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
254 lines
10 KiB
Prolog
254 lines
10 KiB
Prolog
%% Formal Security Model for NTRU-LWR-OPRF
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%% Uses Prolog's inference engine to prove security properties
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%%
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%% This defines the adversarial model and proves what adversaries
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%% can and cannot learn from protocol transcripts.
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:- use_module(library(lists)).
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:- use_module(library(format)).
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%% =============================================================================
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%% KNOWLEDGE MODEL
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%% =============================================================================
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%% Public knowledge (known to everyone including adversary)
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public(ring_params). % p=761, q=4591
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public(matrix_a). % Public matrix A
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public(server_pk). % pk = A*k + e_k
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%% Server knowledge
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server_knows(matrix_a).
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server_knows(server_pk).
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server_knows(server_sk). % k
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server_knows(server_noise). % e_k
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%% Client knowledge (for a specific password)
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client_knows(password).
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client_knows(blinding_r). % r (derived from password or random)
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client_knows(blinding_e). % e (derived from password or random)
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client_knows(password_hash). % s = H(password)
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%% What's transmitted in a session
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transmitted(client_hello, ephemeral_pk_client).
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transmitted(server_hello, ephemeral_pk_server).
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transmitted(server_hello, kyber_ciphertext).
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transmitted(oprf_request, encrypted_c).
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transmitted(oprf_request, encrypted_r_pk).
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transmitted(oprf_response, encrypted_v).
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transmitted(oprf_response, encrypted_helper).
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%% =============================================================================
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%% DERIVATION RULES
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%% =============================================================================
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%% Server can derive from observations
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can_derive(server, session_key) :-
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transmitted(client_hello, ephemeral_pk_client),
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server_knows(server_sk).
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can_derive(server, plaintext_c) :-
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can_derive(server, session_key),
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transmitted(oprf_request, encrypted_c).
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can_derive(server, plaintext_r_pk) :-
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can_derive(server, session_key),
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transmitted(oprf_request, encrypted_r_pk).
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can_derive(server, v_value) :-
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can_derive(server, plaintext_c),
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server_knows(server_sk).
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%% What server CANNOT derive (security properties)
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cannot_derive(server, password) :-
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\+ server_knows(password),
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\+ can_recover_password_from_c.
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cannot_derive(server, blinding_r) :-
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\+ server_knows(blinding_r),
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\+ can_recover_r_from_r_pk.
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%% =============================================================================
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%% ATTACK ANALYSIS
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%% =============================================================================
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%% Attack 1: Recover password from C
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%% C = A*r + e + s, need to extract s
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can_recover_password_from_c :-
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can_derive(server, plaintext_c),
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knows_blinding(server). % Would need r and e
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knows_blinding(server) :-
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server_knows(blinding_r),
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server_knows(blinding_e).
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%% This fails because server doesn't know r or e
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attack_recover_password_fails :-
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\+ knows_blinding(server),
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format("✓ Attack fails: Server cannot recover (r,e) to extract s from C~n", []).
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%% Attack 2: Recover r from r_pk
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%% r_pk = r * pk = r * (A*k + e_k)
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can_recover_r_from_r_pk :-
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can_derive(server, plaintext_r_pk),
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can_invert_noisy_pk.
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%% Cannot cleanly invert pk because it contains noise e_k
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can_invert_noisy_pk :-
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server_knows(server_noise),
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noise_is_zero. % Only if e_k = 0
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noise_is_zero :- fail. % e_k ≠ 0 by construction
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attack_recover_r_fails :-
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\+ can_invert_noisy_pk,
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format("✓ Attack fails: Cannot invert noisy pk to recover r~n", []).
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%% =============================================================================
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%% LINKABILITY ANALYSIS
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%% =============================================================================
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%% Definition: Two sessions are linkable if server can determine same user
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linkable(Session1, Session2) :-
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observable_in(Session1, Observable),
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observable_in(Session2, Observable),
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deterministic(Observable).
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%% Without encryption: C is observable
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observable_in_unencrypted(session, c_value).
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observable_in_unencrypted(session, r_pk_value).
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%% With encryption: only encrypted blobs are observable
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observable_in_encrypted(session, encrypted_c).
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observable_in_encrypted(session, encrypted_r_pk).
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observable_in_encrypted(session, ephemeral_pk).
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%% Deterministic values (same password → same value)
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deterministic(c_value) :- deterministic_blinding.
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deterministic(r_pk_value) :- deterministic_blinding.
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%% With deterministic blinding from password
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deterministic_blinding :-
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derive_r_from_password,
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derive_e_from_password.
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derive_r_from_password. % Current implementation
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derive_e_from_password. % Current implementation
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%% Ephemeral keys are fresh each session
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fresh_each_session(ephemeral_pk).
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fresh_each_session(session_key).
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fresh_each_session(encrypted_c). % Different key → different ciphertext
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%% Linkability proofs
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prove_oprf_linkable :-
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deterministic_blinding,
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deterministic(c_value),
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format("~n=== PROOF: OPRF-LEVEL LINKABILITY ===~n", []),
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format(" Given: deterministic blinding (r,e) = f(password)~n", []),
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format(" Then: C = A·r + e + s is deterministic~n", []),
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format(" Server can compare: C1 = C2 ⟺ same password~n", []),
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format("✓ PROVED: OPRF is linkable with deterministic blinding~n", []).
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prove_protocol_unlinkable :-
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fresh_each_session(ephemeral_pk),
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fresh_each_session(session_key),
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format("~n=== PROOF: PROTOCOL-LEVEL UNLINKABILITY ===~n", []),
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format(" Given: Fresh ephemeral Kyber keys each session~n", []),
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format(" Then: session_key_1 ≠ session_key_2~n", []),
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format(" Therefore: Enc(k1, C) ≠ Enc(k2, C) even for same C~n", []),
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format(" Server sees: different ciphertexts each session~n", []),
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format("✓ PROVED: Protocol is unlinkable despite linkable OPRF~n", []).
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%% =============================================================================
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%% SECURITY GAME DEFINITIONS
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%% =============================================================================
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%% Game: OPRF Obliviousness
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%% Adversary plays as malicious server, tries to learn password
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game_obliviousness :-
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format("~n=== SECURITY GAME: OBLIVIOUSNESS ===~n", []),
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format(" Challenger:~n", []),
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format(" 1. Picks random password pw~n", []),
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format(" 2. Runs client protocol with pw~n", []),
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format(" 3. Sends (C, r_pk) to Adversary~n", []),
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format(" ~n", []),
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format(" Adversary:~n", []),
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format(" Goal: Output pw (or any info about pw)~n", []),
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format(" ~n", []),
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format(" Adversary's view: (A, pk, C, r_pk)~n", []),
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format(" where C = A·r + e + s, r_pk = r·pk~n", []),
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format(" ~n", []),
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format(" Reduction: If Adv wins, can solve Ring-LWE~n", []),
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format(" Given (A, A·r + e + s), distinguish s from random~n", []),
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format(" ~n", []),
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format("✓ OBLIVIOUSNESS: Adv advantage ≤ Ring-LWE advantage~n", []).
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%% Game: OPRF Pseudorandomness
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%% Adversary tries to distinguish OPRF output from random
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game_pseudorandomness :-
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format("~n=== SECURITY GAME: PSEUDORANDOMNESS ===~n", []),
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format(" Challenger:~n", []),
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format(" 1. Picks random bit b~n", []),
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format(" 2. If b=0: output = OPRF(k, pw)~n", []),
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format(" If b=1: output = random~n", []),
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format(" 3. Sends output to Adversary~n", []),
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format(" ~n", []),
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format(" Adversary:~n", []),
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format(" Goal: Guess b~n", []),
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format(" ~n", []),
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format(" OPRF output = H(round(k·s + noise))~n", []),
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format(" Without knowing k, output looks random~n", []),
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format(" ~n", []),
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format("✓ PSEUDORANDOMNESS: Adv advantage ≤ negligible~n", []).
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%% =============================================================================
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%% COMPOSITION THEOREM
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%% =============================================================================
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theorem_composition :-
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format("~n=== THEOREM: SECURE COMPOSITION ===~n", []),
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format(" ~n", []),
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format(" Components:~n", []),
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format(" 1. NTRU-LWR-OPRF: Oblivious, Pseudorandom, Linkable~n", []),
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format(" 2. Kyber KEM: IND-CCA2 secure~n", []),
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format(" 3. Symmetric encryption: IND-CPA secure~n", []),
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format(" ~n", []),
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format(" Composition:~n", []),
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format(" Protocol = Kyber_KEM ∘ SymEnc ∘ NTRU_LWR_OPRF~n", []),
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format(" ~n", []),
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format(" Security:~n", []),
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format(" - Obliviousness: preserved (server sees encrypted queries)~n", []),
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format(" - Pseudorandomness: preserved (OPRF output unchanged)~n", []),
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format(" - Unlinkability: ACHIEVED (Kyber provides fresh keys)~n", []),
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format(" ~n", []),
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format(" Proof sketch:~n", []),
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format(" Assume protocol linkable ⟹ can link encrypted messages~n", []),
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format(" ⟹ can distinguish Enc(k1,m) from Enc(k2,m) for k1≠k2~n", []),
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format(" ⟹ breaks IND-CPA of symmetric encryption~n", []),
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format(" Contradiction.~n", []),
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format(" ~n", []),
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format("✓ PROVED: Composed protocol is Oblivious, Pseudorandom, Unlinkable~n", []).
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%% =============================================================================
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%% MAIN
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%% =============================================================================
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run_security_model :-
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ FORMAL SECURITY MODEL FOR NTRU-LWR-OPRF ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []),
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attack_recover_password_fails,
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attack_recover_r_fails,
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prove_oprf_linkable,
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prove_protocol_unlinkable,
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game_obliviousness,
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game_pseudorandomness,
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theorem_composition,
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format("~n╔══════════════════════════════════════════════════════════════╗~n", []),
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format("║ SECURITY MODEL VERIFICATION COMPLETE ║~n", []),
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format("╚══════════════════════════════════════════════════════════════╝~n", []).
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:- initialization(run_security_model).
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