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

Journal of Applied Physics

Improving the success rate of quantum algorithm attacking RSA encryption system

Yumin Dong, Hengrui Liu, Yanying Fu, Xuanxuan Che, Cheng Liu, Lina Sun, Guangrui Wen

Published Jul 10, 2023• 7 citations
Publisher

Abstract

Abstract The task of Shor’s factorization algorithm (SFA) is to find a non-trivial factor of a given composite number N. In the algorithm, it is explicitly required that the constructor f (x) = a x mod N cycle is an even number, otherwise it returns to the algorithm header to recalculate. According to the analysis of the principle of SFA, the characteristics of RSA public key cryptosystem and a large number of calculation results, the random selection of a value is related to the obtained period. Aiming at this defect, a new optimization scheme is proposed on the basis of the previous optimization, and the odd cycle obtained when a is a perfect square number randomly becomes effective. When the arbitrarily chosen value of a obtains a period that is a multiple of 3, by modifying the decomposition method, the requirements for the period are relaxed without affecting the algorithm complexity. Using the classic algorithm, we experimentally prove the effectiveness of the improved algorithm, and the new algorithm significantly reduces the probability of repeated operations.

DOI
10.1063/5.0153709

Related papers

RSA Prime Factorization on IBM Qiskit

Matthew Evans Audric Rengkung, Arya Wicaksana

The advancement of quantum computing in recent years poses severe threats to the RSA public-key cryptosystem. The RSA cryptosystem fundamentally relies its security on the computational hardness of number theory problems: prime factorization (integer factoring). Shor’s quantum factoring algorithm could theoretically answer the computational problem in polynomial time. This paper contributes to the experiment and demonstration of Shor’s quantum factoring algorithm for RSA prime factorization using IBM Qiskit. The performance of the quantum program is evaluated based on user time and the success probability. The results show that a more significant public modulus N in the RSA public key improves factorization’s computational hardness, requiring more quantum bits to solve. A further enhancement on implementing Shor’s oracle function is essential in increasing success probability and reducing the number of shots required.

Journal of Internet Services and Information Security•May 30, 2023•8 citations

Factoring Safe Semiprimes with a Single Quantum Query

Frédéric Grosshans, Thomas Lawson, François Morain +1 more

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to $1/2$, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.

arXiv:1511.04385•Nov 13, 2015•9 citations

A new quantum algorithm for computing RSA ciphertext period

Yahui Wang, Songyuan Yan, Huanguo Zhang

Shor proposed a quantum polynomial-time integer factorization algorithm to break the RSA public-key cryptosystem. In this paper, we propose a new quantum algorithm for breaking RSA by computing the order of the RSA ciphertext C . The new algorithm has the following properties: 1) recovering the RSA plaintext M from the ciphertext C without factoring n ; 2) avoiding the even order of the element; 3) having higher success probability than Shor’s; 4) having the same complexity as Shor’s.

Wuhan University Journal of Natural Sciences•Jan 9, 2017•4 citations

Quantum polynomial-time fixed-point attack for RSA

Yahui Wang, Huanguo Zhang, Houzhen Wang

Security analysis of public-key cryptosystems is of fundamental significance for both theoretical research and applications in cryptography. In particular, the security of widely used public-key cryptosystems merits deep research to protect against new types of attacks. It is therefore highly meaningful to research cryptanalysis in the quantum computing environment. Shor proposed a well-known factoring algorithm by finding the prime factors of a number n = pq, which is exponentially faster than the best known classical algorithm. The idea behind Shor's quantum factoring algorithm is a straightforward programming consequence of the following proposition: to factor n, it suffices to find the order r; once such an r is found, one can compute gcd(ar/2± 1, n) = p or q. For odd values of r it is assumed that the factors of n cannot be found (since ar/2is not generally an integer). That is, the order r must be even. This restriction can be removed, however, by working from another angle. Based on the quantum inverse Fourier transform and phase estimation, this paper presents a new polynomial-time quantum algorithm for breaking RSA, without explicitly factoring the modulus n. The probability of success of the new algorithm is greater than 4φ(r)/π2r, exceeding that of the existing quantum algorithm for attacking RSA based on factorization. In constrast to the existing quantum algorithm for attacking RSA, the order r of the fixed point C for RSA does not need to be even. It changed the practices that cryptanalysts try to recover the private-key, directly from recovering the plaintext M to start, a ciphertext-only attack attacking RSA is proposed.

China Communications•Feb 1, 2018•33 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

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
Abstract The task of Shor’s factorization algorithm (SFA) is to find a non-trivial factor of a given composite number N. In the algorithm, it is explicitly required that the constructor f (x) = a x mod N cycle is an even number, otherwise it returns to the algorithm header to recalculate. According to the analysis of the principle of SFA, the characteristics of RSA public key cryptosystem and a large number of calculation results, the random selection of a value is related to the obtained period. Aiming at this defect, a new optimization scheme is proposed on the basis of the previous optimization, and the odd cycle obtained when a is a perfect square number randomly becomes effective. When the arbitrarily chosen value of a obtains a period that is a multiple of 3, by modifying the decomposition method, the requirements for the period are relaxed without affecting the algorithm complexity. Using the classic algorithm, we experimentally prove the effectiveness of the improved algorithm, and the new algorithm significantly reduces the probability of repeated operations.
The advancement of quantum computing in recent years poses severe threats to the RSA public-key cryptosystem. The RSA cryptosystem fundamentally relies its security on the computational hardness of number theory problems: prime factorization (integer factoring). Shor’s quantum factoring algorithm could theoretically answer the computational problem in polynomial time. This paper contributes to the experiment and demonstration of Shor’s quantum factoring algorithm for RSA prime factorization using IBM Qiskit. The performance of the quantum program is evaluated based on user time and the success probability. The results show that a more significant public modulus N in the RSA public key improves factorization’s computational hardness, requiring more quantum bits to solve. A further enhancement on implementing Shor’s oracle function is essential in increasing success probability and reducing the number of shots required.
Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to 1/21/21/2, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.
Security analysis of public-key cryptosystems is of fundamental significance for both theoretical research and applications in cryptography. In particular, the security of widely used public-key cryptosystems merits deep research to protect against new types of attacks. It is therefore highly meaningful to research cryptanalysis in the quantum computing environment. Shor proposed a well-known factoring algorithm by finding the prime factors of a number n = pq, which is exponentially faster than the best known classical algorithm. The idea behind Shor's quantum factoring algorithm is a straightforward programming consequence of the following proposition: to factor n, it suffices to find the order r; once such an r is found, one can compute gcd(ar/2± 1, n) = p or q. For odd values of r it is assumed that the factors of n cannot be found (since ar/2is not generally an integer). That is, the order r must be even. This restriction can be removed, however, by working from another angle. Based on the quantum inverse Fourier transform and phase estimation, this paper presents a new polynomial-time quantum algorithm for breaking RSA, without explicitly factoring the modulus n. The probability of success of the new algorithm is greater than 4φ(r)/π2r, exceeding that of the existing quantum algorithm for attacking RSA based on factorization. In constrast to the existing quantum algorithm for attacking RSA, the order r of the fixed point C for RSA does not need to be even. It changed the practices that cryptanalysts try to recover the private-key, directly from recovering the plaintext M to start, a ciphertext-only attack attacking RSA is proposed.
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 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.