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.
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.
Abstract We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.
We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.
Simulation of Shor Algorithm for Discrete Logarithm Problems With Comprehensive Pairs of Modulo p and Order q
The discrete logarithm problem (DLP) over finite fields, commonly used in classical cryptography, has no known polynomial-time algorithm on classical computers. However, Shor has provided its polynomial-time algorithm on quantum computers. Nevertheless, there are only few examples simulating quantum circuits that operate on general pairs of modulopand orderq. In this paper, we constructed such quantum circuits and solved DLPs for all 1,860 possible pairs ofpandqup to 32 qubits using a quantum simulator with PRIMEHPC FX700. From this, we obtained and verified values of the success probabilities, which had previously been heuristically analyzed by [Ekerå, arXiv:1905.09084 (2019)]. As a result, the detailed waveform shape of the success probability of Shor's algorithm for solving the DLP, known as a periodic function of orderq, was clarified. Additionally, we generated 1,015 quantum circuits for larger pairs ofpandq, extrapolated the circuit sizes obtained, and compared them forp=2048bits between safeprime groups and Schnorr groups. While in classical cryptography, the cipher strength of safe-prime groups and Schnorr groups is the same ifpis equal, we quantitatively demonstrated how much the strength of the latter decreases to the bit length ofpin the former when using Shor's quantum algorithm. In particular, it was experimentally and theoretically shown that when a basic adder is used in the addition circuit, the cryptographic strength of a Schnorr group withp=2048bits under Shor's algorithm is almost equivalent to that of a safe-prime group withp=1024bits.
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.
We present our perspectives on the current status of, integer factoring and discrete logarithm, algorithms underlying the strength of classical public-key cryptography protocols used in key-exchange, encryption and digital signatures. We present an expository overview of the algebraic, number-theoretic and algorithmic principles behind the canonical, powerful algorithms used in the factoring of large, composite RSA modulii. These cover the two main ideas of group order based methods, and quadratic congruence based methods. We provide the resource estimates for the record breaking factoring based on the general number field sieve. Next we present an overview of the algorithmic theory of quantum computation behind, the polynomial time Shor's quantum algorithm for, period finding in hidden subgroup structure problems. This algorithm has played a key role in advancing post-quantum cryptography, and also in the exciting, contemporary developments in quantum computing, networking, sensing, and materials science as a whole. Latest advances in quantum computing rest on a variety of physics, engineering, algorithms, compilation, hardware, software techniques. There are several tradeoffs based on number of qubits, coherence times, control issues, errors and fault tolerance. In this environment, nuances of mathematical ideas, behind every step of the intricate arithmetical and algebraic computations under quantum models plays a significant role. We provide an overview of some of the recent ideas behind such algorithms. We conclude with a recent deep question on the ultimate feasibility of ultra large-scale, error-free, discrete, quantum computations required typically in cryptanalysis.
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.