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Turkish Journal of Mathematics and Computer Science

Fundamental Structure of Shor’s Quantum Algorithm for Factoring Integers

Turgut Hanoymak, Akram Chehrazi

Published Dec 31, 2019• 0 citations
OpenAlex

Abstract

One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.

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One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.
The process to find the prime factors of a large number, the “factoring problem” is believed to be a very hard problem. For this reason, it is the cornerstone of modern cryptographic schemes, like RSA cryptosystem. In 1994, Professor Peter Shor proposed a new polynomial-time quantum algorithm that finds the prime factors of a number with many digits. This was a bolt from the blue for the security of transactions and electronic communications and became an example of how quantum computing changes our perception of security and safety. In this paper Shor's Algorithm is presented and an implementation, a way to factor number 21 is described. In addition, some reliability issues of quantum devices were considered in order to explore the potentiality of Shor's algorithm.
The complexity of public key algorithms like RSA and variants of Elgamal is based on the involvement of exponentiation in encryption and decryption. RSA is based on integer factorization and variants of Elgamal or based on discrete logarithms and are formed as hard problems. To achieve the perceived level of security, the required key length for public key cryptosystem needs to be determined. Quantum computers are capable of conducting an exponential number of simultaneous computations because of which they can find factors in polynomial time. Shor’s algorithm is popular algorithm for factoring huge integers in polynomial time on quantum computers. This paper presents RSA algorithm followed by various possible attacks for factoring and demonstrates Shor’s algorithm for factoring and discuss results for interpretations.
Quantum integer factorization is a groundbreaking field that explores the use of quantum algorithms, notably Shor's algorithm, to efficiently factorize large integers into their prime components. Its potential implications on modern encryption techniques, such as the Rivest-Shamir-Adleman (RSA) algorithm, are of considerable interest. This paper's central aim is to contrast the computational time and implementation of the Integer Factorization Problem (IFP) between classical and quantum computers. The inner workings of Quantum Shor's algorithm are broken down step by step, and the estimated processing time is contrasted with the classical General Number Field Sieve (GNFS) approach. Given the exponential speedup provided by quantum computing, this comparison helps assess Shor's algorithm's possible advantage over GNFS in factoring large numbers. Direct comparisons with GNFS are currently difficult due to Shor's algorithm's reliance on large-scale quantum computer development for practical execution speeds. This paper focuses on solving integer factorization up to2n2^{n}2nbits.
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer NNN is achieved in a number of steps equal to the number kkk of its prime factors, -- e.g., if NNN is the product of two primes, two quantum measurements are enough, regardless of the number of digits nnn of the number NNN. Since kkk is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.
The computational difficulty of factoring large integers forms the basis of security for RSA public-key cryptography. The best-known factoring algorithms for classical computers run in sub-exponential time. The integer factorization problem can be reduced to the Boolean Satisfiability problem (SAT). While this reduction has proved to be useful for studying SAT solvers, large integers have not been factored via such a reduction. Shor's quantum factoring algorithm factors integers in expected polynomial time. Large-scale fault-tolerant quantum computers capable of implementing Shor's algorithm are not yet available, preventing relevant benchmarking experiments. Recently, several authors have attempted quantum factorizations via reductions to SAT or similar NP-hard problems. While this approach may shed light on algorithmic approaches for quantum solutions to NP-hard problems, in this paper we study and question its practicality. We find no evidence that this is a viable path toward factoring large numbers, even for scalable fault-tolerant quantum computers, as well as for various quantum annealing or other special purpose quantum hardware.
RSA cryptography is a principle part in today's cyber-security frameworks, which intensely depends on the diffi-culty of factorizing large integers. Finding a factor of large prime number is not a small feature compared to the time complexity for computation on a classical computer is far beyond comprehension. Back in 1994, Peter Shor proposed a quantum algorithm to factorize integers much more efficiently than a classical computer. This can effectively break asymmetric cryptosystems such as RSA or Diffie-Hellman, depicted as the most famous quantum algorithm. It leverages the efficacy proclaimed by quantum bits and clearly shows how quantum computers can outperform classical ones in specific tasks. In this paper, we show how Shor's algorithm factorizes an integer into its two prime factors. Shor uses both classical and quantum computation to perform this task in polynomial time, using Euclidean algorithm and find the greatest common divisor (gcd) of two numbers in the classical part. Thereafter, the power of Quantum Fourier Transform (QFT) and Quantum Phase Estimation (QPE) is utilised for period-finding using quantum bits. We have implemented this algorithm on IBM's Quantum Lab and successfully factorized the number 15 into its prime factors as 5 and 3, by using 5 qubits. This paper provides an overview of the implementation of Shor's algorithm involving QPE, QFT, Modular Exponentiation and the corresponding individual implementations, the only prerequisite being, to know how a qubit works.