Shor's Algorithm
Introduction
Shor's algorithm, named after mathematician Peter W. Shor, is a quantum algorithm (an algorithm that runs on a realistic model of quantum computation) for integer factorization formulated in 1994. With the ability to break RSA encryption, it sparked interest in both quantum computing and quantum cryptography.
Theoretical Background
Shor's algorithm is based on the principles of quantum mechanics, the branch of physics dealing with the smallest particles of the universe. It also relies heavily on number theory, specifically the concept of integer factorization.
Quantum Computing Basics
Quantum computing is a type of computation that utilizes quantum bits, or qubits, instead of the traditional binary bits used in classical computing. Qubits can exist in a superposition of states, allowing them to perform multiple calculations simultaneously. This property is what gives quantum computers their potential for solving certain types of problems, like integer factorization, exponentially faster than classical computers.
Shor's Algorithm
Shor's algorithm uses a quantum Fourier transform to compute the periodicity of a function, which is then used to find the factors of a large number. The algorithm consists of two main parts: a quantum part that finds the period of a function, and a classical part that uses this period to compute the factors of the number.
Quantum Fourier Transform
The Quantum Fourier Transform (QFT) is a linear transformation on quantum bits and is the quantum analogue of the discrete Fourier transform. The QFT is part of many quantum algorithms, including Shor's algorithm, and is used to transform quantum states from the computational basis to the Fourier basis.
Period Finding
The period finding part of Shor's algorithm is where the quantum computer is used. This part of the algorithm finds the period of the function a^x mod N, where a is a randomly chosen number less than N, and N is the number we wish to factor. The output of this function repeats every r numbers, where r is the period. The quantum computer is able to find the period r efficiently using the quantum Fourier transform.
Factorization
Once the period r of the function a^x mod N is found, it can be used to find the factors of N. If r is even, and a^r/2 is not equal to -1 mod N, then the greatest common divisor of N and a^r/2 ± 1 will give a nontrivial factor of N. If r is odd, or if a^r/2 is equal to -1 mod N, then the algorithm is repeated with a new randomly chosen a.
Impact on Cryptography
Shor's algorithm has significant implications for cryptography, particularly public key cryptography systems like RSA. RSA relies on the fact that factoring large numbers is computationally difficult for classical computers. However, Shor's algorithm can factor large numbers exponentially faster than classical algorithms, effectively breaking RSA encryption if a large enough quantum computer can be built.
Current State of Quantum Computing
While the theory behind Shor's algorithm is well established, practical implementation is still a challenge due to the current state of quantum computing. As of now, quantum computers with enough qubits to run Shor's algorithm against cryptographic key sizes do not yet exist. However, the field is rapidly advancing, and the possibility of such a computer being built in the future cannot be ruled out.