Principle: Multiplying large primes is easy, factoring the result is hard

Encryption:
1. Choose two large prime numbers: p, q (each ~2048 bits)
2. Multiply them: N = p × q (public key)
3. Encrypt message: C = M^e mod N

Decryption (requires knowing p, q):
1. Calculate private key from p, q
2. Decrypt: M = C^d mod N

Security assumption:
Factoring N into p × q is computationally infeasible
Classical computer: Trillions of years for RSA-4096
Quantum computer: Minutes (Shor's algorithm)

# Simplified Shor's Algorithm

def shors_algorithm(N, quantum_computer):
    """
    Factor N = p × q using quantum computer
    Classical: O(exp(n^(1/3))) - exponential time
    Quantum: O(n^3) - polynomial time
    """

    # Step 1: Choose random a < N
    a = random.randint(2, N-1)

    # Step 2: Quantum period-finding (the magic step)
    # Create superposition of all values
    quantum_state = create_superposition(range(N))

    # Apply function f(x) = a^x mod N
    quantum_state = apply_modular_exponentiation(quantum_state, a, N)

    # Quantum Fourier Transform (finds period r)
    r = quantum_fourier_transform(quantum_state)

    # Step 3: Classical post-processing
    p = gcd(a^(r/2) - 1, N)
    q = N / p

    return p, q  # Factors found!

# Runtime:
# RSA-4096 factoring:
# Classical: 2^128 operations ≈ 10^38 years
# Quantum (1M qubits): ~8 minutes

Circuit for RSA-4096 Factoring:
├─ Logical qubits needed: ~20,000
├─ Physical qubits (with error correction): ~1,000,000
├─ Quantum gates: ~10^9 operations
├─ Coherence time required: 8 minutes (480 seconds)
├─ Error rate tolerance: <0.001% per gate
└─ Result: Factors p and q

IBM Condor-X (2053): First quantum computer to meet all requirements

IBM Quantum Cloud Infrastructure:

User Layer:
├─ Web API (RESTful, GraphQL)
├─ SDK (Qiskit, Cirq, Q#)
└─ Authentication: OAuth 2.0, API keys

Orchestration Layer:
├─ Job queue: 1M+ jobs/day
├─ Resource allocation: Kubernetes for quantum
├─ Scheduling: Priority-based (academia, commercial, government)
└─ Estimated runtime: Provided pre-execution

Quantum Processing Unit (QPU):
├─ 47 QPU facilities globally
├─ Each: 1,048,576 qubits (transmon superconducting)
├─ Cryogenics: Dilution refrigerator (10 mK)
├─ Control systems: 10M+ FPGA controllers
└─ Error correction: Real-time (surface code)

Classical Co-Processing:
├─ GPU clusters (for classical parts of algorithm)
├─ 100 Gbps interconnect to QPU
└─ Result validation and storage

Attack Vector:
1. Attacker accessed government-tier quantum cloud account (stolen credentials)
2. Submitted Shor's algorithm job disguised as "Quantum simulation research"
3. Bypassed 2048-bit safety limit via kernel-level exploit
4. Job executed: Factor RSA-4096 public keys
5. Runtime: 8 minutes per key
6. Keys factored: 10,000 high-value targets (SSL certificates, Bitcoin wallets, government keys)

Total execution time: 55 hours (parallelized across multiple QPUs)

06:47 UTC: First quantum factoring job submitted
06:55 UTC: RSA-4096 private key extracted (SSL certificate for major bank)
07:30 UTC: 47 bank SSL certificates compromised
08:15 UTC: Attacker begins man-in-the-middle attacks on banking traffic
10:00 UTC: $2.4B stolen from compromised accounts

Targets expanded:
├─ Cryptocurrency wallets: $14.7T Bitcoin/Ethereum stolen
├─ Government secrets: NSA/CIA encrypted archives decrypted
├─ Corporate espionage: Trade secrets from 2,400 companies
├─ Medical records: 847M patient records exposed
└─ Military communications: 15 years of encrypted traffic decrypted

Cascading failures:
├─ Cryptocurrency markets: Collapsed (theft + panic)
├─ Banking systems: Frozen (trustworthiness destroyed)
├─ Government leaks: Wikileaks × 1000 (40 years of secrets)
├─ Internet trust: Broken (HTTPS unreliable)
└─ Global economy: Paralyzed