding:
  1. Server prepares: (V₀ᵢ, V₁ᵢ) = evaluations for bit=0 and bit=1
  2. Client selects: Vᵢ = V_{bᵢ}ᵢ via 1-of-2 OT
  3. Server learns nothing about bᵢ

The selection is hidden by OT security. Server evaluates both options but cannot determine which the client chose.

2.2 Why It's Slow

256 bits × 2 evaluations × OT overhead = ~86ms

The OT protocol requires multiple rounds and exponentiations per bit.

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3. The LEAP Solution: Vector-OLE

3.1 VOLE Correlation

Vector Oblivious Linear Evaluation (VOLE) provides correlations of the form:

Sender holds:    (Δ, b)
Receiver holds:  (a, c)
Correlation:     c = a·Δ + b

Where the receiver knows a (the input) and c (the output), but NOT Δ (the key). The sender knows Δ and b, but NOT a or c.

3.2 VOLE-Based OPRF

The LEAP construction uses VOLE to achieve oblivious evaluation:

Setup:
  - Server key: k ∈ Rq (small)
  - Public: A ∈ Rq

Protocol:
  1. Client and Server run VOLE with:
     - Receiver (Client) input: s (password-derived)
     - Sender (Server) input: k (secret key)
     
  2. VOLE outputs:
     - Client receives: y = k·s + Δ (masked evaluation)
     - Server receives: random shares (learns nothing about s)
     
  3. Client unblinds using their share of Δ

3.3 Why Fingerprinting Fails

Unlike split-blinding where client sends (C, Cᵣ):

• In VOLE, the "blinding" is jointly computed
• Server contributes to the randomness but cannot recover A·s + e
• The protocol transcript is computationally indistinguishable from random

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4. The Spring PRF with LWR

4.1 Learning With Rounding (LWR)

Instead of using reconciliation helpers (which leak information), Spring uses LWR:

PRF_k(x) = ⌊A·k·x⌋_p   (rounded to modulus p < q)

The rounding absorbs the error locally:

• No helper bits needed from server
• Client can compute the rounding independently
• Error is "hidden" in the rounded-away bits

4.2 Spring PRF Construction

Spring_k(x):
  1. Expand x to ring element: s = H(x)
  2. Compute: v = A·k·s mod q
  3. Round: y = ⌊v·p/q⌋ mod p
  4. Output: H'(y)

4.3 Error Budget

For LWR with parameters (n, q, p):

• Rounding error: ≤ q/(2p)
• LWE error contribution: ≤ n·β²
• Security requires: n·β² < q/(2p)

With n=256, q=65537, p=256, β=3:

• Error bound: 256·9 = 2304
• Rounding threshold: 65537/512 ≈ 128

Problem: 2304 > 128. Need larger q or smaller p.

With q=65537, p=16:

• Rounding threshold: 65537/32 ≈ 2048
• Still marginal. Need careful parameter selection.

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5. Our Implementation: LEAP-Style VOLE OPRF

5.1 Simplified VOLE for Ring-LWE

We implement a simplified VOLE using Ring-LWE:

struct VoleCorrelation {
    delta: RingElement,      // Server's key contribution
    b: RingElement,          // Server's randomness
    a: RingElement,          // Client's input (derived from password)
    c: RingElement,          // Client's output: c = a·Δ + b
}

5.2 The Protocol

Round 1 (Client → Server): Commitment
  - Client computes: com = H(r) where r is fresh randomness
  - Client sends: com
  
Round 2 (Server → Client): Challenge
  - Server generates: ρ (random challenge)
  - Server sends: ρ
  
Round 3 (Client → Server): Masked Input
  - Client computes: C = A·(s + H(r||ρ)) + e'
  - Client sends: C
  
Round 4 (Server → Client): Evaluation
  - Server computes: V = k·C
  - Server sends: V (no helper needed with LWR)
  
Client Finalization:
  - Client computes: W = (s + H(r||ρ))·B
  - Client rounds: y = ⌊W·p/q⌋
  - Output: H(y)

5.3 Security Analysis

Unlinkability:

• Server sees C = A·(s + H(r||ρ)) + e'
• Cannot compute C - ? to get fingerprint because:
  • r is committed before ρ is known
  • H(r||ρ) is unpredictable without knowing r
  • Client doesn't send the "unblinding key"

Determinism:

• Final output depends only on k·A·s (deterministic)
• The H(r||ρ) terms cancel in the LWR computation
• Same password → same PRF output

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6. Comparison

Property	Split-Blinding	OT-Based	VOLE-LEAP
Speed	~0.1ms	~86ms	~0.75ms
Rounds	1	256 OT	2-4
Unlinkability	❌ Fingerprint	✅ OT hiding	✅ VOLE hiding
Error Handling	Reconciliation	Per-bit	LWR rounding
Communication	~2KB	~100KB	~4KB

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7. Open Questions

1. Parameter Selection: Exact (n, q, p, β) for 128-bit security with LWR
2. VOLE Efficiency: Optimal silent VOLE instantiation for Ring-LWE
3. Constant-Time: Ensuring all operations are timing-safe
4. Threshold: Extending to t-of-n threshold OPRF

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References

1. Heimberger et al., "LEAP: A Fast, Lattice-based OPRF", Eurocrypt 2025
2. Banerjee et al., "Spring PRF from Rounded Ring Products", 2024
3. Shan et al., "Ring-LPR OPRF", 2025
4. Boyle et al., "Efficient Pseudorandom Correlation Generators", Crypto 2019