Xiu-er algorithm: Decoding the "Immortal myth" of RSA encryption

RSA encryption VS. Runway algorithm

As the Terminator of RSA cryptography-"Too many operations, unable to read"-The Shor's algorithm is not found by brute force, but by the parallelism of quantum computing, which can quickly decompose the number of conventions, This breaks the foundation of the RSA algorithm (assuming that we cannot effectively decompose a known integer). At the same time, the display algorithm shows that the problem of factoring can be solved efficiently on quantum computers, so a quantum computer large enough to crack RSA.

RSA encryption "Once" is powerful, because it is the difficulty of factoring large integers determines the reliability of RSA algorithm. It is easy to multiply two prime numbers, but it is very difficult to find the mass factor of a large number. This is the reliance of a large number of modern technology, RSA encryption is the rapid popularity of its simplicity.

However, there is a technology that can make RSA encryption useless. The runway algorithm can crack RSA, but how can it really make a real effect?

We are not suggesting that you try all possible quality factors at the same time.

Instead, use a (relatively) concise statement:

If we quickly find the cycle of the following periodic function,

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

We can hack RSA encryption.

Sul Five steps away

So, how does the runway algorithm work? In the five footwork, only one step is needed to use a quantum computer, and the other steps can be solved by traditional methods.

The first step:

Using the traditional greatest common divisor decomposition (GCD) algorithm, that is, the Euclidean method. N is the factor you need to try, and M is a random positive integer less than n.

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

Step Two:

Find Cycle P

M mod n, m^2 mod n, m^3 mod n

This is a step in the use of quantum computing.

Step Three:

If the period p is odd, go back to the first step and select another random integer. If not, continue to the next step.

Fourth Step:

Inspection

If it is established, proceed to the fifth step; instead, return to the first step.

Fifth Step:

Solution

The value of an extraordinary factor n is solved, and then you can decipher RSA encryption.

How is the second step achieved?

However, how does a quantum computer find a function cycle? Why is that so important?

Let's take a look at the cycle P:

M mod n, m^2 mod n, m^3 mod n

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

The process of discovery cycles relies on the ability of quantum computers to simultaneously compute many states, the "superposition" of states, so that we can find the cycles of the equations.

We need to do this:

1, apply Hadamard gate to create a quantum superposition state 2, quantum transformation to make the equation effective 3, the implementation of quantum Fourier transform

Similar to the traditional case, after these conversions, a measured value will produce a value for the approximate equation period (you can get "crest", as in Fourier transform, and the accuracy will be higher). Using the quantum Fourier transform, we can solve the problem of sorting and factoring, both of which are the same. The quantum Fourier transform allows a quantum computer to perform phase estimation (approximation of the eigenvalues of unitary operators).

When you complete the quantum section (step two), you can check the validity of the cycle, and then use another traditional greatest common divisor algorithm to get the key quality factor.

Interestingly, since this technique is not about finding all the potential factorization, but finding the potential cycle, you don't have to try a lot of random numbers until you find a successful mass factor n. If p is odd, then you have to go back to the first step, where

K is a qualitative factor that differs from N. Therefore, even if you double the key length (N), looking for factorization does not appear to be slowing down. RSA is not secure, and the same double key length does not help you withstand the onslaught of quantum computing, while guaranteeing security.

"Cracking RSA-2048 (2048-bit) keys can take 1 billion years of traditional computers, and quantum computers can take only 100 seconds to complete. "--dr Krysta Svore, Microsoft Research

Quantum Fourier transforms are used to build quantum circuits, making the physical realization of the algorithm a quantum computer one of the most relaxed tasks.

Quantum Fourier transform: 've seen

The core of the algorithm is the discovery order, which can reduce the problem of abelian subgroups, which can be solved by using the quantum Fourier transform. --nist Quantum World

Quantum Fourier transform is the key to many quantum algorithms. It does not speed up the search for traditional Fourier transforms, but can perform a Fourier transform within a quantum amplitude. The quantum Fourier transform can be rapidly processed on a quantum computer with exponential growth. Although the scope of direct mapping of classical Fourier transforms is exceeded, quantum computers can do other things as well. For example, solving the problem of hidden subgroups (that is, solving discrete logarithm problems), or counting problems (solving this problem can solve many other forms of cryptography in modern cryptography). More importantly, Quantum Fourier transforms can be applied to machine learning, chemistry, material science, or analog quantum systems.

There is only one step in the runway algorithm that needs to be done on a quantum computer, and everything else can be done on a normal supercomputer. When the quantum computer finishes running the subroutine, it returns the result to the supercomputer so that it can continue with the calculation process. Quantum computers may never exist alone, but they always work with supercomputers, and by doing so they can hack RSA keys.

Because of the limited space, many of the mathematical details and the process of proof will not be mentioned, if you are interested in these mathematical explanations, if you have linear algebra, group theory, advanced mathematics, you can look at these:

Quantum Computer Science

Quantum Information and Quantum computation

NIST Quantum Zoo?? -? A list of all quantum algorithms

Xiu-er algorithm: Decoding the "Immortal myth" of RSA encryption