Threat Model

The security analysis of HASC employs a comprehensive threat model that encompasses both traditional Byzantine adversaries and sophisticated adaptive attackers within the context of multi-layer blockchain architectures.

1. Adversarial Capabilities

Consider a probabilistic polynomial-time adversary A with the following formal capabilities:

A = (Init, Corrupt, Message, Execute)

Where:

Init: Initial network state
Corrupt: Node corruption function
Message: Network message control
Execute: Attack execution function

The adversarial model assumes Byzantine behavior bounded by:

Theorem 4.1 (Byzantine Threshold): Given a network of size n, the system maintains security against f Byzantine nodes where:

f < ⌊(n-1)/3⌋ ∧ ∑{v∈H} W(v) > 2·∑{v∈B} W(v)

Where:

H: Honest validator set
B: Byzantine validator set
W(v): Validator weight function

Proof: By contradiction, assume f ≥ ⌊(n-1)/3⌋. Then:

∃ partition P = {P₁, P₂}

where:

|P₁| ≤ ⌊(n-1)/3⌋
|P₂| ≤ ⌊(n-1)/3⌋

This violates the weight constraint:

∑{v∈H} W(v) > 2·∑{v∈B} W(v)

Therefore, f < ⌊(n-1)/3⌋ must hold.

2. Security Assumptions

The security framework operates under the following cryptographic assumptions:

Assumption 1 (Hash Function):
∀x,y: Pr[H(x) = H(y)] ≤ negl(λ)

Assumption 2 (Digital Signatures):
∀msg: Pr[Forge(sig, msg)] ≤ negl(λ)

Where:

λ: Security parameter
negl(): Negligible function