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Quantum Computing and Factorization Problems

Process of factorization of numbers involves finding a prime number that results in multiplication of two digit numbers. This process seems easy when finding the results for smaller numbers but it creates a problem where numbers are in the form of power ‘n’ that is exponentially. In order to solve this problem we need a strong and fast algorithm who can compute fast results.

Factorization techniques play a very important role in the field of mathematics cryptography and number theory. Cryptography is the most important part in internet based communication because sensitive information is being encrypted and then it is being delivered to the sender. So developing efficient methods for factorization is challenging in the field of computer science.

Limitation of classical algorithm 

Classical algorithm also known as cryptography algorithm which is used in the encryption and decryption of messages and here it is used in the calculation of certain problems but in the case of large problems their speed is not enough to calculate or in order to get a solution. Here society needs the power of quantum computing.

Large number factorisation is crucial in cryptography and other sectors such as finance, yet traditional techniques like the General Number Field Sieve (GNFS) and Quadratic Sieve (QS) are computationally costly. As the magnitude of the number grows, so does the complexity of these techniques, making it practically impossible for classical computers to factor huge numbers in a reasonable amount of time. This difficulty serves as the foundation for encryption systems such as RSA, which rely on factorization’s hardness to provide security. However, advancements in processing power or new mathematical methodologies could risk the security of these systems, exposing sensitive data.

Classical computing also has difficulty with optimisation and search difficulties. For example, determining the most effective route in a transportation network or optimizing resource allocation often.

Shor’s algorithm is used to find the factor of large numbers as it possesses the property of quantum computing which can calculate faster than the normal computer because of its exponential computing power.  Classical computers depend on brute force methods in order to calculate the factors of big numbers which becomes  expensive sometimes and it’s time consuming, while shor’s algorithm with the help of unique property like  superposition and entanglement, to perform factorization more efficiently

Quantum computers are advanced technology  that will completely change the problem solving task  in areas such as artificial intelligence.core differences of quantum computing  come from the fact that the traditional computers, unlike quantum computers which make use of quantum physics concepts, what is referred to “bits” (i.e. 1s and 0s) to do computing. Instead quantum computers work on quantum bits or qubits which in essence means that they may be in more than one state at a given time. This property therefore presents a technology where a very complex problem can be handled faster than  traditional computers, Now let’s see the properties of quantum computer




Property of Quantum Computers

1. Superposition :  It’s a conditional property of a system which  allows it to stand in multiple states at the same time.  In classical systems a bit is either o or 1 while in a quantum computing system it can be both at the same time. This technique allows computing systems to perform fast calculations.

2. Entanglement : it’s a condition in which a quantum system becomes dependent on each other for example while executing this method the state of one qbit depends on the state of another even though distance between them is far from each other and the system is called as dependency while in classical computers both are independent.

3. Interference : it’s a condition in which two states of a computing system are combined and produce the best result by manipulating the pattern in order to increase the chance of reaching the correct solution and decreasing the probability of  wrong solution.


Quantum factorization is a crucial problem in cryptography as it has a significant impact for cryptography and security. Shor’s algorithm in polynomial time for discrete logarithm and factor calculation has become a threat to the classical encryption system. Qc can be useful in running factorization in polynomial time.However practical implementation is still a major challenge.


The factorization challenge opens a new field of research in quantum computing  protocol security including various techniques such as key distribution and development of quantum resistant cryptography.

Let’s discuss the algorithm of shor’s

Shor’s algorithm : In 1994 peter shor developed an algorithm which can solve factorization problems in an efficient way. It was designed to solve factors of large numbers faster than classical algorithms.

Algorithm operates in two stages. 

  1. Fourier transformation : This method is the most effective in the field of mathematics.in order to solve a problem is divided into the sub problem for which the solution is known by its formula and after getting to that form we apply the formula and get the desired result.  

This technique is used to find periodicity in the function of the factored number.

  1. Classical post – processing : Once the period of the quantum part is determined with the help of an algorithm , classic arithmetic is used in order to derive the factors of the derived number.


In short, a classical algorithm will take hundreds years to calculate a factor containing several digits where with the help of shor’s algorithm it will be calculated in polynomial time. These two properties will lead to breaking the encryption of an effective algorithm called RSA , as any message encrypted by shor can easily be decrypted by shor’s algorithm because calculation is very fast.

Conclusion 
Factorization is a very complex task when we have to deal with a large number of digits in order to find the correct solution in the field of mathematics and specially in coding logic with respect to cryptography.
Factorization is transformed by Shor’s algorithm, a quantum computing innovation that uses the strength of quantum systems to solve the issue tenfold quicker than with traditional approaches.

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