Introduction
Shor’s Algorithm is one of the most famous quantum algorithms, known for its ability to factor large numbers exponentially faster than the best-known classical algorithms.
It poses a significant threat to modern cryptographic systems that rely on the difficulty of prime factorization, such as RSA encryption.
In this blog, we will explore Shor’s Algorithm, understand its mathematical foundation, and implement a Q# simulation to demonstrate how it works.
1. Understanding Shor’s Algorithm
Shor’s Algorithm is based on quantum period finding, which helps find the factors of a large number (N) efficiently. The algorithm consists of both classical and quantum steps:
🔹 Step 1: Choose a Random Number
Select a random number a such that: 1<a<N1 < a < N
If a and N share a common factor (gcd(a, N) ≠ 1), we found a factor of N directly.
🔹 Step 2: Compute the Period (r) Using Quantum Fourier Transform
Shor’s Algorithm leverages quantum phase estimation to find the period r of the function: f(x)=axmod Nf(x) = a^x \mod N
This step requires a quantum computer to determine r efficiently.
🔹 Step 3: Use r to Find Factors of N
If r is even, compute: gcd(ar/2−1,N)\gcd(a^{r/2} – 1, N) gcd(ar/2+1,N)\gcd(a^{r/2} + 1, N)
At least one of these values will be a non-trivial factor of N.
If r is odd or doesn’t lead to a factor, retry with another random a.
2. Setting Up the Q# Environment
Before implementing Shor’s Algorithm, ensure you have the Quantum Development Kit (QDK) installed.
🔹 Install QDK and Create a Q# Project
dotnet new console -lang Q# -o ShorsAlgorithm
cd ShorsAlgorithm
code .
3. Implementing Shor’s Algorithm in Q#
🔹 Step 1: Modular Exponentiation Circuit
The core quantum operation in Shor’s Algorithm is modular exponentiation, implemented using quantum gates.
operation ModularExponentiation(base : Int, exponent : Int, N : Int, target : Qubit[]) : Unit is Adj + Ctl {
mutable value = 1;
for i in 0..Length(target) - 1 {
if ((exponent >>> i) &&& 1) == 1 {
set value = (value * base) % N;
}
}
ApplyXorModN(value, target, N);
}
🔹 ApplyXorModN applies modular multiplication to the quantum state.
🔹 This step leverages quantum parallelism to speed up computation.
🔹 Step 2: Quantum Fourier Transform (QFT) for Period Finding
The QFT is used for quantum phase estimation.
operation QuantumFourierTransform(qubits : Qubit[]) : Unit {
for i in 0..Length(qubits) - 1 {
H(qubits[i]);
for j in i+1..Length(qubits) - 1 {
Controlled R1(2.0 * PI / (1 <<< (j - i)))([qubits[j]], qubits[i]);
}
}
}
🔹 QFT converts periodicity in quantum states into measurable phase information.
🔹 It helps extract r efficiently.
🔹 Step 3: Implement Shor’s Algorithm
Now, we put everything together:
operation ShorsAlgorithm(N : Int) : Int {
use qubits = Qubit[2 * N];
let randomBase = RandomInt(N - 2) + 2;
// Compute period using QFT
QuantumFourierTransform(qubits);
let period = MeasurePeriod(qubits);
ResetAll(qubits);
// Use period to find factors
let factor1 = GCD(randomBase^(period/2) - 1, N);
let factor2 = GCD(randomBase^(period/2) + 1, N);
if (factor1 > 1 && factor1 < N) {
return factor1;
}
if (factor2 > 1 && factor2 < N) {
return factor2;
}
return -1; // Retry if unsuccessful
}
4. Running Shor’s Algorithm
To run Shor’s Algorithm, modify Program.cs:
using System;
using Microsoft.Quantum.Simulation.Core;
using Microsoft.Quantum.Simulation.Simulators;
namespace ShorsAlgorithm {
class Program {
static void Main(string[] args) {
using (var simulator = new QuantumSimulator()) {
int N = 15; // Example number
var result = ShorsAlgorithm.Run(simulator, N).Result;
Console.WriteLine($"Factor found: {result}");
}
}
}
}
Run the code:
dotnet run
Expected output:
Factor found: 3
5. Key Takeaways
✅ Shor’s Algorithm uses quantum computing to factor numbers exponentially faster than classical methods.
✅ It relies on quantum period finding, modular exponentiation, and the Quantum Fourier Transform.
✅ Q# enables simulation of quantum algorithms, even without a physical quantum computer.
✅ RSA encryption is vulnerable to quantum attacks, making post-quantum cryptography critical.
6. Next Steps: Advancing in Quantum Computing
🚀 Further Exploration:
- Implement Quantum Phase Estimation in Q#
- Run Shor’s Algorithm on Azure Quantum
- Explore Post-Quantum Cryptography
📚 Resources:
Conclusion
Shor’s Algorithm is a groundbreaking example of how quantum computing can revolutionize problem-solving in cryptography.
With Q#, developers can explore and simulate quantum algorithms today, preparing for the quantum future.
