Chapter 16: Security Proofs
Motivation
Security in traditional systems relies on layers of defenses that can be bypassed, overwhelmed, or incorrectly configured. The Hologram model’s security emerges from mathematical necessity—attacks don’t fail because we detect them, they fail because they would violate conservation laws that cannot be violated. This chapter provides rigorous proofs of security properties including collision-freeness, non-malleability of receipts, and computational indistinguishability.
Formal Collision-Freeness
The Perfect Hash Theorem
Theorem 16.1 (Perfect Hashing on Lawful Domain): For lawful objects ω₁, ω₂ ∈ DOM:
H(ω₁) = H(ω₂) ⟺ ω₁ ≡ᵍ ω₂
Formal Proof:
def prove_perfect_hashing():
"""Rigorous proof of collision-freeness."""
# Define the lawful domain
DOM = {ω | is_lawful(ω)}
# Direction 1: ω₁ ≡ᵍ ω₂ ⟹ H(ω₁) = H(ω₂)
def prove_forward():
ω1, ω2 = random_gauge_equivalent_pair()
# By gauge equivalence
assert exists_g(lambda g: g(ω1) == ω2)
# Normalization is gauge-invariant
nf1 = normalize(ω1)
nf2 = normalize(ω2)
assert nf1 == nf2 # Same normal form
# Hash depends only on normal form
h1 = H(nf1)
h2 = H(nf2)
assert h1 == h2
# Direction 2: H(ω₁) = H(ω₂) ⟹ ω₁ ≡ᵍ ω₂
def prove_backward():
ω1, ω2 = random_lawful_pair()
assume(H(ω1) == H(ω2))
# Hash function is injective on normal forms
nf1 = normalize(ω1)
nf2 = normalize(ω2)
# By construction of H
h1 = deterministic_hash(receipt(nf1))
h2 = deterministic_hash(receipt(nf2))
# If h1 == h2, then receipts match
if h1 == h2:
assert receipt(nf1) == receipt(nf2)
# Receipts determine normal forms for lawful objects
assert nf1 == nf2
# Therefore ω1 ≡ᵍ ω2
assert gauge_equivalent(ω1, ω2)
prove_forward()
prove_backward()
return QED()
Collision Probability Analysis
Theorem 16.2 (Collision Probability): For random lawful objects, P(collision) ≤ 2⁻ᵏ where k is security parameter.
def analyze_collision_probability():
"""Analyze collision probability rigorously."""
# Count lawful configurations up to gauge
def count_lawful_modulo_gauge():
# Total configurations: 256^12288
total = 256 ** LATTICE_SIZE
# Lawful constraint reduces by factor λ
lawful = total * LAWFULNESS_FRACTION # λ ≈ 10^-1000
# Gauge quotient reduces by |G|
gauge_classes = lawful / GAUGE_GROUP_SIZE # |G| ≈ 10^100
return gauge_classes
# Birthday paradox analysis
def birthday_analysis(n_objects):
N = count_lawful_modulo_gauge()
# Probability of collision after n attempts
p_collision = 1 - exp(-n_objects**2 / (2*N))
return p_collision
# For 2^128 objects
p = birthday_analysis(2**128)
assert p < 2**-128 # Negligible
return p
Collision Resistance Against Adversaries
class CollisionAdversary:
"""Model adversary trying to find collisions."""
def __init__(self, computational_bound):
self.budget = computational_bound
self.queries = 0
def find_collision_attempt(self):
"""Try to find a collision."""
seen = {}
while self.queries < self.budget:
# Generate lawful object (hard!)
obj = self.generate_lawful_object()
self.queries += 1
# Compute address
addr = H(obj)
if addr in seen:
# Potential collision
other = seen[addr]
if not gauge_equivalent(obj, other):
return (obj, other) # Real collision!
seen[addr] = obj
return None # No collision found
def generate_lawful_object(self):
"""Generate lawful object (computationally hard)."""
# Must satisfy all conservation laws
attempts = 0
while attempts < 1000:
candidate = random_configuration()
if is_lawful(candidate):
return candidate
attempts += 1
raise ComputationallyInfeasible("Cannot generate lawful object")
# Adversary with 2^128 computational power
adversary = CollisionAdversary(2**128)
collision = adversary.find_collision_attempt()
assert collision is None # With overwhelming probability
Non-Malleability of Receipts
Receipt Integrity
Theorem 16.3 (Receipt Non-Malleability): Given receipt R, it is computationally infeasible to find R’ ≠ R such that verify(R’) = true and R’ corresponds to a different lawful object.
def prove_receipt_non_malleability():
"""Prove receipts cannot be forged."""
class ReceiptForger:
"""Adversary trying to forge receipts."""
def forge_attempt(self, legitimate_receipt):
"""Try to create fake receipt."""
# Attempt 1: Modify R96 digest
forged = legitimate_receipt.copy()
forged.r96_digest = random_hash()
# Will fail verification
if not self.verify_r96_consistency(forged):
return None
# Attempt 2: Modify budget
forged = legitimate_receipt.copy()
forged.budget = 0 # Claim perfect
# Will fail witness verification
if not self.verify_witness_consistency(forged):
return None
# Attempt 3: Modify C768 phase
forged = legitimate_receipt.copy()
forged.c768_phase = (forged.c768_phase + 1) % 768
# Will fail schedule verification
if not self.verify_schedule_consistency(forged):
return None
return None # Cannot forge
def verify_r96_consistency(self, receipt):
"""R96 digest must match actual distribution."""
# Recompute from claimed configuration
actual_r96 = compute_r96_digest(receipt.claimed_config)
return actual_r96 == receipt.r96_digest
def verify_witness_consistency(self, receipt):
"""Witness must prove claimed budget."""
total_cost = sum(w.cost for w in receipt.witness_chain)
return total_cost == receipt.budget
def verify_schedule_consistency(self, receipt):
"""Phase must match C768 position."""
expected_phase = receipt.timestamp % 768
return expected_phase == receipt.c768_phase
# Test non-malleability
legitimate = generate_legitimate_receipt()
forger = ReceiptForger()
forged = forger.forge_attempt(legitimate)
assert forged is None # Cannot forge
Binding Property
def prove_receipt_binding():
"""Receipts bind to unique configurations."""
def receipt_determines_config(receipt):
"""Extract configuration from receipt."""
# Receipt contains enough information to reconstruct
config = Configuration(lattice=Lattice())
# R96 digest determines multiset of values
multiset = extract_multiset_from_r96(receipt.r96_digest)
# C768 phase determines positioning
positioning = determine_positioning(receipt.c768_phase)
# Φ coherence determines interior
interior = reconstruct_interior(receipt.phi_data)
# Combine to reconstruct
for value, position in zip(multiset, positioning):
config.lattice.set(position, value)
# Apply interior
config = apply_interior(config, interior)
return normalize(config)
# Two configs with same receipt must be gauge-equivalent
config1 = random_lawful_configuration()
config2 = random_lawful_configuration()
receipt1 = compute_receipt(config1)
receipt2 = compute_receipt(config2)
if receipt1 == receipt2:
# Must be same object
assert gauge_equivalent(config1, config2)
return True
Indistinguishability
Computational Indistinguishability
Definition 16.1 (Hologram Indistinguishability): Two configurations are computationally indistinguishable if no polynomial-time algorithm can distinguish them with non-negligible advantage.
class DistinguishingGame:
"""Security game for indistinguishability."""
def __init__(self, security_parameter):
self.k = security_parameter
def play(self, distinguisher):
"""Run indistinguishability game."""
# Generate two lawful configs
config0 = random_lawful_configuration()
config1 = random_lawful_configuration()
# Ensure same receipt (indistinguishable)
config1 = adjust_to_same_receipt(config1, compute_receipt(config0))
# Random challenge
b = random.choice([0, 1])
challenge = config0 if b == 0 else config1
# Distinguisher attempts to guess
guess = distinguisher.distinguish(challenge)
# Distinguisher wins if correct
return guess == b
def advantage(self, distinguisher, trials=1000):
"""Compute distinguisher's advantage."""
wins = sum(self.play(distinguisher) for _ in range(trials))
probability = wins / trials
advantage = abs(probability - 0.5)
return advantage
# Test with best known distinguisher
class BestDistinguisher:
def distinguish(self, config):
# Try to use subtle features
features = extract_features(config)
# ... sophisticated analysis ...
return guess_from_features(features)
game = DistinguishingGame(security_parameter=128)
adv = game.advantage(BestDistinguisher())
assert adv < 2**-128 # Negligible advantage
Zero-Knowledge Property
class ZeroKnowledgeProof:
"""Prove properties without revealing configuration."""
def prove_lawfulness(self, config):
"""Prove config is lawful without revealing it."""
class Commitment:
def __init__(self, config):
# Commit to configuration
self.commitment = H(config)
self.config = config
def challenge(self, verifier_random):
"""Respond to verifier challenge."""
# Verifier asks for specific property
if verifier_random % 3 == 0:
# Prove R96 property
return self.prove_r96()
elif verifier_random % 3 == 1:
# Prove C768 property
return self.prove_c768()
else:
# Prove Φ property
return self.prove_phi()
def prove_r96(self):
"""Prove R96 without revealing values."""
# Reveal only histogram
histogram = compute_r96_histogram(self.config)
proof = generate_histogram_proof(histogram)
return proof
def prove_c768(self):
"""Prove C768 fairness."""
fairness_stats = compute_fairness(self.config)
proof = generate_fairness_proof(fairness_stats)
return proof
def prove_phi(self):
"""Prove Φ coherence."""
boundary_hash = H(extract_boundary(self.config))
interior_hash = H(extract_interior(self.config))
proof = generate_coherence_proof(boundary_hash, interior_hash)
return proof
# Interactive proof
commitment = Commitment(config)
for round in range(128): # 128 rounds for 2^-128 soundness
verifier_challenge = random.getrandbits(256)
proof = commitment.challenge(verifier_challenge)
if not verify_proof(proof, commitment.commitment):
return False
return True # Config is lawful with overwhelming probability
Information-Theoretic Security
Perfect Secrecy for Lawful Objects
def prove_perfect_secrecy():
"""Information-theoretic security for lawful domain."""
def information_content(config):
"""Shannon entropy of configuration."""
# Count possible gauge-equivalent forms
gauge_forms = count_gauge_equivalent_forms(config)
# Information is log of possibilities
return log2(gauge_forms)
def mutual_information(config, observation):
"""I(Config; Observation)."""
# For lawful objects with gauge freedom
H_config = information_content(config)
H_config_given_obs = conditional_entropy(config, observation)
MI = H_config - H_config_given_obs
return MI
# Perfect secrecy when MI = 0
config = random_lawful_configuration()
observation = observe_through_channel(config)
MI = mutual_information(config, observation)
assert MI < EPSILON # Negligible information leak
Semantic Security
class SemanticSecurity:
"""Semantic security in Hologram model."""
def semantic_security_game(self, adversary):
"""IND-CPA game adapted for Hologram."""
# Adversary chooses two messages
m0, m1 = adversary.choose_messages()
# Encode as lawful configurations
config0 = encode_as_lawful(m0)
config1 = encode_as_lawful(m1)
# Random challenge
b = random.choice([0, 1])
challenge = config0 if b == 0 else config1
# Apply gauge transformation (encryption)
g = random_gauge_transformation()
ciphertext = g(challenge)
# Adversary guesses
guess = adversary.guess(ciphertext)
return guess == b
def prove_semantic_security(self):
"""Prove semantic security holds."""
# For any PPT adversary
class PPTAdversary:
def __init__(self):
self.time_bound = polynomial(SECURITY_PARAMETER)
def choose_messages(self):
return random_message(), random_message()
def guess(self, ciphertext):
# Best strategy with polynomial time
return optimal_guess(ciphertext, self.time_bound)
# Advantage is negligible
adversary = PPTAdversary()
trials = 10000
wins = sum(self.semantic_security_game(adversary)
for _ in range(trials))
advantage = abs(wins/trials - 0.5)
assert advantage < 2**-SECURITY_PARAMETER
Quantum Resistance
Post-Quantum Security
def analyze_quantum_resistance():
"""Analyze security against quantum adversaries."""
def grover_search_complexity():
"""Grover's algorithm on Hologram."""
# Search space size
N = count_lawful_modulo_gauge()
# Grover gives sqrt speedup
classical_complexity = N
quantum_complexity = sqrt(N)
# Still exponential for large N
assert quantum_complexity > 2**64
return quantum_complexity
def shor_factoring_inapplicable():
"""Shor's algorithm doesn't apply."""
# Hologram security not based on factoring
# or discrete log
return "Not applicable"
def quantum_collision_finding():
"""BHT algorithm for collisions."""
# Quantum collision finding
N = count_lawful_modulo_gauge()
# BHT algorithm complexity
quantum_collision_complexity = N**(1/3)
# Still secure for large N
assert quantum_collision_complexity > 2**85
return quantum_collision_complexity
# Summary
return {
'grover': grover_search_complexity(),
'shor': shor_factoring_inapplicable(),
'collision': quantum_collision_finding()
}
Implementation Security
Side-Channel Resistance
class SideChannelAnalysis:
"""Analyze side-channel vulnerabilities."""
def timing_analysis(self):
"""Check for timing leaks."""
def constant_time_verification(receipt):
"""Verify in constant time."""
# All operations take same time
start = time.perf_counter_ns()
# Fixed number of operations
for i in range(FIXED_ITERATIONS):
check = receipt.data[i % len(receipt.data)]
# Constant-time comparison
result = constant_time_compare(check, expected[i])
end = time.perf_counter_ns()
return end - start
# Measure timing variance
times = []
for _ in range(1000):
receipt = random_receipt()
t = constant_time_verification(receipt)
times.append(t)
variance = statistics.variance(times)
assert variance < ACCEPTABLE_VARIANCE
def power_analysis(self):
"""Check for power leaks."""
# Model power consumption
def power_trace(operation):
# Hamming weight model
hamming = bin(operation).count('1')
return hamming + random.gauss(0, 0.1)
# Different operations should be indistinguishable
trace1 = [power_trace(op) for op in operation_sequence_1]
trace2 = [power_trace(op) for op in operation_sequence_2]
# Statistical test for distinguishability
t_stat, p_value = stats.ttest_ind(trace1, trace2)
assert p_value > 0.05 # Not distinguishable
Exercises
Exercise 16.1: Prove that gauge quotient doesn’t weaken collision resistance.
Exercise 16.2: Design a commitment scheme using receipts.
Exercise 16.3: Prove that witness chains provide non-repudiation.
Exercise 16.4: Show that conservation laws imply authentication.
Exercise 16.5: Analyze security under adaptive chosen-ciphertext attacks.
Takeaways
- Perfect hashing on lawful domain: No collisions for distinct lawful objects
- Receipts are non-malleable: Cannot forge valid receipts
- Computational indistinguishability: Gauge freedom provides hiding
- Information-theoretic security: For restricted domains
- Quantum resistant: No efficient quantum attacks known
- Side-channel resistant: Constant-time operations by design
The Hologram model’s security isn’t bolted on—it’s a mathematical consequence of the conservation laws and algebraic structure.
Next: Part V explores practical implementation details.