Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Transactions on Quantum Engineering

Shor's Algorithm Using Efficient Approximate Quantum Fourier Transform

Kento Oonishi, Noboru Kunihiro

Published Jan 1, 2023• 17 citations
Publisher

Abstract

Shor's algorithm solves the integer factoring and discrete logarithm problems in polynomial time. Therefore, the evaluation of Shor's algorithm is essential for evaluating the security of currently used public-key cryptosystems because the integer factoring and discrete logarithm problems are crucial for the security of these cryptosystems. In this article, a new approximate quantum Fourier transform is proposed, and it is applied to Rines and Chuang's implementation. The proposed implementation requires one-third the number of$T$gates of the original. Moreover, it requires one-fourth the$T$-depth of the original. Finally, a$T$-scheduling method for running the circuit with the smallest KQ is presented.

DOI
10.1109/tqe.2023.3319044

Related papers

Shors Algorithm Using Efficient Approximate Quantum Fourier Transform

Kento Oonishi, Kunihiro Noboru

Shor's algorithm solves the integer factoring and discrete logarithm problems in polynomial time. Therefore, the evaluation of Shor's algorithm is essential for evaluating the security of currently used public-key cryptosystems because the integer factoring and discrete logarithm problems are crucial for the security of these cryptosystems. In this article, a new approximate quantum Fourier transform is proposed, and it is applied to Rines and Chuang's implementation. The proposed implementation requires one-third the number of T gates of the original. Moreover, it requires one-fourth of the T-depth of the original. Finally, a T-scheduling method for running the circuit with the smallest KQ (where K is the number of logical qubits and Q is the circuit depth) is presented.

•Sep 25, 2023•0 citations

An Evaluation of Computational Complexity of Shor’s Algorithm on Surface Codes for Factorization

Kento Oonishi, Noboru Kunihiro

Shor’s algorithm is a quantum algorithm that can efficiently solve the problems of integer factorization and discrete logarithms, which potentially break currently used public-key cryptosystems such as RSA and elliptic curve cryptography. However, a significant challenge in solving these problems is the noise inherent in quantum computers. Previous research advanced fault-tolerant techniques such as surface codes to mitigate this noise in quantum computers. Analyzing the security of currently used public-key cryptosystems on fault-tolerant quantum computers is essential. In this paper, we estimate the computational cost required to solve integer factorization using Shor’s algorithm on surface codes. We evaluated computational resources required for surface codes and magic state distillation and derived the computational cost of Shor’s algorithm based on these evaluations. Our computational complexity analysis indicates that there is no significant difference in the order of complexity. However, it underscores the paramount importance of optimizing the coefficient part for future efficiency improvements.

IEEE Access•Jan 1, 2026•2 citations

Quantum Algorithm for Attacking RSA Based on the eth Root

Yahui WANG, Huanguo Zhang, Houzhen Wang

The emergence of some quantum algorithms has brought a serious threat to modern cryptography,among which Shor’s algorithm is the most important threatening algorithm for cryptanalysis currently.Shor’s algorithm can solve the integer factorization problem (IFP) and discrete logarithm problem (DLP) in polynomial-time,which makes the current widely used RSA,ElGamal and ECC public key cryptosystem unsafe any more under the quantum computing environment.Therefore,it is necessary to research the cryptanalysis in the quantum computing environment.Solving the IFP is the core idea of Shor’s algorithm for attacking RSA,but breaking RSA does not have to be solved by solving the IFP.A quantum algorithm is designed to attack the RSA cryptosystem starting from the angle of non-factorization.Focusing on the characteristics of RSA public key cryptosystem,using the quantum Fourier transform,the RSA plaintext $M$ can be got by calculating the ${e^{{th}}}$ root modulus $n$ .That is,without solving the IFP,RSA is broken.Different from the previous practices that cryptanalysts try to recover the private-key,a ciphertext-only attack algorithm for RSA,directly from recovering the plaintext $M$ to start, is presented.Results show that the probability of success of the new algorithm is higher than that of Shor’s algorithm attacking RSA.At the same time,the new algorithm does not recover the RSA plaintext from the ciphertext without factoring the modulus $n$ ,and avoids the restriction that the order of ciphertext $C$ modules $n$ is even.

•Jan 1, 2018•0 citations

A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform

Rupert C. D. Young, Philip M. Birch, Chris R. Chatwin

The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.

Pattern Recognition and Tracking•Apr 27, 2018•4 citations

Analysis of RSA and Shor’s algorithm for cryptography: A quantum perspective

Seema Kute, Chitra G. Desai, Mukti E. Jadhav

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.

AIP conference proceedings•Jan 1, 2024•5 citations

Implementation of the Shor Algorithm on a Quantum Circuit

Alexei Petrenko, Sergei Petrenko

This chapter analyzes the applicability of Fourier transformations to determine the periodic structure of finite-length sequences. The classical factorization problem is reformulated to solve it on modern quantum computers with 16 or more qubits. The chapter presents Shor's algorithm for factorization of natural numbers. Performing two basic quantum operations is required to implement the described Shor algorithm, namely exponentiation and quantum discrete Fourier transformation. The chapter estimates the amount of quantum resources needed to solve some asymmetric cryptographic problems using derivatives of the Shor algorithm, with different parameters of these problems, and compares them with the complexity of solving the iterative problem when searching for the key of a symmetric cryptosystem. Using the example of solving the problem of finding the exponent (multiplicative order) of a simple integer modulo by another simple integer, the chapter shows the main features of modeling and engineering implementation of the quantum Shor algorithm.

River Publishers eBooks•Feb 7, 2023•0 citations
Shor's algorithm solves the integer factoring and discrete logarithm problems in polynomial time. Therefore, the evaluation of Shor's algorithm is essential for evaluating the security of currently used public-key cryptosystems because the integer factoring and discrete logarithm problems are crucial for the security of these cryptosystems. In this article, a new approximate quantum Fourier transform is proposed, and it is applied to Rines and Chuang's implementation. The proposed implementation requires one-third the number ofTTTgates of the original. Moreover, it requires one-fourth theTTT-depth of the original. Finally, aTTT-scheduling method for running the circuit with the smallest KQ is presented.
Shor's algorithm solves the integer factoring and discrete logarithm problems in polynomial time. Therefore, the evaluation of Shor's algorithm is essential for evaluating the security of currently used public-key cryptosystems because the integer factoring and discrete logarithm problems are crucial for the security of these cryptosystems. In this article, a new approximate quantum Fourier transform is proposed, and it is applied to Rines and Chuang's implementation. The proposed implementation requires one-third the number of T gates of the original. Moreover, it requires one-fourth of the T-depth of the original. Finally, a T-scheduling method for running the circuit with the smallest KQ (where K is the number of logical qubits and Q is the circuit depth) is presented.
Shor’s algorithm is a quantum algorithm that can efficiently solve the problems of integer factorization and discrete logarithms, which potentially break currently used public-key cryptosystems such as RSA and elliptic curve cryptography. However, a significant challenge in solving these problems is the noise inherent in quantum computers. Previous research advanced fault-tolerant techniques such as surface codes to mitigate this noise in quantum computers. Analyzing the security of currently used public-key cryptosystems on fault-tolerant quantum computers is essential. In this paper, we estimate the computational cost required to solve integer factorization using Shor’s algorithm on surface codes. We evaluated computational resources required for surface codes and magic state distillation and derived the computational cost of Shor’s algorithm based on these evaluations. Our computational complexity analysis indicates that there is no significant difference in the order of complexity. However, it underscores the paramount importance of optimizing the coefficient part for future efficiency improvements.
The emergence of some quantum algorithms has brought a serious threat to modern cryptography,among which Shor’s algorithm is the most important threatening algorithm for cryptanalysis currently.Shor’s algorithm can solve the integer factorization problem (IFP) and discrete logarithm problem (DLP) in polynomial-time,which makes the current widely used RSA,ElGamal and ECC public key cryptosystem unsafe any more under the quantum computing environment.Therefore,it is necessary to research the cryptanalysis in the quantum computing environment.Solving the IFP is the core idea of Shor’s algorithm for attacking RSA,but breaking RSA does not have to be solved by solving the IFP.A quantum algorithm is designed to attack the RSA cryptosystem starting from the angle of non-factorization.Focusing on the characteristics of RSA public key cryptosystem,using the quantum Fourier transform,the RSA plaintext MMM can be got by calculating the eth{e^{{th}}}eth root modulus nnn .That is,without solving the IFP,RSA is broken.Different from the previous practices that cryptanalysts try to recover the private-key,a ciphertext-only attack algorithm for RSA,directly from recovering the plaintext MMM to start, is presented.Results show that the probability of success of the new algorithm is higher than that of Shor’s algorithm attacking RSA.At the same time,the new algorithm does not recover the RSA plaintext from the ciphertext without factoring the modulus nnn ,and avoids the restriction that the order of ciphertext CCC modules nnn is even.
The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.
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.
This chapter analyzes the applicability of Fourier transformations to determine the periodic structure of finite-length sequences. The classical factorization problem is reformulated to solve it on modern quantum computers with 16 or more qubits. The chapter presents Shor's algorithm for factorization of natural numbers. Performing two basic quantum operations is required to implement the described Shor algorithm, namely exponentiation and quantum discrete Fourier transformation. The chapter estimates the amount of quantum resources needed to solve some asymmetric cryptographic problems using derivatives of the Shor algorithm, with different parameters of these problems, and compares them with the complexity of solving the iterative problem when searching for the key of a symmetric cryptosystem. Using the example of solving the problem of finding the exponent (multiplicative order) of a simple integer modulo by another simple integer, the chapter shows the main features of modeling and engineering implementation of the quantum Shor algorithm.