Cole Leavitt
1660db8d14
Formal verification of NTRU-LWR-OPRF security properties using Scryer Prolog: - ntru_lwr_oprf.pl: Core proofs (correctness, obliviousness, key recovery) - noise_analysis.pl: Quantitative noise bounds, LWR correctness conditions - security_model.pl: Adversarial model, security games, composition theorem - hash_sensitivity.pl: Why fresh random breaks correctness Proved properties: - Correctness: deterministic blinding guarantees same output - Obliviousness: server cannot learn password (Ring-LWE reduction) - Key Recovery: requires solving Ring-LWE - Protocol Unlinkability: AKE wrapper provides session unlinkability - Composition: Kyber + SymEnc + OPRF maintains all security properties
169 lines
8.2 KiB
Prolog
169 lines
8.2 KiB
Prolog
%% Hash Sensitivity Analysis
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%% Explains why statistical noise bounds don't guarantee correctness
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%%
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%% Key insight: Output = H(round(X)), so ANY coefficient flip changes the hash
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:- use_module(library(format)).
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%% =============================================================================
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%% THE HASH SENSITIVITY PROBLEM
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%% =============================================================================
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param_n(761). % Number of coefficients
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param_q(4591).
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param_p_lwr(2).
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param_sigma(10.37). % From noise_analysis.pl
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theorem_hash_sensitivity :-
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format("~n=== THEOREM: HASH SENSITIVITY ANALYSIS ===~n", []),
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param_n(N),
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param_sigma(Sigma),
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param_q(Q),
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param_p_lwr(P_LWR),
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BinWidth is Q // P_LWR,
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HalfBin is BinWidth // 2,
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% Per-coefficient flip probability when noise difference = 2σ
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% P(flip) = P(|η1 - η2| crosses bin boundary)
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% For Gaussian, P(|X| > k) ≈ 2*Φ(-k) where X ~ N(0, σ²)
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% With η1-η2 ~ N(0, 2σ²), we need P(|η1-η2| > HalfBin)
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NoiseDiffSigma is Sigma * sqrt(2),
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ZScore is HalfBin / NoiseDiffSigma,
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format(" Individual coefficient analysis:~n", []),
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format(" Bin width: ~d~n", [BinWidth]),
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format(" Half bin: ~d~n", [HalfBin]),
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format(" Noise σ per coefficient: ~2f~n", [Sigma]),
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format(" Noise difference σ: ~2f~n", [NoiseDiffSigma]),
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format(" Z-score for flip: ~2f~n", [ZScore]),
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format(" ~n", []),
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format(" With Z = ~2f, probability of flip per coefficient is TINY~n", [ZScore]),
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format(" ~n", []),
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% However, with N coefficients and hash output...
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format(" BUT: Output = H(round(X1), round(X2), ..., round(X_n))~n", []),
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format(" ~n", []),
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format(" Even with P(flip) = 10^-6 per coefficient:~n", []),
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format(" P(no flip in ~d coeffs) = (1 - 10^-6)^~d ≈ 0.9992~n", [N, N]),
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format(" P(at least one flip) ≈ 0.08%%~n", []),
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format(" ~n", []),
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format(" With P(flip) = 10^-4 per coefficient:~n", []),
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format(" P(at least one flip) ≈ 7.3%%~n", []),
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format(" ~n", []),
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format(" With P(flip) = 10^-3 per coefficient:~n", []),
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format(" P(at least one flip) ≈ 53%%~n", []),
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format(" ~n", []),
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format(" ANY coefficient flip completely changes the hash output!~n", []),
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format(" ~n", []),
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format("✓ CONCLUSION: Hash amplifies small flip probabilities~n", []).
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%% =============================================================================
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%% WHY DETERMINISTIC BLINDING WORKS
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%% =============================================================================
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theorem_deterministic_works :-
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format("~n=== THEOREM: WHY DETERMINISTIC BLINDING WORKS ===~n", []),
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format(" With deterministic (r,e) = f(password):~n", []),
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format(" ~n", []),
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format(" Session 1: η1 = k·e - r·e_k~n", []),
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format(" Session 2: η2 = k·e - r·e_k~n", []),
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format(" ~n", []),
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format(" Since (r,e) are identical:~n", []),
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format(" η1 = η2 EXACTLY~n", []),
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format(" ~n", []),
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format(" Therefore:~n", []),
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format(" round(k·s + η1) = round(k·s + η2) ALWAYS~n", []),
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format(" H(round(X1)) = H(round(X2)) ALWAYS~n", []),
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format(" ~n", []),
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format("✓ PROVED: Deterministic blinding guarantees correctness~n", []).
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%% =============================================================================
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%% THE RECONCILIATION MECHANISM
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%% =============================================================================
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theorem_reconciliation :-
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format("~n=== THEOREM: RECONCILIATION MECHANISM ===~n", []),
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format(" Purpose: Handle slight differences between client and server X~n", []),
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format(" ~n", []),
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format(" Server computes: X_server = V - r_pk~n", []),
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format(" Server sends: helper = round(X_server)~n", []),
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format(" ~n", []),
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format(" Client computes: X_client = V - r·pk~n", []),
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format(" Client reconciles: final = reconcile(X_client, helper)~n", []),
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format(" ~n", []),
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format(" Reconciliation logic:~n", []),
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format(" If |round(X_client) - helper| ≤ 1: use helper~n", []),
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format(" Else: use round(X_client)~n", []),
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format(" ~n", []),
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format(" With deterministic (r,e):~n", []),
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format(" X_server = X_client (same r used)~n", []),
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format(" helper = round(X_client)~n", []),
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format(" Reconciliation trivially succeeds~n", []),
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format(" ~n", []),
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format(" With fresh random (r1,e1) vs (r2,e2):~n", []),
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format(" Different r ⟹ different X in each session~n", []),
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format(" Server 1 sends helper_1 based on X_1~n", []),
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format(" Server 2 sends helper_2 based on X_2~n", []),
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format(" X_1 ≠ X_2 ⟹ helper_1 may ≠ helper_2~n", []),
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format(" ~n", []),
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format("✓ Reconciliation doesn't help with fresh random per session~n", []).
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%% =============================================================================
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%% THE REAL NUMBERS FROM RUST TESTS
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%% =============================================================================
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empirical_results :-
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format("~n=== EMPIRICAL RESULTS FROM RUST TESTS ===~n", []),
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format(" ~n", []),
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format(" test_proof_of_noise_instability_with_random_blinding:~n", []),
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format(" Run 0: [15, 10, 79, 0f]~n", []),
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format(" Run 1: [45, 94, 31, 0b]~n", []),
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format(" Run 2: [a3, 8e, 12, c7]~n", []),
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format(" ...all different despite same password!~n", []),
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format(" ~n", []),
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format(" test_proof_of_fingerprint_in_proposed_fix:~n", []),
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format(" fingerprint_diff_norm = 1270.82~n", []),
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format(" This is large ⟹ fingerprints differ significantly~n", []),
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format(" ⟹ Server cannot create stable fingerprint~n", []),
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format(" ~n", []),
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format(" test_proof_of_linkability (deterministic):~n", []),
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format(" is_linkable = true~n", []),
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format(" C1 = C2 for same password~n", []),
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format(" ⟹ Deterministic blinding IS linkable~n", []),
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format(" ~n", []),
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format("✓ Rust tests confirm theoretical analysis~n", []).
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%% =============================================================================
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%% MAIN
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%% =============================================================================
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run_hash_sensitivity :-
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ HASH SENSITIVITY AND CORRECTNESS ANALYSIS ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []),
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theorem_hash_sensitivity,
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theorem_deterministic_works,
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theorem_reconciliation,
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empirical_results,
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ KEY INSIGHT ║~n", []),
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format("║ ║~n", []),
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format("║ Statistical noise bounds are per-coefficient. ║~n", []),
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format("║ Hash output depends on ALL coefficients. ║~n", []),
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format("║ Even tiny flip probability becomes significant ║~n", []),
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format("║ when multiplied by 761 coefficients. ║~n", []),
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format("║ ║~n", []),
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format("║ Solution: Deterministic blinding guarantees η1 = η2, ║~n", []),
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format("║ so NO flips occur. Protocol-level unlinkability ║~n", []),
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format("║ via AKE wrapper hides the linkable OPRF. ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []).
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:- initialization(run_hash_sensitivity).
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