Academic literature on the topic 'Fermat factoring algorithm'

The development of public-key cryptography generation using the factoring method is very important in practical cryptography applications. In cryptographic applications, the urgency of factoring is very risky because factoring can crack public and private keys, even though the strength in cryptographic algorithms is determined mainly by the key strength generated by the algorithm. However, solving the composite number to find the prime factors is still very rarely done. Therefore, this study will compare the Fermat factorization algorithm and Pollard rho by finding the key generator public key algorithm's prime factor value. Based on the series of test and analysis factoring integer algorithm using Fermat's Factorization and Pollards' Rho methods, it could be concluded that both methods could be used to factorize the public key which specifically aimed to identify the prime factors. During the public key factorizing process within 16 bytes – 64 bytes, Pollards' Rho's average duration was significantly faster than Fermat's Factorization.

The security of many public-key cryptosystems and protocols relies on the difficulty of factoring a large positive integer n into prime factors. The Fermat factoring method is a core of some modern and important factorization methods, such as the quadratic sieve and number field sieve methods. It factors a composite integer n=pq in polynomial time if the difference between the prime factors is equal to ∆=p-q≤n^(0.25) , where p>q. The execution time of the Fermat factoring method increases rapidly as ∆ increases. One of the improvements to the Fermat factoring method is based on studying the possible values of (n mod 20). In this paper, we introduce an efficient algorithm to factorize a large integer based on the possible values of (n mod 20) and a precomputation strategy. The experimental results, on different sizes of n and ∆, demonstrate that our proposed algorithm is faster than the previous improvements of the Fermat factoring method by at least 48%.

RSA, a public key cryptosystem, was proposed to protect the information in the insecure channel. The security of RSA relies on the difficulty of factoring the modulus which is the product of two large primes. We proposed Modified Fermat Factorization Version 2 (MFFV2) modified from Modified Fermat Factorization (MFF) to break RSA. The key of MFFV2 is to decrease the number of times of MFF for computing an integers square root. However, MFFV2 is still time-consuming to some extent due to computation time of the subtraction of two integers for all iterations. Thus, this paper aims to propose Modified Fermat Factorization Version 3 (MFFV3) to increase the computation speed when compared with MFFV2. For MFFV3, we can ignore computing the difference between two integers when we know that the subtractions result is certainly not a perfect square. Hence, we develop the Differences Least Significant Digit Table (DLSDT), the information table used to analyze the least significant digit of the subtractions result. Experimental results show that the computation time of MFFV3 for factoring the modulus is substantially reduced in comparison to MFF and MFFV2 respectively.

Abstract. The security of the RSA cryptosystem is directly proportional to the size of its modulus, n. The modulus n is a multiplication of two very large prime numbers, notated as p and q. Since modulus n is public, a cryptanalyst can use factorization algorithms such as Euler’s and Pollard’s algorithms to derive the private keys, p and q. Brute force is an algorithm that searches a solution to a problem by generating all the possible candidate solutions and testing those candidates one by one in order to get the most relevant solution. Random search is a numerical optimization algorithm that starts its search by generating one candidate solution randomly and iteratively compares it with other random candidate solution in order to get the most suitable solution. This work aims to compare the performance of brute force algorithm and random search in factoring the RSA modulus into its two prime factors by experimental means in Python programming language. The primality test is done by Fermat algorithm and the sieve of Eratosthenes.

This study aims to analyze the performance and security of the RSA algorithm in combination with the key generation method of enhanced and secured RSA key generation scheme (ESRKGS). ESRKGS is an improvement of the RSA improvisation by adding four prime numbers in the property embedded in key generation. This method was applied to instant messaging using TCP sockets. The ESRKGS+RSA algorithm was designed using standard RSA development by modified the private and public key pairs. Thus, the modification was expected to make it more challenging to factorize a large number n into prime numbers. The ESRKGS+RSA method required 10.437 ms faster than the improvised RSA that uses the same four prime numbers in conducting key generation processes at 1024-bit prime number. It also applies to the encryption and decryption process. In the security testing using Fermat Factorization on a 32-bit key, no prime number factor was found. The test was processed for 15 hours until the test computer resource runs out.

In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors.

From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.

From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.

Most cryptographic systems are based on an underlying difficult problem. The RSA cryptosystem and many other cryptosystems rely on the fact that factoring a large composite number into two prime numbers is a hard problem. The are many algorithms for factoring integers. This chapter presents some of the basic algorithms for integer factorization like the Trial Division, Fermat's Algorithm. Pollard's Rho Method, Pollard's p-1 method and the Elliptic Curve Method. The Number Field Sieve algorithm along with Special Number field Sieve and the General Number Field Sieve are also used in factoring large numbers. Other factoring algorithms discussed in this chapter are the Continued Fractions Algorithms and the Quadratic Sieve Algorithm.