This post is part of my 21-day quantum computing learning journey, focusing on Shor’s Algorithm and its revolutionary impact on cryptography. Today we’re exploring how quantum computing threatens our current security infrastructure and what the future holds for post-quantum cryptography.

Progress: 16/21 days completed. Shor’s Algorithm: ✓. RSA Analysis: ✓. Post-Quantum

Shor’s Algorithm: The Quantum Threat to Modern Cryptography

Shor’s algorithm, developed by Peter Shor in 1994, represents one of the most significant theoretical breakthroughs in quantum computing. This algorithm can efficiently factor large integers and solve discrete logarithm problems - the mathematical foundations that secure much of our digital world today.

While classical computers would need billions of years to factor a 2048-bit RSA key, a sufficiently powerful quantum computer running Shor’s algorithm could accomplish this in mere hours. This isn’t science fiction - it’s a mathematical certainty that’s driving a global race to develop quantum-resistant cryptography.

The Mathematical Foundation: Period Finding

At its core, Shor’s algorithm solves the period-finding problem. Given a function f(x) = a^x mod N, it finds the period r such that f(x+r) = f(x). This periodicity is the key to factorization.

• Classical approach: Exponential time complexity O(√N)
• Quantum approach: Polynomial time complexity O((log N)³)

Implementing Shor’s Algorithm Components

import numpy as np
from qiskit import QuantumCircuit, execute, Aer
from qiskit.visualization import plot_histogram
from math import pi, gcd
from fractions import Fraction

def quantum_period_finding(a, N, n_qubits):
    """
    Quantum period finding subroutine for Shor's algorithm
    a: base number
    N: modulus
    n_qubits: number of qubits for precision
    """
    print(f"Finding period of f(x) = {a}^x mod {N}")
    print("=" * 50)
    
    # Create quantum circuit
    qc = QuantumCircuit(2 * n_qubits, n_qubits)
    
    # Step 1: Initialize superposition in first register
    for i in range(n_qubits):
        qc.h(i)
    
    qc.barrier()
    
    # Step 2: Apply controlled modular exponentiation
    # (Simplified implementation for demonstration)
    for i in range(n_qubits):
        # Apply controlled operations U^(2^i) where U|y⟩ = |ay mod N⟩
        power = 2**i
        qc.cp(2 * pi * power * a / N, i, n_qubits + (i % n_qubits))
    
    qc.barrier()
    
    # Step 3: Apply inverse QFT to first register
    qft_inverse = create_qft_circuit(n_qubits).inverse()
    qc.compose(qft_inverse, range(n_qubits), inplace=True)
    
    qc.barrier()
    
    # Step 4: Measure first register
    qc.measure(range(n_qubits), range(n_qubits))
    
    return qc

def create_qft_circuit(n_qubits):
    """Create Quantum Fourier Transform circuit"""
    qc = QuantumCircuit(n_qubits)
    
    def qft_rotations(circuit, n):
        if n == 0:
            return circuit
        n -= 1
        circuit.h(n)
        for qubit in range(n):
            circuit.cp(pi/2**(n-qubit), qubit, n)
        qft_rotations(circuit, n)
    
    qft_rotations(qc, n_qubits)
    
    # Swap qubits to reverse order
    for qubit in range(n_qubits//2):
        qc.swap(qubit, n_qubits-qubit-1)
    
    return qc

def classical_post_processing(measurement_results, N, n_qubits):
    """
    Classical post-processing to extract period from quantum measurements
    """
    print("\nClassical Post-Processing:")
    print("=" * 30)
    
    periods = []
    
    for bitstring, count in measurement_results.items():
        if count > 50:  # Only consider frequent measurements
            # Convert binary to decimal
            y = int(bitstring, 2)
            
            if y != 0:
                # Use continued fractions to find period
                phase = y / (2**n_qubits)
                frac = Fraction(phase).limit_denominator(N)
                period_candidate = frac.denominator
                
                print(f"Measurement: |{bitstring}⟩ ({count} times)")
                print(f"  y = {y}, phase = {phase:.4f}")
                print(f"  Period candidate: {period_candidate}")
                
                # Verify period candidate
                if period_candidate < N and verify_period(2, N, period_candidate):
                    periods.append(period_candidate)
                    print(f"  ✓ Valid period found: {period_candidate}")
                else:
                    print(f"  ✗ Invalid period")
    
    return periods

def verify_period(a, N, r):
    """Verify if r is a valid period for a^x mod N"""
    return pow(a, r, N) == 1

def shor_factorization_demo():
    """Demonstrate Shor's algorithm for factoring N = 15"""
    print("Shor's Algorithm Demo - Factoring N = 15")
    print("=" * 50)
    
    N = 15  # Number to factor
    a = 2   # Chosen base (gcd(a,N) = 1)
    n_qubits = 4  # Qubits for period finding
    
    print(f"Factoring N = {N} using base a = {a}")
    print(f"Classical complexity: O(√{N}) = O({int(np.sqrt(N))})")
    print(f"Quantum complexity: O((log {N})³) = O({int(np.log2(N)**3)})")
    
    # Create quantum circuit
    qc = quantum_period_finding(a, N, n_qubits)
    
    print("\nQuantum Circuit:")
    print(qc.draw())
    
    # Simulate
    backend = Aer.get_backend('qasm_simulator')
    job = execute(qc, backend, shots=1024)
    result = job.result()
    counts = result.get_counts()
    
    print("\nQuantum Measurement Results:")
    for state, count in sorted(counts.items(), key=lambda x: x[1], reverse=True)[:5]:
        print(f"|{state}⟩: {count} times ({count/1024:.1%})")
    
    # Classical post-processing
    periods = classical_post_processing(counts, N, n_qubits)
    
    # Factor using found periods
    print("\nFactorization:")
    for r in periods:
        if r % 2 == 0:  # Period must be even
            factor1 = gcd(pow(a, r//2) - 1, N)
            factor2 = gcd(pow(a, r//2) + 1, N)
            
            if factor1 > 1 and factor1 < N:
                print(f"Period r = {r} → Factor found: {factor1}")
                print(f"Verification: {N} = {factor1} × {N//factor1}")
                return factor1, N//factor1
            elif factor2 > 1 and factor2 < N:
                print(f"Period r = {r} → Factor found: {factor2}")
                print(f"Verification: {N} = {factor2} × {N//factor2}")
                return factor2, N//factor2
    
    print("No factors found in this run. Try again or use more qubits.")
    return None, None

Breaking RSA Encryption: The Practical Impact

RSA encryption, used in HTTPS, secure email, and digital signatures, relies on the difficulty of factoring large composite numbers. Here’s how Shor’s algorithm threatens RSA:

RSA Security Levels

• RSA-1024: Vulnerable to quantum computers with ~2000 logical qubits
• RSA-2048: Requires ~4000 logical qubits
• RSA-4096: Needs ~8000 logical qubits

Conclusion

Shor’s algorithm represents both a threat and an opportunity. While it challenges our current cryptographic infrastructure, it also drives innovation in post-quantum cryptography and quantum-safe communication protocols.

The quantum threat is real, but it’s not immediate. Organizations have time to prepare, but they must start now. The transition to post-quantum cryptography will be one of the largest cryptographic migrations in history, requiring careful planning and execution.

As we advance toward practical quantum computers, the cryptographic arms race continues. The future of digital security lies not in avoiding the quantum threat, but in embracing quantum-resistant solutions that can coexist with both classical and quantum technologies.

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Reference:

1. Nielsen, M. A., & Chuang, I. L. (2024). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press
2. IBM Qiskit Team. Shor’s Algorithm Tutorial
3. NIST (2022). Post-Quantum Cryptography Standardization
4. Shor, P. W. (1997). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing.