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+# 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