# UC Security Proof for Lattice-Based OPAQUE

This document provides a formal security proof for the opaque-lattice implementation in the Universal Composability (UC) framework.

## Table of Contents

1. [Overview](#1-overview)
2. [Preliminaries](#2-preliminaries)
3. [Ideal Functionalities](#3-ideal-functionalities)
4. [Protocol Description](#4-protocol-description)
5. [Simulator Construction](#5-simulator-construction)
6. [Security Proof](#6-security-proof)
7. [Concrete Security Bounds](#7-concrete-security-bounds)

---

## 1. Overview

### 1.1 Protocol Summary

opaque-lattice implements a post-quantum secure OPAQUE protocol using:
- **Ring-LPR OPRF**: Oblivious PRF based on Ring Learning Parity with Rounding
- **ML-KEM (Kyber768)**: Key encapsulation for authenticated key exchange
- **ML-DSA (Dilithium3)**: Digital signatures for server authentication
- **VOPRF Extension**: Verifiable OPRF with Lyubashevsky-style sigma protocol

### 1.2 Security Goals

We prove the protocol realizes the ideal functionality F_aPAKE (asymmetric Password-Authenticated Key Exchange) with the following properties:

| Property | Description |
|----------|-------------|
| **Password Obliviousness** | Server learns nothing about password during OPRF |
| **Forward Secrecy** | Compromise of long-term keys doesn't reveal past session keys |
| **Server Compromise Resistance** | Attacker cannot offline-attack passwords after server compromise |
| **Quantum Resistance** | Security holds against quantum adversaries |
| **Verifiability** | Client can verify server used consistent OPRF key |

### 1.3 Security Model

We work in the UC framework of Canetti [Can01] with:
- **Global Random Oracle Model (GROM)**: Hash functions H₁, H₂, H₃ modeled as random oracles
- **Adaptive Corruptions**: Adversary can corrupt parties at any point
- **Static Compromise**: Adversary learns all internal state upon corruption

---

## 2. Preliminaries

### 2.1 Notation

| Symbol | Meaning |
|--------|---------|
| λ | Security parameter (128 bits) |
| R | Ring Z[x]/(x^n + 1) where n = 256 |
| R_q | Ring R modulo q (q = 4 in our construction) |
| ⌊·⌋₁ | Deterministic rounding: ⌊x⌋₁ = ⌊x/2⌋ mod 2 |
| k ←$ S | Sample k uniformly from set S |
| negl(λ) | Negligible function in λ |
| poly(λ) | Polynomial function in λ |

### 2.2 Computational Assumptions

**Definition 2.1 (Ring-LPR Problem)**
For a ∈ R₂, s ∈ R₂, the Ring Learning Parity with Rounding problem states that:
```
(a, ⌊a·s mod 4⌋₁) ≈_c (a, ⌊u⌋₁)
```
where u ←$ R₄ is uniform random.

**Definition 2.2 (Dihedral Coset Problem)**
Given quantum states encoding cosets of a hidden subgroup in the dihedral group D_n, find the hidden subgroup generator. Time complexity: O(e^n) even for quantum computers.

**Theorem 2.1 (Security Reduction Chain)**
```
Ring-LPR → LPR → LWR → G-EDCP → DCP
```
Each reduction is polynomial-time. The DCP problem is believed quantum-hard with time complexity O(e^n).

**Definition 2.3 (ML-KEM Security)**
ML-KEM (Kyber768) is IND-CCA2 secure under the Module-LWE assumption.

**Definition 2.4 (ML-DSA Security)**
ML-DSA (Dilithium3) is EUF-CMA secure under the Module-LWE and Module-SIS assumptions.

### 2.3 Building Blocks

**PRF Construction (Ring-LPR)**
```
F_k(x) = H₂(⌊k · H₁(x) mod 4⌋₁)
```
where:
- H₁: {0,1}* → R₂ (hash-to-ring)
- H₂: R₁ → {0,1}^512 (ring-to-output hash)
- k ∈ R₂ (secret key)

**Key Commitment**
```
Commit(k; r) = H₃(k || r)
```
where r ←$ {0,1}^256 is randomness.

---

## 3. Ideal Functionalities

### 3.1 Ideal VOPRF Functionality F_VOPRF

```
Functionality F_VOPRF

Parameters: Output length l, security parameter λ

Initialization:
- On (Init, sid) from server S:
  - If first Init for sid: set T_sid(·) = ⊥, tx(S) = 0
  - Send (Init, sid, S) to adversary A

- On (Param, S, π) from A:
  - If params[S] undefined: set params[S] = π

Key Commitment:
- On (Commit, sid) from S:
  - Sample random table key k_sid
  - Store commitment c = H(k_sid)
  - Send (Committed, sid, c) to A

Offline Evaluation:
- On (OfflineEval, sid, c, p) from party P:
  - If params[i] = c for some server i:
    - If T_sid(i, p) undefined: T_sid(i, p) ←$ {0,1}^l
    - Send (OfflineEval, sid, T_sid(i, p)) to P
  - Else: ignore

Online Evaluation:
- On (Eval, sid, ssid, S, p) from user U:
  - Record ⟨ssid, S, U, p⟩
  - Send (Eval, sid, ssid, U, S) to A

- On (SndrComplete, sid, ssid) from S:
  - Increment tx(S)
  - Send (SndrComplete, sid, ssid, S) to A

- On (RcvCmplt, sid, ssid, U, π) from A:
  - Retrieve ⟨ssid, S, U, p⟩
  - Ignore if:
    - No such record exists
    - Honest server S has params[S] = π but tx(S) = 0
    - S honest but π ≠ params[S]
  - If T_sid(i, p) undefined: T_sid(i, p) ←$ {0,1}^l
  - Send (Eval, sid, ssid, T_sid(i, p)) to U
  - Decrement tx(S) if params[S] = π

Verification:
- On (Verify, sid, c, p, y, proof) from U:
  - Check if y = T_sid(i, p) for params[i] = c
  - Send (Verified, sid, valid/invalid) to U
```

### 3.2 Ideal AKE Functionality F_AKE

```
Functionality F_AKE

Initialization:
- Maintain session table sessions[·]
- Maintain corruption set Corrupt

Key Exchange:
- On (NewSession, sid, P, P', role) from party P:
  - Record (sid, P, P', role, ⊥)
  - Send (NewSession, sid, P, P', role) to A

- On (TestPwd, sid, P, pw') from A:
  - If P ∈ Corrupt: return ⊥
  - Retrieve (sid, P, P', role, pw)
  - If pw' = pw: mark session compromised
  - Return (compromised/not-compromised)

- On (NewKey, sid, P, sk) from A:
  - Retrieve (sid, P, P', role, pw)
  - If P' ∈ Corrupt or session compromised:
    - Output (sid, sk) to P
  - Else if (sid, P', P, role', k') exists with k' ≠ ⊥:
    - Output (sid, k') to P
  - Else:
    - Sample k ←$ {0,1}^λ
    - Record k, output (sid, k) to P
```

### 3.3 Ideal aPAKE Functionality F_aPAKE

```
Functionality F_aPAKE

Parameters: Security parameter λ

Registration:
- On (Register, sid, U, S, pw) from user U:
  - Compute file ← F_OPRF.Eval(pw)
  - Store (U, file) for server S
  - Send (Registered, sid, U) to S

Login:
- On (Login, sid, U, S, pw) from U:
  - Initiate OPRF evaluation with S
  - If pw matches stored file:
    - Derive session key sk
    - Output (LoginComplete, sid, sk) to U, S
  - Else:
    - Output (LoginFailed, sid) to U

Server Compromise:
- On (Corrupt, S) from A:
  - Add S to Corrupt set
  - Send all stored files to A
  - Note: Offline attacks still require online OPRF interaction
```

---

## 4. Protocol Description

### 4.1 Registration Protocol

```
Client(pw)                                    Server(oprf_key)
═══════════════════════════════════════════════════════════════
1. salt ←$ {0,1}^256
2. pw_hash = Argon2id(pw, salt)
3. (state, blind) = OPRF.Blind(pw_hash)
                          ─────── blind ───────────►
4.                                               eval = OPRF.Eval(oprf_key, blind)
                          ◄────── eval ─────────
5. rw = OPRF.Finalize(state, eval)
6. (pk_U, sk_U) = KEM.KeyGen()
7. (pk_auth, sk_auth) = SIG.KeyGen()  
8. envelope = AEAD.Enc(rw, sk_U || sk_auth)
9. record = (pk_U, pk_auth, envelope, salt)
                          ─────── record ──────────►
10.                                              Store(U, record)
```

### 4.2 Login Protocol (KE1 → KE2 → KE3)

```
Client(pw)                                    Server(oprf_key, record)
═══════════════════════════════════════════════════════════════
KE1: Client → Server
1. pw_hash = Argon2id(pw, record.salt)
2. (state, blind) = OPRF.Blind(pw_hash)
3. (ek_C, dk_C) = KEM.KeyGen()
                    ───── KE1: (blind, ek_C) ─────►

KE2: Server → Client
4.                                       eval = OPRF.Eval(oprf_key, blind)
5.                                       (ct, ss_S) = KEM.Encap(ek_C)
6.                                       K_session = KDF(ss_S, transcript)
7.                                       mac_S = MAC(K_session, transcript)
8.                                       sig = SIG.Sign(sk_S, transcript)
                    ◄─── KE2: (eval, ct, mac_S, sig, envelope) ───

KE3: Client → Server  
9. Verify sig with pk_S
10. rw = OPRF.Finalize(state, eval)
11. (sk_U, sk_auth) = AEAD.Dec(rw, envelope)
12. ss_C = KEM.Decap(dk_C, ct)
13. K_session = KDF(ss_C, transcript)
14. Verify mac_S
15. mac_C = MAC(K_session, transcript)
                    ───── KE3: mac_C ─────►
16.                                      Verify mac_C
17. Output K_session                     Output K_session
```

### 4.3 VOPRF Extension

For verifiable evaluation, server additionally:

```
Server with committed key (k, c, r):
1. Compute eval = F_k(blind)
2. Generate ZK proof π proving:
   - Knowledge of k such that c = Commit(k; r)
   - eval = F_k(blind)
3. Return (eval, π)

Client:
1. Verify π against commitment c
2. If valid: proceed with finalization
3. If invalid: abort
```

---

## 5. Simulator Construction

### 5.1 Simulator for F_VOPRF

```
Simulator SIM_VOPRF

On (Init, sid, S) from F_VOPRF:
  - Sample random key k_sim
  - Compute c = Commit(k_sim; r_sim)
  - Send (Param, S, c) to F_VOPRF
  - Record (S, k_sim, r_sim, c)

On (Eval, sid, ssid, U, S) from F_VOPRF:
  If S honest:
    - Wait for adversary to deliver
    - On delivery: send (SndrComplete, sid, ssid) to F_VOPRF
    - Send (RcvCmplt, sid, ssid, U, c_S) to F_VOPRF
  
  If S corrupted:
    - Extract adversary's evaluation blind_A
    - Query F_VOPRF for T_sid(c_A, blind_A)
    - Program H₂ to output T_sid value
    - Simulate protocol messages

On H₂ query (x, y, c) from A:
  If ∃ honest S with params[S] = c:
    - Send (OfflineEval, sid, c, H₁⁻¹(x)) to F_VOPRF
    - Receive ρ from F_VOPRF
    - Return ρ
  Else:
    - Return random value or use table
```

### 5.2 Simulator for F_aPAKE

```
Simulator SIM_aPAKE

Registration Simulation:
  On (Register, sid, U, S, pw) from F_aPAKE:
    - Simulate OPRF blind message
    - Receive adversarial evaluation
    - Extract any password guesses
    - Complete registration

Login Simulation (Honest Client, Honest Server):
  - Simulate KE1 with random blind
  - Simulate KE2 with random evaluation
  - On (TestPwd, sid, U, pw') from A:
    - Forward to F_aPAKE
    - If compromised: program keys accordingly
  - Generate session key from F_aPAKE
  - Program MAC/KDF oracles consistently

Login Simulation (Corrupted Server):
  - Extract server's OPRF key from state
  - Use real OPRF evaluation
  - Monitor password test queries
  - Enforce "one online guess per session"

Login Simulation (Corrupted Client):
  - Extract client's password from state  
  - Use real protocol execution
  - Provide adversary with session key
```

---

## 6. Security Proof

### 6.1 Main Theorem

**Theorem 6.1 (UC Security)**
The opaque-lattice protocol UC-realizes F_aPAKE in the (F_VOPRF, F_RO)-hybrid model, assuming:
1. Ring-LPR is pseudorandom (Definition 2.1)
2. ML-KEM is IND-CCA2 secure
3. ML-DSA is EUF-CMA secure
4. AEAD is IND-CPA and INT-CTXT secure
5. HKDF is a secure PRF

The advantage of any PPT adversary A in distinguishing real from ideal execution is:
```
Adv(A) ≤ q_pwd · Adv_LPR + q_KEM · Adv_IND-CCA + q_SIG · Adv_EUF-CMA 
        + q_AEAD · Adv_AEAD + q_sessions · negl(λ)
```
where q_* denotes the number of respective queries.

### 6.2 Proof by Game Sequence

**Game 0 (Real Protocol)**
The real execution of opaque-lattice with adversary A.

**Game 1 (Random Oracle Instrumentation)**
Replace hash functions H₁, H₂, H₃ with random oracles maintained by simulator.
- Indistinguishable by random oracle assumption

**Game 2 (OPRF Simulation)**
Replace real OPRF evaluations with queries to F_VOPRF.
- For honest server: outputs are random (Ring-LPR pseudorandomness)
- For corrupted server: extract key, compute real evaluation

*Lemma 6.1:* |Pr[Game 2] - Pr[Game 1]| ≤ q_oprf · Adv_LPR

**Game 3 (KEM Simulation)**
Replace KEM encapsulation with F_KEM ideal functionality.
- Honest parties: shared secret is random
- Corrupted parties: extract/inject values

*Lemma 6.2:* |Pr[Game 3] - Pr[Game 2]| ≤ q_kem · Adv_IND-CCA

**Game 4 (Signature Simulation)**
Replace signatures with F_SIG ideal functionality.
- Verify signatures using committed public key
- Reject any forgeries

*Lemma 6.3:* |Pr[Game 4] - Pr[Game 3]| ≤ q_sig · Adv_EUF-CMA

**Game 5 (Envelope Simulation)**
Replace AEAD with ideal encryption.
- Envelope contents are hidden until rw is known
- Tampering detected by INT-CTXT

*Lemma 6.4:* |Pr[Game 5] - Pr[Game 4]| ≤ q_aead · Adv_AEAD

**Game 6 (Password Test Restriction)**
Enforce that adversary must make explicit TestPwd query to F_aPAKE.
- Each online session allows at most one password test
- Offline dictionary attack requires OPRF evaluation

*Lemma 6.5:* |Pr[Game 6] - Pr[Game 5]| ≤ negl(λ)

**Game 7 (Ideal Execution)**
Execute with F_aPAKE and simulator SIM.
- Session keys are random unless compromised
- Password never revealed to honest parties

*Lemma 6.6:* Game 6 ≡ Game 7

### 6.3 Verifiability Proof

**Theorem 6.2 (VOPRF Soundness)**
For any PPT adversary A, the probability that A produces a valid proof π for an evaluation y = F_k(x) where k differs from the committed key is negligible.

*Proof Sketch:*
1. By binding property of commitment: A cannot open to different k
2. By soundness of sigma protocol: A cannot forge proofs
3. By Fiat-Shamir security: Non-interactive proofs are sound in ROM

**Theorem 6.3 (VOPRF Zero-Knowledge)**
The sigma protocol proof reveals nothing about k beyond the validity of the statement.

*Proof Sketch:*
1. Construct simulator S that generates accepting proofs without k
2. S samples response z uniformly, computes mask m = z - e·k_dummy
3. By rejection sampling analysis: real and simulated distributions are statistically close
4. Distinguishing advantage bounded by 2^(-λ)

---

## 7. Concrete Security Bounds

### 7.1 Parameter Selection

| Parameter | Value | Security Level |
|-----------|-------|----------------|
| Ring dimension n | 256 | 128-bit post-quantum |
| Ring modulus q | 4 | Minimal for rounding |
| KEM security | Kyber768 | NIST Level 3 |
| Signature security | Dilithium3 | NIST Level 3 |
| Hash output | 512 bits | Collision resistance |
| Commitment nonce | 256 bits | Binding security |

### 7.2 Concrete Advantages

Assuming λ = 128 security parameter:

| Component | Advantage Bound |
|-----------|-----------------|
| Ring-LPR PRF | 2^(-128) (DCP hardness) |
| ML-KEM IND-CCA | 2^(-128) (MLWE hardness) |
| ML-DSA EUF-CMA | 2^(-128) (MLWE+SIS hardness) |
| AEAD (AES-GCM) | 2^(-128) |
| HKDF-SHA512 | 2^(-256) |
| Commitment binding | 2^(-128) (collision resistance) |
| ZK soundness | 2^(-128) (sigma protocol) |

### 7.3 Attack Complexity

| Attack | Complexity | Mitigation |
|--------|------------|------------|
| Offline dictionary | Requires OPRF oracle | One guess per session |
| Online brute force | O(2^128) sessions | Rate limiting |
| Quantum OPRF attack | O(e^256) | DCP hardness |
| Server compromise | No offline attack | OPRF obliviousness |
| Forward secrecy break | O(2^128) per session | Ephemeral KEM keys |

---

## References

[Can01] R. Canetti. "Universally Composable Security: A New Paradigm for Cryptographic Protocols." FOCS 2001.

[JKX18] S. Jarecki, H. Krawczyk, J. Xu. "OPAQUE: An Asymmetric PAKE Protocol Secure Against Pre-Computation Attacks." Eurocrypt 2018.

[Lyu09] V. Lyubashevsky. "Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures." ASIACRYPT 2009.

[Lyu12] V. Lyubashevsky. "Lattice Signatures without Trapdoors." EUROCRYPT 2012.

[Sha25] Z. Shan et al. "Fast Post-Quantum Private Set Intersection from Ring-LPR OPRF." J. Syst. Arch. 2025.

[Alb21] M. Albrecht et al. "Round-optimal Verifiable OPRFs from Ideal Lattices." PKC 2021.

[Fal22] J. Faller. "Composable OPRFs via Garbled Circuits." Master's Thesis, 2022.

[RFC9807] RFC 9807. "The OPAQUE Augmented PAKE Protocol." 2024.

---

## Appendix A: Proof Details

### A.1 Ring-LPR Pseudorandomness

**Lemma A.1** For uniformly random k ∈ R₂ and arbitrary x ∈ R₂:
```
{(x, F_k(x))} ≈_c {(x, U)}
```
where U is uniform random output.

*Proof:*
1. F_k(x) = H₂(⌊k·x mod 4⌋₁)
2. By Ring-LPR assumption: ⌊k·x mod 4⌋₁ ≈_c ⌊u⌋₁ for random u
3. H₂ is a random oracle: output is uniformly distributed
4. Combining: F_k(x) is computationally indistinguishable from random

### A.2 Sigma Protocol Analysis

**Commitment:**
```
t = H(m || m·a)
```
where m ←$ R_q with small coefficients.

**Challenge:**
```
e = H(c || t || x || y)[0:16]
```
(128-bit challenge via Fiat-Shamir)

**Response:**
```
z = m + e·k
```
with rejection if ||z||_∞ > B.

**Rejection Probability:**
By Lemma 4.1 of [Lyu12], if m is sampled from discrete Gaussian with σ > 12·||k||:
```
Pr[rejection] ≤ 2^(-100)
```

**Soundness:**
If adversary produces accepting proofs for (c, x, y₁) and (c, x, y₂) with y₁ ≠ y₂:
```
z₁ - z₂ = e₁·k - e₂·k = (e₁ - e₂)·k
```
Since e₁ ≠ e₂ with overwhelming probability, we can extract k.

**Zero-Knowledge:**
Simulator chooses z uniformly, computes t = H(z - e·k_dummy || ...), programs RO.
Statistical distance from real: 2^(-λ) by rejection sampling lemma.

---

## Appendix B: Implementation Notes

### B.1 Constant-Time Implementation

All operations on secret data must be constant-time:
- Ring multiplication: coefficient-by-coefficient, no early termination
- Rounding: table lookup with constant access pattern
- Comparison: bitwise operations only

### B.2 Side-Channel Mitigations

- **Timing attacks**: All branches on secret data eliminated
- **Cache attacks**: No secret-dependent memory access patterns
- **Power analysis**: Balanced operations where possible

### B.3 Zeroization

All secret values are zeroized after use:
- OPRF keys: `RingLprKey` implements `ZeroizeOnDrop`
- Session keys: explicit zeroize before deallocation
- Intermediate values: scoped to minimize lifetime