shor

1. Factorization using Shor's
Algorithm
Quantum Computing Approach to
Integer Factorization

2. What is Shor's Algorithm?
• • Shor's algorithm is a quantum algorithm for
integer factorization.
• • Developed by Peter Shor in 1994.
• • Efficiently finds the prime factors of a
composite number.
• • Runs exponentially faster than the best-
known classical algorithms.
• • Has significant implications for cryptography
(e.g., RSA).

3. Classical vs Quantum Factorization
• • Classical algorithms (like trial division) are
slow for large numbers.
• • Quantum algorithms leverage superposition
and entanglement.
• • Shor's algorithm can factor in polynomial
time: O((log N)^3).
• • Breaks RSA encryption if large-scale
quantum computers become practical.

4. Steps in Shor’s Algorithm
• 1. Choose a composite number N to factor.
• 2. Choose a random number a < N.
• 3. Compute gcd(a, N); if > 1, it’s a factor.
• 4. Use quantum period-finding to find r, the
period of a^x mod N.
• 5. If r is even and a^(r/2) ≠ -1 mod N, compute
gcd(a^(r/2) ± 1, N).
• 6. These gcds give non-trivial factors of N.

5. Example: Factoring 15
• 1. N = 15, choose a = 2.
• 2. gcd(2, 15) = 1 → continue.
• 3. Use quantum computer to find r such that
2^r ≡ 1 mod 15.
• 4. r = 4 (2^4 = 16 ≡ 1 mod 15).
• 5. Compute gcd(2^(r/2) ± 1, 15) = gcd(4 ± 1,
15).
• 6. Factors: gcd(5, 15) = 5 and gcd(3, 15) = 3.

6. Quantum Period Finding Circuit
• • Superposition: Apply Hadamard gates to
initialize qubits.
• • Modular exponentiation: Apply unitary
operator for a^x mod N.
• • Quantum Fourier Transform: Extract period
from interference pattern.
• • Measurement: Collapse state to get period r
with high probability.

7. Applications & Limitations
• • Applications:
• - Breaking RSA encryption
• - Cryptanalysis
• • Limitations:
• - Requires scalable quantum hardware
• - Quantum error correction still a challenge
• - Practical only for small numbers with
current tech

8. Conclusion
• • Shor's algorithm revolutionizes factorization
via quantum computing.
• • Demonstrates real advantage of quantum
over classical.
• • Promotes the need for quantum-safe
cryptographic systems.
• • Still a theoretical threat pending quantum
hardware advancements.