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2. 3. 2026

Security Cryptography

Comparing Digital and Quantum Computers, Part 4: Attack with Quantum Computer

Everyone in the IT world is wondering how big a threat quantum computers pose to cryptography and how to deal with the problem. This series of articles tries to explain the problem in a popular way. After discrete logarithm problem, and its solution on digital computers, it is useful to understand the impact of quantum computers.

Principles of attacks to current asymmetric algorithms using quantum computersu

Breaking current algorithms for asymmetric cryptography requires the use of QFT, the Quantum Fourier Transformation. Current asymmetric algorithms are built on certain assumptions and complex problems that we have not been able to solve yet.

The RSA algorithm is built around the factorization problem. That is, if we have a number n , we are unable to determine the numbers p and q from which it was formed. In the case of RSA, p and q are large primes to protect against brute force attacks. Factoring such numbers is extremely computationally demanding.

DH, DSA, ECDH, ECDSA algorithms are algorithms that are built on the discrete logarithm problem. In simple terms, we can usually calculate the power of a number. But there is a certain area of mathematics where roots are complicated. This is modular arithmetic, which uses numbers in certain groups (groups). We have all encountered this area in school, it is about counting with remainders. It is possible to imagine it as counting on a clock that has a prime number of entries (counting modulo a prime number). Raising a number to a power of some value tells us the number of steps, but because we are rotating in a circle, we can often get to the same values. What were the original values and the number we were exponentiating?

In the above cases, QFT can help us. Its principle is already known today, but it is a function that allows us to search for a certain repetition period. The original Fourier transform, after conversion into the form of a quantum algorithm, allowed Peter Shor to propose a procedure for attacking the RSA algorithm in 1995. Later, this principle was also used to attack other forms of asymmetric cryptography. In the case of factorization, the procedure searches for the period of the function, in the case of elliptic curves, a linear function (linear relationships between individual elements), which allows us to calculate the parameters of the secret key.

If we talk about specific algorithms, for RSA we will look for an order (modular basis). It is possible to imagine it as a swimming pool, which has a length and width corresponding to the length of the number n . We make its surface oscillate, so that we have waves of all sizes and frequencies. A quantum computer evaluates this surface using QFT at once. The places where there are the largest deviations also indicate the most probable combinations of numbers. The algorithm tries to find out which combinations of results are the most probable, but the result must then be verified on a regular digital computer. This is called post-processing. The goal of new variants of the algorithm is to reduce the demand for the number of qubits and increase the probability of a successful result.

The situation is a little different for elliptic curves. Here, the basis of a linear function is sought. The curve has certain parameters and usually a specified starting point. If we know the target point, QFT can be used to find all possible combinations of multiples that lead to a given result. Based on the given values, the algorithm then finds the probability for the given values, because for certain multiples it is not possible to reach the given points. The idea of such a solution is much more complicated. Graph paper is commonly used to draw the function, but in this case the paper will be extremely flexible. Since it is limited by a prime number to a certain length and width, we need a certain adjustment to draw an infinite curve. This means connecting the top and bottom sides (a cylinder is formed) and then connecting the sides to each other (a toroid is formed, something like an inflated tire). If we know the starting point and the end point, QFT finds all possible parameters when it is possible to get from the starting point to the end point. Something like a taut string that copies this curve from the starting point to the end point. There are an enormous number of these strings. Certain combinations of tones (repetition frequencies) can be played on all of these strings. Those that are not tuned play falsely. False frequencies are automatically discarded, the rest are sorted out by the QFT function. This function looks for strings with the purest tone, that is, it looks for suitable combinations of parameters. However, unlike RSA, it is necessary to use not one, but two registers for the calculation, which is why the number of qbits is larger. As with RSA, post-processing is also necessary after the calculation. Since better and better procedures are being sought today, the goal of new variants of the algorithm is to reduce the demand on the number of qbits and increase the probability of a successful result.

Miniaturization of quantum computers and its limits

Current quantum computers require hundreds to thousands of physical qubits per logical qubit. This is due to noise and the need to ensure adequate error correction using quantum correction codes. Most people in computer science are used to Moore's Law (more precisely, observations) and are reassured by the idea that it is only necessary to wait a few years. However, there is a significant difference between digital and quantum computers. Moore's Law was built on automation, increasing density and negligible noise, and currently (more precisely since 2015) it is also reaching its limits. In the case of quantum computers, the situation is significantly worse. Each individual qubit is simply not a transistor, it cannot be easily reduced. It needs control, isolation and calibration. At the same time, adding qubits does not necessarily mean an increase in performance. The reason is the mutual influence in the entire system, each qubit means more cabling, cooling, noise and therefore more error correction.

The observation of the development is too short for a possible estimate of the rules of development. Probably the best metric in this area is called Quantum Volume. It describes the increase based on the number of qubits, circuit depth and error rate, yet even that cannot predict the development in an adequate way. So although the number of qubits is seemingly growing exponentially, this does not have a corresponding effect on the logical qubits, performance and significant improvement in the area of error reduction. Scaling is currently extremely difficult, the production of quantum computers is not an automated serial line, which we are used to with silicon. It is more of a manufactory with tens to hundreds of highly qualified specialists.

Miniaturization itself then encounters several physical problems. One problem is the uncertainty principle, known as the Heisenberg principle. Simply put, we have the momentum of a particle and its position, but we can only know one of the items. Another view is similar, we have time and energy, again we can only know one of the items. Any measurement requires energy, the smaller the particle, the higher it is. And of course, we influence the particle by measuring. Another problem concerns the addressing of the value caused by the energy used. So the smaller the given element, the higher the energy it needs for manipulation. This of course also affects the surrounding systems and creates unwanted noise. In addition to the problems mentioned, the distance between the individual qubits must be greater than the wavelength of the controlling electromagnetic fields, today microwaves are usually used. Otherwise, the individual physical qubits would interact with each other. Then we have another problem, which is the ratio of the volume to the surface of the qubit. The smaller the qbit, i.e. the smaller its volume, the more pronounced this ratio to the surface is, and below a certain size, the behavior of the qubit is mainly determined by its surface area. Because the most significant noise generation occurs on the surface (interaction with the environment), the surface area significantly affects the error rate of a given physical qubit. Below a size of about 10 nm, the surface determines the ability to withstand noise. We can continue beyond the size limits based on Planck units (Planck length 1.6∙10⁻³⁵ m and Planck time 5.4∙10⁻⁴⁴ s) , i.e. the smallest length and shortest time. Below this limit, the concept of a qubit ceases to have meaning. More precisely, the question is whether anything else has meaning below this limit, but these are questions that probably not many people can answer. We will probably never be able to reach this limit. Finally, as with classical computers, there is the Landauer limit. This specifies that the destruction of any information generates heat. For this reason, it is better not to erase information, but to count it. For quantum computers, heat is another source of noise that threatens their ability to perform calculations. Moreover, the smaller the circuit, the worse the heat is dissipated. The Landauer limit thus creates a natural barrier that will always have an impact on the design of the computer and the threat of interruption of the calculation by spontaneous collapse.

This paragraph often mentions the terms physical and logical qubit, but what is the difference? A physical qubit is an implementation in hardware, a physical element with quantum properties. A logical qubit is a construction over several physical qubits using quantum correction codes. It is significantly more stable, because the coupling between the physical representations allows the creation of a more stable logical representation with the help of the aforementioned codes. In addition to its significant impact on reliability, this property also allows a certain freedom in marketing product representations.

To be continued in the next section How Quantum Computers Works (March 9th 2026)

References:

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
Source: https://arxiv.org/
Semiclassical Fourier Transform for Quantum Computation
Source: https://arxiv.org/
Circuit for Shor’s algorithm using 2n+3 qubits
Source: https://arxiv.org/
Surface codes: Towards practical large-scale quantum computation
Source: https://arxiv.org/
How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits
Source: https://arxiv.org/

Autor článku:

Jan Dušátko

Jan Dušátko has been working with computers and computer security for almost a quarter of a century. In the field of cryptography, he has cooperated with leading experts such as Vlastimil Klíma or Tomáš Rosa. Currently he works as a security consultant, his main focus is on topics related to cryptography, security, e-mail communication and Linux systems.

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1.1. These General Terms and Conditions are, unless otherwise agreed in writing in the contract, an integral part of all contracts relating to training organised or provided by the trainer, Jan Dušátko, IČ 434 797 66, DIČ 7208253041, with location Pod Harfou 938/58, Praha 9 (next as a „lector“).
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