When building cryptosystems, cryptographers focus on finding problems that are not believed to be solvable in polynomial-time. Some of the most popular problems they have found are the Discrete Logarithm Problem and Integer Factoring. The former is used in Diffie-Hellman Key Exachange (DHK) and El Gamal encryption, while the latter is used in RSA. El Gamal and DHK are both very popular, but RSA is more prevalent due to its efficiency.\n Nevertheless, it is plausible that in the next few decades, all three of these systems will likely be useless due to the advances made by Peter Shor in quantum computing. This paper will explain the details of how Shor’s algorithm works and how it accomplishes the above. It will also feature a redesign of the proof of Jeffrey Miller (1975) that efficiently reduces from order finding in a group of order N to factoring N. Hopefully, doing so will aid future students in their studies of quantum algorithms and Post-Quantum Cryptography.
When building cryptosystems, cryptographers focus on finding problems that are not believed to be solvable in polynomial-time. Some of the most popular problems they have found are the Discrete Logarithm Problem and Integer Factoring. The former is used in Diffie-Hellman Key Exachange (DHK) and El Gamal encryption, while the latter is used in RSA. El Gamal and DHK are both very popular, but RSA is more prevalent due to its efficiency.\n Nevertheless, it is plausible that in the next few decades, all three of these systems will likely be useless due to the advances made by Peter Shor in quantum computing. This paper will explain the details of how Shor’s algorithm works and how it accomplishes the above. It will also feature a redesign of the proof of Jeffrey Miller (1975) that efficiently reduces from order finding in a group of order N to factoring N. Hopefully, doing so will aid future students in their studies of quantum algorithms and Post-Quantum Cryptography.
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 cryptography is a promising field for securing information against quantum computers. Shor's algorithm, a quantum algorithm, can factor large prime numbers in polynomial time and solve the discrete logarithm problem, which is the basis for classical cryptographic algorithms like Rivest-Shamir-Adleman algorithm (RSA), Diffie-Hellman key exchange (DHKE), and Elliptic Curve Diffie-Hellman algorithm(ECDH). These algorithms pose a significant threat to their security and reliability. Post-quantum cryptography techniques are being implemented by all industries to address the threat posed by quantum supremacy. The chapter discusses the potential threat of Shor's algorithm on classical cryptographic algorithms and discusses post-quantum cryptographic algorithms like lattice-based, multivariate, code-based, and Hash-based cryptography. It also discusses recent advancements that have improved the security and usefulness of these algorithms.
This thesis contains a collection of efficient quantum algorithms and classical pre- and post-processing algorithms for various number- and group-theoretical problems with concrete cryptanalytical applications. More specifically, the focus is on the integer factoring problem (IFP), the discrete logarithm problem (DLP), and the order-finding problem (OFP), and on certain important variations of these problems. It is for example explained how the OFP and complete IFP may be solved efficiently in a single run of a reduced version of Shor's algorithm for the OFP with very high success probability, as evidenced by a lower bound that is 1 − o(1), how the short DLP in groups of unknown order may be solved efficiently and with an advantage over Shor's algorithm for the DLP (that furthermore requires the order to be known), as evidenced by a lower bound on the single-run success probability that is 1 − o(1), how the RSA IFP may be solved efficiently and with an advantage over Shor's algorithm for the IFP, by reducing the RSA IFP to a short DLP, how the DLP may heuristically be solved efficiently, and with high success probability that is 1 − o(1), in a single run of a reduced version of Shor’s algorithm for the DLP that admits efficient implementation, how the DLP may be solved in groups of unknown order, such as Schnorr groups, by jointly solving an OFP and a DLP at lower overall cost but higher per-run cost compared to first solving the OFP and then the DLP, how tradeoffs — between the number of group operations that are evaluated quantumly and the number of runs that are required — may be achieved in all of the aforementioned algorithms, and how all of the aforementioned algorithms may be simulated efficiently classically, for arbitrary large cryptographically relevant problem instances, when the solution to the instance is known, enabling the number of runs required when performing tradeoffs to be tightly estimated. It is furthermore for example explained how Regev's recent quantum algorithm for the IFP may be extended to the DLP and OFP, yielding i) a reduction in the circuit size and depth by a factor compared to Shor for these problems when working in , for N an n-bit integer, at the expense of performing runs, and ii) slight improvements of Regev's orginal algorithm for the IFP, how the above algorithms may be simulated efficiently classically, for large special-form problem instances that are classically tractable, and that Regev’s post-processing, and the post-processing for our extensions, are robust to an arbitrarily large fraction < 1 of the runs being bad. The above has implications for cryptography based on the intractability of the IFP and DLP — including but not limited to for widely used schemes such as RSA, DSA and Diffie–Hellman. The algorithms in this thesis are all derived from, and make notable improvements to, the seminal works of Shor. Some of the contributions are joint works with other authors.
Many cryptographic algorithms depend on computational complexity assumptions. Notorious cases are the RSA algorithm for public key criptography or the Diffie-Hellman key exchange protocol, to publicly agree on a common secret key. Both algorithms are known to be broken by quantum computing as well as those that can be reduced to a discrete logarithm problem. These are key algorithms in our digital society and are at the basis of everyday tasks, specially those that rely on digital signatures. The RSA algorithm, in particular, is probably the most used algorithm and is its assumed security the one that guards most of the e-commerce nowadays. In this case, it is the time complexity of finding the prime factors of a large number, that grows worse than polinomially with the size of the number, the main guardian of our cyberinfrastructure. The fact that a quantum computer can solve this problem in polynomial time using Shor's algorithm is seen as a potentially major disruption and has prompted security agencies to advice the progressive deprecation of these algorithms.
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.
Today we witness an explosive growth in attention to Q-computing. Q-computing technologies, along with artificial intelligence (AI) and machine learning (ML) technologies, cloud and foggy computing, as well as technologies for collecting and streaming processing of Big Data and ETL, are constantly leading the lists of ""end-to-end"" information technologies for the digital economy of technologically developed countries of the world. One of the main reasons for this is the potential ability of quantum computers to solve some computational problems more efficiently than any of the most modern classical computers of the von Neumann architecture (supercomputers). The most expressive and interesting, from an applied point of view, examples of such problems are integer factorization, effectively performed by Shor's quantum algorithm, as well as record search in an unordered database, effectively solved by Grover's algorithm. This monograph contains the best practice for solving problems of quantum cryptanalysis to improve cyber security and resilience of the digital economy. The book discusses well-known and author's software implementations of promising quantum Shor algorithms, Grover, Simon et al. Shor's algorithm provides exponential acceleration of solving factorization problems, discrete logarithm problems (DLPs) and elliptic curve discrete logarithm problems (ECDLPs). The mentioned tasks are widely used in TLS, SSH or IPsec cryptographic applications of Internet/Intranet and IIoT/IoT networks, communication protocols based on Diffie–Hellman key agreements (dependent on the strength of the DLP or ECDLP), digital signature algorithms (DSA, ECDSA, RSA-PSS), public key encryption algorithms (El Gamal, RSA-OAEP), etc. In other words, Shor's quantum algorithm is potentially capable of violating these algorithms, and with them all the mechanisms of public-key cryptography deployed in cyberspace.