Shor's Algorithm

Extracted Attributes

• Field

  Quantum Computing, Cryptography, Number Theory

• Impact

  Accelerates threat to modern encryption standards (e.g., RSA, SHA-256), drives urgency for Post-Quantum Cryptography

• Purpose

  Efficiently find prime factors of an integer

• Key Components

  Quantum Fourier Transform (QFT), Modular Arithmetic, Quantum Parallelism

• Related Concepts

  Shor's Code (Quantum Error Correction)

• Quantum Advantage

  Runs in polynomial time relative to the logarithm of the number being factored, significantly faster than classical algorithms

• Other Applications

  Discrete logarithm problem, period-finding problem

• Practical Challenges

  Requires a large number of qubits, susceptible to noise, needs quantum error correction

• Time Complexity (Quantum)

  O((log N)^2 (log log N)(log log log N)) or O((log N)^2 (log log N)) using fastest multiplication

Timeline

• Shor's algorithm for finding prime factors of an integer was developed by Peter Shor. (Source: Summary)

  1994-XX-XX

Shor's algorithm

Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. It is one of the few known quantum algorithms with compelling potential applications and strong evidence of superpolynomial speedup compared to best known classical (non-quantum) algorithms. On the other hand, factoring numbers of practical significance requires far more qubits than available in the near future. Another concern is that noise in quantum circuits may undermine results, requiring additional qubits for quantum error correction. Shor proposed multiple similar algorithms for solving the factoring problem, the discrete logarithm problem, and the period-finding problem. "Shor's algorithm" usually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer N {\displaystyle N} , Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in log ⁡ N {\displaystyle \log N} . It takes quantum gates of order O ( ( log ⁡ N ) 2 ( log ⁡ log ⁡ N ) ( log ⁡ log ⁡ log ⁡ N ) ) {\displaystyle O\!\left((\log N)^{2}(\log \log N)(\log \log \log N)\right)} using fast multiplication, or even O ( ( log ⁡ N ) 2 ( log ⁡ log ⁡ N ) ) {\displaystyle O\!\left((\log N)^{2}(\log \log N)\right)} utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey and van der Hoeven, thus demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is significantly faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time: O ( e 1.9 ( log ⁡ N ) 1 / 3 ( log ⁡ log ⁡ N ) 2 / 3 ) {\displaystyle O\!\left(e^{1.9(\log N)^{1/3}(\log \log N)^{2/3}}\right)} .

Web Search Results

• Why is Shor's algorithm such a keystone application of quantum ...

  Shor's algorithm is a quantum algorithm for integer factorization, which is the process of finding the prime factors of a given number. The algorithm consists of classical parts and a quantum subroutine. The steps of Shor’s Algorithm are: [...] Shor's algorithm is a quantum algorithm for integer factorization, which is the process of finding the prime factors of a given number. The algorithm consists of classical parts and a quantum subroutine. The steps of Shor’s Algorithm are: [...] ### Understanding Shor's Algorithm Understanding Shor’s algorithm means understanding the basic principles that underpin quantum computing, including superposition and entanglement. This algorithm also creates a segue into going deeper into topics like Quantum Fourier Transform to start building up the quantum equivalents of how bits and gates work in our classical world.Â

• What is Shor's Algorithm? - Utimaco

  ## Breadcrumb # What is Shor’s Algorithm? Definition: In 1994, Peter Shor, the MIT Maths Professor, devised a quantum algorithm for generating prime factors of large numbers much more efficiently than classical computers. Shor's algorithm is a quantum computer algorithm that can solve prime factors of an integer in polynomial time. It allows us to factorize into prime numbers in O(logN ^3) time and O(logN) space. ## Shor’s Algorithm Explained

• What is Shor's Algorithm - QuEra Computing

  How Shor's Algorithm Works -------------------------- The core of Shor's Algorithm is the Quantum Fourier Transform (QFT), a quantum analog of the classical Fourier Transform. The QFT allows the algorithm to find the period of a specific mathematical function related to the number being factored. Once the period is found, the prime factors can be efficiently extracted using classical methods. [...] In fact, the size of the quantum circuit resulted in the discovery of one of the earliest Quantum Error Correction(QEC) codes, also by Peter Shor, known as Shor’s Code. To avoid confusion, “Shor’s Algorithm” refers to the factoring algorithm, and “Shor’s Code” refers to the quantum error correction code (QECC) that would make it possible to implement Shor’s Algorithm on fault-tolerant quantum computers (FTQC). FTQC will use newer, more sophisticated codes, but many of these potential codes are [...] Shor's factoring algorithm finds one of two unknown variables that are crucial for efficiently factoring an integer. With two unknowns in one equation, finding both values quickly becomes classically intractable as the target integer gets larger. There are classical algorithms to find one of those values, but they become increasingly inefficient as the target integer gets larger. With Shor's quantum algorithm finding that value efficiently, it then becomes considerably easier to find the other

• Shor's Factorization Algorithm - GeeksforGeeks

  Shor’s Algorithm depends on: Modular Arithmetic Quantum Parallelism Quantum Fourier Transformation The Algorithm stands as: Given an odd composite number N, find an integer d, strictly between 1 and N, that divides N. Shor’s Algorithm consists of the following two parts: [...] Shor’s Factorization Algorithm is proposed by Peter Shor. It suggests that quantum mechanics allows the factorization to be performed in polynomial time, rather than exponential time achieved after using classical algorithms. This could have a drastic impact on the field of data security, a concept based on the prime factorization of large numbers. [...] Many polynomial-time algorithms for integer multiplication (e.g., Euclid’s Algorithm) do exist, but no polynomial-time algorithm for factorization exists. So, Shor came up with an algorithm i.e. Shor’s Factorization Algorithm, an algorithm for factorizing non-prime integers N of L bits. Quantum algorithms are far much better than classical algorithms because they are based on Quantum Fourier Transform.

• Shor's Factoring Algorithm (for Dummies) Step-by-Step with example

  However in 1994, American mathematician Peter Shor devised an Algorithm for Quantum computers that could be a breakthrough in factoring large integers. Shor’s Factoring Algorithm is one of the best algorithms for factorization. The reason why it is so popular is the fact that given enough advancements in quantum computation, this algorithm can be used to break encryption. We will talk more about this later. # Table of Content # Some Math Terminologies