Shur algorithm: Cracking RSA encryption's "immortal myth"

Shur algorithm: Cracking RSA encryption's "immortal myth"

RSA encryption was once regarded as the most reliable encryption algorithm until the appearance of the xiur algorithm broke the myth of RSA.

RSA encryption VS xiuer Algorithm

As the terminator of RSA encryption technology, Shor's algorithm, which is "too many operations and cannot be read", does not use brute force cracking to find the final password, instead, we can use the parallelism of quantum computing to quickly break down the common number, thus breaking the foundation of the RSA algorithm (that is, we cannot effectively break down a known integer ). At the same time, the xiuer algorithm shows that the factorization problem can be effectively solved on quantum computers, so a large enough quantum computer can crack RSA.

The reason why RSA encryption was "once" powerful is that the difficulty of Factorization large integers determines the reliability of the RSA algorithm. It is easy to multiply two prime numbers, but it is very difficult to find the prime factor of a large number. This is the reliance of a large number of modern technologies. RSA encryption is rapidly popular with its simplicity.

However, there is a technology that makes RSA encryption useless. Shur algorithm can crack RSA, but how can we make it really take effect?

We do not recommend that you try all possible quality factors at the same time.

Instead, use (relatively) concise statements:

If we can quickly find the cycle of the following cyclic function,

F (x) = m ^ x (mod N)

We can crack RSA encryption.

Step 5

So how does the xiur algorithm work? In the five-step method, only one step requires the use of quantum computers, and other steps can be solved using traditional methods.

Step 1:

The traditional gcd algorithm is used to divide the moving phase. N is the factor you need to try, and m is a random positive integer smaller than N.

If gcd (m, N) = 1, continue. Once you use gcd to find a factor, you can get an extraordinary factor and end it.

Step 2:

Locate period P

M mod N, m ^ 2 mod N, m ^ 3 mod N

This is a step towards using quantum computing.

Step 3:

If the period P is an odd number, return to the first step and select another random integer. If not, continue to the next step.

Step 4:

Inspection

If so, proceed to Step 5. Otherwise, go back to step 1.

Step 5:

Solution

Get a value of extraordinary prime factor N, and then you can crack RSA encryption.

How is the second step implemented?

However, how do quantum computers find function cycles? Why is this so important?

Let's take a look at the period P:

M mod N, m ^ 2 mod N, m ^ 3 mod N

(Because this is an exponential function, we can convert a complex prime number into a hyperbolic sine, cosine, and then get a period)

This discovery cycle process depends on the ability of quantum computers to compute many states simultaneously, that is, the superposition of States. Therefore, we can find the cycle of the equation.

We need to do this:

1. Use Hadamard gate to create a quantum superposition state

2. Quantum transformation takes effect for the equation

3. perform quantum Fourier Transformation

Similar to the conventional situation, after these transformations, a measurement value will generate a value of the approximate equation period (you can obtain the "peak", just like in the Fourier transformation, but the accuracy will be higher ). Using Quantum Fourier transformation, we can solve the sorting and factor problems, which are the same. Quantum Fourier transformation allows a quantum computer to perform phase estimation (approximate value of the feature value of the youoperator ).

When you complete the quantum part (step 2), you can check the validity of the cycle, and then use another traditional maxcompute algorithm to obtain the key quality factor.

Interestingly, because this technology is not about finding all potential prime factors, but about finding a potential cycle, you don't have to try a lot of random numbers until you find a successful prime factor N. If P is an odd number, you have to go back to Step 1. Here

K is a qualitative factor different from N. Therefore, even if you double the key length (N), the search for a prime factor will not slow down. RSA is insecure, and doubling the key length cannot help you resist the surging quantum computing attacks and ensure security.

"Cracking the RSA-2048 (2048-bit) key may take 1 billion years for a traditional computer, and it takes 100 seconds for a quantum computer to complete ."

-- Dr. Krysta Svore, Microsoft Research Institute

Quantum Fourier transformation is used to establish a quantum line, which makes the physical implementation of the Shur algorithm one of the most easy tasks of quantum computers.

Quantum Fourier Transformation: Blue

The core of the Shur algorithm is the discovery sequence, which can reduce the hidden subgroups of Abel and solve the problem by using quantum Fourier transformation. -- NIST quantum world

Quantum Fourier transformation is the key to many quantum algorithms. It does not accelerate the search for the Traditional Fourier transformation, but can execute a Fourier transformation within a quantum amplitude. On a quantum computer, quantum Fourier transformation can be processed exponentially and rapidly. Although it is beyond the scope of direct ing of classical Fourier Transformations, quantum computers can do other things. For example, solving the problem of hidden subgroups (that is, solving the problem of discrete logarithm) or counting (solving this problem can solve many other forms of passwords in modern cryptography ). More importantly, quantum Fourier transform can be applied to machine learning, chemistry, material science, or analog subsystems.

Only one step of the Shur algorithm needs to be completed on a quantum computer, and other steps can be completed on a common supercomputer. After running the subroutine, the quantum computer returns the result to the supercomputer to continue the computation. Quantum computers may never exist independently. Instead, they always work with super computers to execute tasks. With such cooperation, they can crack RSA keys.

Due to limited space, many mathematical details and evidentiary processes will not be repeated. If you are interested in these mathematical interpretations, if you have knowledge of linear algebra, group theory, and advanced mathematics, you can look at these:

Quantum Computer Science

Quantum Information and Quantum Computation

NIST Quantum Zoo  -  a list Of all Quantum algorithms