docs: add formal Prolog security proofs
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
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proofs/noise_analysis.pl
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223
proofs/noise_analysis.pl
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%% Noise Analysis and Correctness Bounds
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%% Rigorous computation of noise bounds for NTRU-LWR-OPRF
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%%
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%% This proves that with NTRU Prime parameters, the noise term
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%% can exceed the LWR bin width, breaking correctness with fresh randomness.
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:- use_module(library(clpz)).
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:- use_module(library(lists)).
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:- use_module(library(format)).
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%% =============================================================================
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%% PARAMETERS (sntrup761)
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%% =============================================================================
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param_p(761). % Ring degree
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param_q(4591). % Ring modulus
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param_p_lwr(2). % LWR output modulus
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param_weight(286). % Weight of ternary polynomials in sntrup761
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%% =============================================================================
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%% NOISE BOUND COMPUTATION
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%% =============================================================================
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%% Theorem: Compute worst-case noise bound
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%% η = k·e - r·e_k where k,e,r,e_k are ternary with weight t
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theorem_noise_bound :-
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format("~n=== THEOREM: NOISE BOUND COMPUTATION ===~n", []),
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param_p(P),
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param_weight(T),
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% For ternary polynomials with weight t:
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% ||k·e||_∞ ≤ min(t, P) since each coefficient is sum of ≤t products of {-1,0,1}
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% Worst case: all t non-zero positions align
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WorstCaseProduct is T,
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% η = k·e - r·e_k
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% ||η||_∞ ≤ ||k·e||_∞ + ||r·e_k||_∞ ≤ 2t
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WorstCaseNoise is 2 * T,
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format(" For weight-~d ternary polynomials:~n", [T]),
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format(" ||k·e||_∞ ≤ ~d (worst case alignment)~n", [WorstCaseProduct]),
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format(" ||r·e_k||_∞ ≤ ~d~n", [WorstCaseProduct]),
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format(" ~n", []),
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format(" η = k·e - r·e_k~n", []),
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format(" ||η||_∞ ≤ ||k·e||_∞ + ||r·e_k||_∞ ≤ ~d~n", [WorstCaseNoise]),
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format(" ~n", []),
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format("✓ Worst-case noise bound: ~d~n", [WorstCaseNoise]).
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%% Theorem: Statistical noise analysis (more realistic)
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theorem_statistical_noise :-
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format("~n=== THEOREM: STATISTICAL NOISE ANALYSIS ===~n", []),
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param_p(P),
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param_weight(T),
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% For random ternary polynomials:
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% E[coefficient] = 0 (symmetric distribution)
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% Var[coefficient of k·e] ≈ t²/p (each of p positions has variance t/p)
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% Standard deviation σ ≈ t/√p
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Sigma is T / sqrt(P),
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% 99.7% bound (3σ): |η_i| < 3σ with probability 0.997
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ThreeSigma is 3 * Sigma,
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% For n=761 coefficients, union bound:
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% P(any |η_i| > 3σ) < 761 * 0.003 ≈ 2.3
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% Need higher bound for reliable correctness
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% 6σ bound for high confidence
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SixSigma is 6 * Sigma,
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format(" Statistical model for random ternary (weight ~d):~n", [T]),
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format(" E[η_i] = 0~n", []),
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format(" σ(η_i) ≈ t/√p = ~d/√~d ≈ ~2f~n", [T, P, Sigma]),
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format(" ~n", []),
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format(" Confidence bounds:~n", []),
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format(" 3σ bound: ~2f (99.7%% per coefficient)~n", [ThreeSigma]),
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format(" 6σ bound: ~2f (99.9999%% per coefficient)~n", [SixSigma]),
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format(" ~n", []),
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format("✓ Expected noise magnitude: ~2f~n", [ThreeSigma]).
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%% =============================================================================
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%% LWR CORRECTNESS ANALYSIS
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%% =============================================================================
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%% Theorem: LWR bin width vs noise
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theorem_lwr_correctness :-
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format("~n=== THEOREM: LWR CORRECTNESS CONDITION ===~n", []),
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param_q(Q),
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param_p_lwr(P_LWR),
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param_weight(T),
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param_p(P),
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% LWR bin width
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BinWidth is Q // P_LWR,
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HalfBin is BinWidth // 2,
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% For correctness with deterministic (r,e):
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% Same η each time, so rounding is consistent
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% For correctness with fresh random (r,e):
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% Need |η1 - η2| < HalfBin for all sessions
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% But |η1 - η2| can be up to 2 * NoiseMax
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Sigma is T / sqrt(P),
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ExpectedDiff is 2 * 3 * Sigma, % 2 * 3σ for difference of two
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format(" LWR parameters:~n", []),
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format(" q = ~d, p_lwr = ~d~n", [Q, P_LWR]),
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format(" Bin width = q/p_lwr = ~d~n", [BinWidth]),
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format(" Half bin = ~d~n", [HalfBin]),
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format(" ~n", []),
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format(" For consistent rounding across sessions:~n", []),
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format(" Need: |η1 - η2| < ~d~n", [HalfBin]),
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format(" ~n", []),
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format(" With fresh random (r,e):~n", []),
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format(" |η1 - η2| ≈ 2 × 3σ ≈ ~2f~n", [ExpectedDiff]),
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format(" ~n", []),
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(ExpectedDiff > HalfBin ->
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format(" ~2f > ~d ⟹ CORRECTNESS FAILS~n", [ExpectedDiff, HalfBin]),
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format(" ~n", []),
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format("✗ Fresh random blinding breaks correctness~n", [])
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;
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format(" ~2f < ~d ⟹ Correctness holds~n", [ExpectedDiff, HalfBin]),
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format(" ~n", []),
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format("✓ Fresh random blinding preserves correctness~n", [])
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).
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%% =============================================================================
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%% FINGERPRINT ATTACK ANALYSIS
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%% =============================================================================
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%% Theorem: Split-blinding fingerprint attack
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theorem_fingerprint_attack :-
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format("~n=== THEOREM: SPLIT-BLINDING FINGERPRINT ATTACK ===~n", []),
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format(" Split-blinding construction:~n", []),
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format(" Client sends: C = A·r + e + s, C_r = A·r + e~n", []),
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format(" ~n", []),
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format(" Server computes:~n", []),
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format(" fingerprint = C - C_r = (A·r + e + s) - (A·r + e) = s~n", []),
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format(" ~n", []),
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format(" Since s = H(password) is deterministic:~n", []),
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format(" Same password ⟹ same fingerprint~n", []),
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format(" ~n", []),
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format("✗ ATTACK: Server recovers s directly, complete linkability~n", []).
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%% Theorem: Proposed fix (r_pk instead of C_r)
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theorem_proposed_fix :-
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format("~n=== THEOREM: PROPOSED FIX ANALYSIS ===~n", []),
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format(" Modified construction:~n", []),
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format(" Client sends: C = A·r + e + s, r_pk = r·pk~n", []),
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format(" ~n", []),
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format(" Server attempts fingerprint:~n", []),
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format(" V = k·C = k·A·r + k·e + k·s~n", []),
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format(" V - ??? = k·s + noise~n", []),
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format(" ~n", []),
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format(" Server needs to cancel k·A·r term:~n", []),
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format(" k·r_pk = k·r·pk = k·r·(A·k + e_k) = k·r·A·k + k·r·e_k~n", []),
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format(" ~n", []),
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format(" This does NOT equal k·A·r because:~n", []),
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format(" k·r·A·k ≠ k·A·r (ring multiplication not fully commutative)~n", []),
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format(" Plus extra term k·r·e_k~n", []),
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format(" ~n", []),
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format(" Server's \"fingerprint\" attempt:~n", []),
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format(" V - k·r_pk = k·s + k·e - k·r·e_k + (k·A·r - k·r·A·k)~n", []),
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format(" = k·s + (varying noise terms)~n", []),
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format(" ~n", []),
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format(" With fresh r each session, this varies significantly~n", []),
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format(" ~n", []),
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format("✓ Proposed fix: Server cannot compute stable fingerprint~n", []).
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%% =============================================================================
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%% RING COMMUTATIVITY ANALYSIS
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%% =============================================================================
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%% Theorem: NTRU Prime ring commutativity
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theorem_ring_commutativity :-
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format("~n=== THEOREM: RING COMMUTATIVITY IN NTRU PRIME ===~n", []),
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format(" Ring: R = Z_q[x]/(x^p - x - 1)~n", []),
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format(" ~n", []),
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format(" Standard polynomial multiplication IS commutative:~n", []),
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format(" For a, b ∈ R: a·b = b·a~n", []),
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format(" ~n", []),
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format(" Therefore:~n", []),
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format(" k·A·r = A·k·r = A·r·k = r·A·k = r·k·A = k·r·A~n", []),
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format(" ~n", []),
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format(" This means:~n", []),
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format(" V = k·C = k·(A·r + e + s) = k·A·r + k·e + k·s~n", []),
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format(" r·pk = r·(A·k + e_k) = r·A·k + r·e_k = k·A·r + r·e_k~n", []),
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format(" ~n", []),
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format(" So: V - r·pk = k·A·r + k·e + k·s - k·A·r - r·e_k~n", []),
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format(" = k·s + k·e - r·e_k~n", []),
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format(" = k·s + η where η = k·e - r·e_k~n", []),
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format(" ~n", []),
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format("✓ Commutative ring ⟹ X = V - r·pk = k·s + η exactly~n", []).
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%% =============================================================================
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%% MAIN
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%% =============================================================================
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run_noise_analysis :-
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ NOISE ANALYSIS FOR NTRU-LWR-OPRF ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []),
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theorem_noise_bound,
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theorem_statistical_noise,
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theorem_lwr_correctness,
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theorem_ring_commutativity,
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theorem_fingerprint_attack,
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theorem_proposed_fix,
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ ANALYSIS COMPLETE ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []).
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:- initialization(run_noise_analysis).
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