591 lines
18 KiB
Markdown
591 lines
18 KiB
Markdown
# UC Security Proof for Lattice-Based OPAQUE
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This document provides a formal security proof for the opaque-lattice implementation in the Universal Composability (UC) framework.
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## Table of Contents
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1. [Overview](#1-overview)
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2. [Preliminaries](#2-preliminaries)
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3. [Ideal Functionalities](#3-ideal-functionalities)
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4. [Protocol Description](#4-protocol-description)
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5. [Simulator Construction](#5-simulator-construction)
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6. [Security Proof](#6-security-proof)
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7. [Concrete Security Bounds](#7-concrete-security-bounds)
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---
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## 1. Overview
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### 1.1 Protocol Summary
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opaque-lattice implements a post-quantum secure OPAQUE protocol using:
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- **Ring-LPR OPRF**: Oblivious PRF based on Ring Learning Parity with Rounding
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- **ML-KEM (Kyber768)**: Key encapsulation for authenticated key exchange
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- **ML-DSA (Dilithium3)**: Digital signatures for server authentication
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- **VOPRF Extension**: Verifiable OPRF with Lyubashevsky-style sigma protocol
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### 1.2 Security Goals
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We prove the protocol realizes the ideal functionality F_aPAKE (asymmetric Password-Authenticated Key Exchange) with the following properties:
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| Property | Description |
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|----------|-------------|
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| **Password Obliviousness** | Server learns nothing about password during OPRF |
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| **Forward Secrecy** | Compromise of long-term keys doesn't reveal past session keys |
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| **Server Compromise Resistance** | Attacker cannot offline-attack passwords after server compromise |
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| **Quantum Resistance** | Security holds against quantum adversaries |
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| **Verifiability** | Client can verify server used consistent OPRF key |
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### 1.3 Security Model
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We work in the UC framework of Canetti [Can01] with:
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- **Global Random Oracle Model (GROM)**: Hash functions H₁, H₂, H₃ modeled as random oracles
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- **Adaptive Corruptions**: Adversary can corrupt parties at any point
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- **Static Compromise**: Adversary learns all internal state upon corruption
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---
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## 2. Preliminaries
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### 2.1 Notation
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| Symbol | Meaning |
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|--------|---------|
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| λ | Security parameter (128 bits) |
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| R | Ring Z[x]/(x^n + 1) where n = 256 |
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| R_q | Ring R modulo q (q = 4 in our construction) |
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| ⌊·⌋₁ | Deterministic rounding: ⌊x⌋₁ = ⌊x/2⌋ mod 2 |
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| k ←$ S | Sample k uniformly from set S |
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| negl(λ) | Negligible function in λ |
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| poly(λ) | Polynomial function in λ |
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### 2.2 Computational Assumptions
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**Definition 2.1 (Ring-LPR Problem)**
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For a ∈ R₂, s ∈ R₂, the Ring Learning Parity with Rounding problem states that:
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```
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(a, ⌊a·s mod 4⌋₁) ≈_c (a, ⌊u⌋₁)
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```
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where u ←$ R₄ is uniform random.
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**Definition 2.2 (Dihedral Coset Problem)**
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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.
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**Theorem 2.1 (Security Reduction Chain)**
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```
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Ring-LPR → LPR → LWR → G-EDCP → DCP
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```
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Each reduction is polynomial-time. The DCP problem is believed quantum-hard with time complexity O(e^n).
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**Definition 2.3 (ML-KEM Security)**
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ML-KEM (Kyber768) is IND-CCA2 secure under the Module-LWE assumption.
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**Definition 2.4 (ML-DSA Security)**
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ML-DSA (Dilithium3) is EUF-CMA secure under the Module-LWE and Module-SIS assumptions.
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### 2.3 Building Blocks
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**PRF Construction (Ring-LPR)**
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```
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F_k(x) = H₂(⌊k · H₁(x) mod 4⌋₁)
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```
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where:
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- H₁: {0,1}* → R₂ (hash-to-ring)
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- H₂: R₁ → {0,1}^512 (ring-to-output hash)
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- k ∈ R₂ (secret key)
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**Key Commitment**
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```
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Commit(k; r) = H₃(k || r)
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```
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where r ←$ {0,1}^256 is randomness.
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---
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## 3. Ideal Functionalities
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### 3.1 Ideal VOPRF Functionality F_VOPRF
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```
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Functionality F_VOPRF
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Parameters: Output length l, security parameter λ
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Initialization:
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- On (Init, sid) from server S:
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- If first Init for sid: set T_sid(·) = ⊥, tx(S) = 0
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- Send (Init, sid, S) to adversary A
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- On (Param, S, π) from A:
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- If params[S] undefined: set params[S] = π
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Key Commitment:
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- On (Commit, sid) from S:
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- Sample random table key k_sid
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- Store commitment c = H(k_sid)
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- Send (Committed, sid, c) to A
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Offline Evaluation:
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- On (OfflineEval, sid, c, p) from party P:
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- If params[i] = c for some server i:
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- If T_sid(i, p) undefined: T_sid(i, p) ←$ {0,1}^l
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- Send (OfflineEval, sid, T_sid(i, p)) to P
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- Else: ignore
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Online Evaluation:
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- On (Eval, sid, ssid, S, p) from user U:
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- Record ⟨ssid, S, U, p⟩
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- Send (Eval, sid, ssid, U, S) to A
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- On (SndrComplete, sid, ssid) from S:
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- Increment tx(S)
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- Send (SndrComplete, sid, ssid, S) to A
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- On (RcvCmplt, sid, ssid, U, π) from A:
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- Retrieve ⟨ssid, S, U, p⟩
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- Ignore if:
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- No such record exists
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- Honest server S has params[S] = π but tx(S) = 0
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- S honest but π ≠ params[S]
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- If T_sid(i, p) undefined: T_sid(i, p) ←$ {0,1}^l
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- Send (Eval, sid, ssid, T_sid(i, p)) to U
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- Decrement tx(S) if params[S] = π
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Verification:
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- On (Verify, sid, c, p, y, proof) from U:
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- Check if y = T_sid(i, p) for params[i] = c
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- Send (Verified, sid, valid/invalid) to U
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```
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### 3.2 Ideal AKE Functionality F_AKE
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```
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Functionality F_AKE
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Initialization:
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- Maintain session table sessions[·]
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- Maintain corruption set Corrupt
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Key Exchange:
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- On (NewSession, sid, P, P', role) from party P:
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- Record (sid, P, P', role, ⊥)
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- Send (NewSession, sid, P, P', role) to A
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- On (TestPwd, sid, P, pw') from A:
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- If P ∈ Corrupt: return ⊥
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- Retrieve (sid, P, P', role, pw)
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- If pw' = pw: mark session compromised
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- Return (compromised/not-compromised)
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- On (NewKey, sid, P, sk) from A:
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- Retrieve (sid, P, P', role, pw)
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- If P' ∈ Corrupt or session compromised:
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- Output (sid, sk) to P
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- Else if (sid, P', P, role', k') exists with k' ≠ ⊥:
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- Output (sid, k') to P
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- Else:
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- Sample k ←$ {0,1}^λ
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- Record k, output (sid, k) to P
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```
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### 3.3 Ideal aPAKE Functionality F_aPAKE
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```
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Functionality F_aPAKE
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Parameters: Security parameter λ
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Registration:
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- On (Register, sid, U, S, pw) from user U:
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- Compute file ← F_OPRF.Eval(pw)
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- Store (U, file) for server S
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- Send (Registered, sid, U) to S
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Login:
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- On (Login, sid, U, S, pw) from U:
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- Initiate OPRF evaluation with S
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- If pw matches stored file:
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- Derive session key sk
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- Output (LoginComplete, sid, sk) to U, S
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- Else:
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- Output (LoginFailed, sid) to U
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Server Compromise:
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- On (Corrupt, S) from A:
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- Add S to Corrupt set
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- Send all stored files to A
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- Note: Offline attacks still require online OPRF interaction
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```
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---
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## 4. Protocol Description
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### 4.1 Registration Protocol
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```
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Client(pw) Server(oprf_key)
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═══════════════════════════════════════════════════════════════
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1. salt ←$ {0,1}^256
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2. pw_hash = Argon2id(pw, salt)
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3. (state, blind) = OPRF.Blind(pw_hash)
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─────── blind ───────────►
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4. eval = OPRF.Eval(oprf_key, blind)
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◄────── eval ─────────
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5. rw = OPRF.Finalize(state, eval)
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6. (pk_U, sk_U) = KEM.KeyGen()
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7. (pk_auth, sk_auth) = SIG.KeyGen()
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8. envelope = AEAD.Enc(rw, sk_U || sk_auth)
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9. record = (pk_U, pk_auth, envelope, salt)
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─────── record ──────────►
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10. Store(U, record)
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```
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### 4.2 Login Protocol (KE1 → KE2 → KE3)
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```
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Client(pw) Server(oprf_key, record)
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═══════════════════════════════════════════════════════════════
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KE1: Client → Server
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1. pw_hash = Argon2id(pw, record.salt)
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2. (state, blind) = OPRF.Blind(pw_hash)
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3. (ek_C, dk_C) = KEM.KeyGen()
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───── KE1: (blind, ek_C) ─────►
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KE2: Server → Client
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4. eval = OPRF.Eval(oprf_key, blind)
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5. (ct, ss_S) = KEM.Encap(ek_C)
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6. K_session = KDF(ss_S, transcript)
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7. mac_S = MAC(K_session, transcript)
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8. sig = SIG.Sign(sk_S, transcript)
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◄─── KE2: (eval, ct, mac_S, sig, envelope) ───
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KE3: Client → Server
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9. Verify sig with pk_S
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10. rw = OPRF.Finalize(state, eval)
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11. (sk_U, sk_auth) = AEAD.Dec(rw, envelope)
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12. ss_C = KEM.Decap(dk_C, ct)
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13. K_session = KDF(ss_C, transcript)
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14. Verify mac_S
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15. mac_C = MAC(K_session, transcript)
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───── KE3: mac_C ─────►
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16. Verify mac_C
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17. Output K_session Output K_session
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```
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### 4.3 VOPRF Extension
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For verifiable evaluation, server additionally:
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```
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Server with committed key (k, c, r):
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1. Compute eval = F_k(blind)
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2. Generate ZK proof π proving:
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- Knowledge of k such that c = Commit(k; r)
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- eval = F_k(blind)
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3. Return (eval, π)
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Client:
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1. Verify π against commitment c
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2. If valid: proceed with finalization
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3. If invalid: abort
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```
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---
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## 5. Simulator Construction
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### 5.1 Simulator for F_VOPRF
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```
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Simulator SIM_VOPRF
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On (Init, sid, S) from F_VOPRF:
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- Sample random key k_sim
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- Compute c = Commit(k_sim; r_sim)
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- Send (Param, S, c) to F_VOPRF
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- Record (S, k_sim, r_sim, c)
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On (Eval, sid, ssid, U, S) from F_VOPRF:
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If S honest:
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- Wait for adversary to deliver
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- On delivery: send (SndrComplete, sid, ssid) to F_VOPRF
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- Send (RcvCmplt, sid, ssid, U, c_S) to F_VOPRF
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If S corrupted:
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- Extract adversary's evaluation blind_A
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- Query F_VOPRF for T_sid(c_A, blind_A)
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- Program H₂ to output T_sid value
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- Simulate protocol messages
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On H₂ query (x, y, c) from A:
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If ∃ honest S with params[S] = c:
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- Send (OfflineEval, sid, c, H₁⁻¹(x)) to F_VOPRF
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- Receive ρ from F_VOPRF
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- Return ρ
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Else:
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- Return random value or use table
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```
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### 5.2 Simulator for F_aPAKE
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```
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Simulator SIM_aPAKE
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Registration Simulation:
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On (Register, sid, U, S, pw) from F_aPAKE:
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- Simulate OPRF blind message
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- Receive adversarial evaluation
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- Extract any password guesses
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- Complete registration
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Login Simulation (Honest Client, Honest Server):
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- Simulate KE1 with random blind
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- Simulate KE2 with random evaluation
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- On (TestPwd, sid, U, pw') from A:
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- Forward to F_aPAKE
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- If compromised: program keys accordingly
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- Generate session key from F_aPAKE
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- Program MAC/KDF oracles consistently
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Login Simulation (Corrupted Server):
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- Extract server's OPRF key from state
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- Use real OPRF evaluation
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- Monitor password test queries
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- Enforce "one online guess per session"
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Login Simulation (Corrupted Client):
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- Extract client's password from state
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- Use real protocol execution
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- Provide adversary with session key
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```
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---
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## 6. Security Proof
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### 6.1 Main Theorem
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**Theorem 6.1 (UC Security)**
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The opaque-lattice protocol UC-realizes F_aPAKE in the (F_VOPRF, F_RO)-hybrid model, assuming:
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1. Ring-LPR is pseudorandom (Definition 2.1)
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2. ML-KEM is IND-CCA2 secure
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3. ML-DSA is EUF-CMA secure
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4. AEAD is IND-CPA and INT-CTXT secure
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5. HKDF is a secure PRF
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The advantage of any PPT adversary A in distinguishing real from ideal execution is:
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```
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Adv(A) ≤ q_pwd · Adv_LPR + q_KEM · Adv_IND-CCA + q_SIG · Adv_EUF-CMA
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+ q_AEAD · Adv_AEAD + q_sessions · negl(λ)
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```
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where q_* denotes the number of respective queries.
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### 6.2 Proof by Game Sequence
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**Game 0 (Real Protocol)**
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The real execution of opaque-lattice with adversary A.
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**Game 1 (Random Oracle Instrumentation)**
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Replace hash functions H₁, H₂, H₃ with random oracles maintained by simulator.
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- Indistinguishable by random oracle assumption
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**Game 2 (OPRF Simulation)**
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Replace real OPRF evaluations with queries to F_VOPRF.
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- For honest server: outputs are random (Ring-LPR pseudorandomness)
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- For corrupted server: extract key, compute real evaluation
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*Lemma 6.1:* |Pr[Game 2] - Pr[Game 1]| ≤ q_oprf · Adv_LPR
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**Game 3 (KEM Simulation)**
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Replace KEM encapsulation with F_KEM ideal functionality.
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- Honest parties: shared secret is random
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- Corrupted parties: extract/inject values
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*Lemma 6.2:* |Pr[Game 3] - Pr[Game 2]| ≤ q_kem · Adv_IND-CCA
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**Game 4 (Signature Simulation)**
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Replace signatures with F_SIG ideal functionality.
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- Verify signatures using committed public key
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- Reject any forgeries
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*Lemma 6.3:* |Pr[Game 4] - Pr[Game 3]| ≤ q_sig · Adv_EUF-CMA
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**Game 5 (Envelope Simulation)**
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Replace AEAD with ideal encryption.
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- Envelope contents are hidden until rw is known
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- Tampering detected by INT-CTXT
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*Lemma 6.4:* |Pr[Game 5] - Pr[Game 4]| ≤ q_aead · Adv_AEAD
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**Game 6 (Password Test Restriction)**
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Enforce that adversary must make explicit TestPwd query to F_aPAKE.
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- Each online session allows at most one password test
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- Offline dictionary attack requires OPRF evaluation
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*Lemma 6.5:* |Pr[Game 6] - Pr[Game 5]| ≤ negl(λ)
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**Game 7 (Ideal Execution)**
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Execute with F_aPAKE and simulator SIM.
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- Session keys are random unless compromised
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- Password never revealed to honest parties
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*Lemma 6.6:* Game 6 ≡ Game 7
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### 6.3 Verifiability Proof
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**Theorem 6.2 (VOPRF Soundness)**
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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.
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*Proof Sketch:*
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1. By binding property of commitment: A cannot open to different k
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2. By soundness of sigma protocol: A cannot forge proofs
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3. By Fiat-Shamir security: Non-interactive proofs are sound in ROM
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**Theorem 6.3 (VOPRF Zero-Knowledge)**
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The sigma protocol proof reveals nothing about k beyond the validity of the statement.
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*Proof Sketch:*
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1. Construct simulator S that generates accepting proofs without k
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2. S samples response z uniformly, computes mask m = z - e·k_dummy
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3. By rejection sampling analysis: real and simulated distributions are statistically close
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4. Distinguishing advantage bounded by 2^(-λ)
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---
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## 7. Concrete Security Bounds
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### 7.1 Parameter Selection
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| Parameter | Value | Security Level |
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|-----------|-------|----------------|
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| Ring dimension n | 256 | 128-bit post-quantum |
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| Ring modulus q | 4 | Minimal for rounding |
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| KEM security | Kyber768 | NIST Level 3 |
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| Signature security | Dilithium3 | NIST Level 3 |
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| Hash output | 512 bits | Collision resistance |
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| Commitment nonce | 256 bits | Binding security |
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### 7.2 Concrete Advantages
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Assuming λ = 128 security parameter:
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| Component | Advantage Bound |
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|-----------|-----------------|
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| Ring-LPR PRF | 2^(-128) (DCP hardness) |
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| ML-KEM IND-CCA | 2^(-128) (MLWE hardness) |
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| ML-DSA EUF-CMA | 2^(-128) (MLWE+SIS hardness) |
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| AEAD (AES-GCM) | 2^(-128) |
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| HKDF-SHA512 | 2^(-256) |
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| Commitment binding | 2^(-128) (collision resistance) |
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| ZK soundness | 2^(-128) (sigma protocol) |
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### 7.3 Attack Complexity
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| Attack | Complexity | Mitigation |
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|--------|------------|------------|
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| Offline dictionary | Requires OPRF oracle | One guess per session |
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| Online brute force | O(2^128) sessions | Rate limiting |
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| 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
|