New Prime Factorization Algorithm and Its Parallel Computing Strategy (original) (raw)
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International Journal of Electrical and Computer Engineering (IJECE), 2020
Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat's Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to find prime factors. Although the computation loops are quite large, there is no square root computing included into the computation. In this paper, the new efficient factorization algorithm is introduced. Euler's theorem is chosen to apply with FFA to find the addition result between two prime factors. The advantage of the proposed method is that almost of square root operations are left out from the computation while loops are not increased, they are equal to the first method. Therefore, if the proposed method is compared with the FFA-1, it implies that the computation time is decreased, because there is no the square root operation and the loops are same. On the other hand, the loops of the proposed method are less than the second method. Therefore, time is also reduced. Furthermore, the proposed method can be also selected to apply with many methods which are modified from FFA to decrease more cost. 1. INTRODUCTION Nowadays, the significant information is always exchanged via the communication channel connected to computer system such as internet. Generally, this channel is the insecure channel. That means attackers can access data easily by using various techniques. With this reason, security for the information becomes very important. At present, many security algorithms were introduced to protect the secret data sending over insecure channel. Cryptography is one of techniques to defend data from attackers by using encryption and decryption processes. In addition, there are two types about cryptography. The first is symmetric key cryptography using the same key which is called secret key for encryption and decryption processes. The second is asymmetric key cryptography (or public key cryptography) [1] using a pair of keys for encryption and decryption. In addition, one key which is always distributed to keep in the key center is called public key. On the other hand, the other key which is always kept secretly by owner's key is called private key. RSA [2] is the most well-known public key cryptography used for both of digital signature and data encryption. This algorithm is one-way function. That means it is very easy to compute the production of
Reverse Factorization and Comparison of Factorization Algorithms in attack to RSA
Factorization algorithms have a major role in the computer security and cryptography. Most of the widely used cryptographic algorithms, like RSA, are built on the mathematical difficulty of factorization for big prime numbers. This research, proposes a new approach to the factorization by using two new enhancements. The new approach is also compared with six different factorization algorithms and evaluated the performance on a big data environment. The algorithms covered are elliptic curve method, quadratic sieve, Fermat's method, trial division and Pollard rho methods. Success rates are compared over a million of integer numbers with different difficulties. We have implemented our own algorithm for random number generation, which is also explained in the paper. We also empirically show that the new approach has an advantage on the factorization attack to RSA.
A New Deterministic RSA-Factoring Algorithm
The security of many cryptography techniques depends upon the intractability of the integer-factoring problem. However, in the recent years there has been a great deal of progress in the art of factoring, relaying mostly on non-deterministic methods. This research proposes a new deterministic factoring algorithm, that factors RSA n = p * q, the algorithm running time relays on the number of digits of n rather than the value of n. The nature of the problem of factoring based on time, complexity and storage required. The proposed algorithm works on solving these problems by using the idea of long multiplication to limit the possible values of p and q. In order to eliminate the storage problem, depth-first search was used with recursive implementation. In addition, the paper discussed the analysis of the proposed algorithm with their running time and complexity. Finally, the paper concludes with future work improvements to the algorithm.
State of the Art Parallel Approaches For Rsa Public Key Based Cryptosystem
International Journal on Computational Science & Applications, 2015
RSA is one of the most popular Public Key Cryptography based algorithm mainly used for digital signatures, encryption/decryption etc. It is based on the mathematical scheme of factorization of very large integers which is a compute-intensive process and takes very long time as well as power to perform. Several scientists are working throughout the world to increase the speedup and to decrease the power consumption of RSA algorithm while keeping the security of the algorithm intact. One popular technique which can be used to enhance the performance of RSA is parallel programming. In this paper we are presenting the survey of various parallel implementations of RSA algorithm involving variety of hardware and software implementations.
A novel accelerated implementation of RSA using parallel processing
Journal of Discrete Mathematical Sciences and Cryptography, 2019
Past research has evidently proved that public key cryptosystems are usually slower than symmetric key cryptosystems due to the reason that they use one additional cryptographic key and different methods for encryption and decryption. RSA is one of the most common asymmetric key cryptography algorithms. Recent research has focused on speeding up RSA using various techniques. With the introduction of distributed computing, parallelization of algorithms enables them to run on multiple cores concurrently at a time. RSA consists of two resource intensive operations namely Modular Exponentiation of up to 1024-bit exponents and repeated calculation of Greatest common divisor. Thus, RSA lays the perfect base for application of Montgomery Reduction algorithm to optimize the Repeated Modular multiplication in exponentiation. In this paper we proposed a parallel scheme for RSA using a new parallel data structure known as Concurrent Indexed List of character blocks. The aim of our research was to improve the speed of RSA encryption and decryption using parallelism and also make it compatible with leading industry cryptography standards. We have simulated four different approaches namely both parallel and sequential with and without Montgomery. We have also integrated our parallel paradigm with renowned C++ Crypto library and achieved a speed-up of upto four times than sequential approach.
Fermat Factorization using a Multi-Core System
International Journal of Advanced Computer Science and Applications
Factoring a composite odd integer into its prime factors is one of the security problems for some public-key cryptosystems such as the Rivest-Shamir-Adleman cryptosystem. Many strategies have been proposed to solve factorization problem in a fast running time. However, the main drawback of the algorithms used in such strategies is the high computational time needed to find prime factors. Therefore, in this study, we focus on one of the factorization algorithms that is used when the two prime factors are of the same size, namely, the Fermat factorization (FF) algorithm. We investigate the performance of the FF method using three parameters: (1) the number of bits for the composite odd integer, (2) size of the difference between the two prime factors, and (3) number of threads used. The results of our experiments in which we used different parameters values indicate that the running time of the parallel FF algorithm is faster than that of the sequential FF algorithm. The maximum speed up achieved by the parallel FF algorithm is 6.7 times that of the sequential FF algorithm using 12 cores. Moreover, the parallel FF algorithm has near-linear scalability.
A Simple Algorithm for Prime Factorization and Primality Testing
Journal of Mathematics
We propose a new simple and faster algorithm to factor numbers based on the nature of the prime numbers contained in such composite numbers. It is well known that every composite number has a unique representation as a product of prime numbers. In this study, we focus mainly on composite numbers that contain a product of prime numbers that are greater than or equal to 5 which are of the form 6 k + 1 or 6 k + 5 . Therefore, we use the condition that every prime or composite P of primes greater than or equal to 5 satisfies P 2 ≡ 1 mod 24 . This algorithm is very fast especially when the difference in the prime components of a composite number (prime gap) is not so large. When the difference between the factors (prime gap) is not so large, it often requires just a single iteration to obtain the factors.
A Comparative Study of RSA based Cryptographic Algorithms
Iasse, 2004
In 1978 a powerful and practical public-key scheme Hadi Otrokwas produced by RSA; there work was applied using 2 large random odd primes p and q, each roughly of the same size. El-Kassar and Awad modi-…ed the RSA public-key encryption scheme from the domain of natural integers, Z, to two principal ideal domains, namely the domain of Gaussian integers, Z[i], and the domain of the rings of polynomials over …nite …elds, F [x], by extending the arithmetic needed for the modi…cations to these domains. In this work we implement the classical and modi…ed RSA cryptosystem to compare and to test their functionality, realiability and security. To test the security of the algorithms we implement an attack algorithm to solve the integer factorization problem. After factorization is found, the RSA problem could be solved by computing the order ©(n), and then …nding the private key using the extended Euclidean algorithm for integers.
2014 International Conference on Contemporary Computing and Informatics (IC3I), 2014
The security of RSA algorithm depends upon the positive integer N, which is the multiple of two precise large prime numbers. Factorization of such great numbers is a problematic process. There are many algorithms has been implemented in the past years. The offered KNJ-Factorization algorithm contributes a deterministic way to factorize RSA N=p*q. The algorithm limits the search by only considering the prime values. Subsequently prime numbers are odd numbers (apart from 2) accordingly it also requires smaller number steps to factorize RSA. In this paper, the anticipated algorithm is very simple besides it is very easy to understand and implement. The main concept of this KNJ-factorization algorithm is, to check only those factors which are odd and prime. The proposed KNJ-Factorization algorithm works very efficiently on those factors; which are adjoining and close to √N. The proposed factorization method can speed up if we can reduce the time for primality testing. It fundamentally decreases the time complexity.
Advances in composite integer factorization
Mathematical Theory and Modeling, 2013
In this research we propose a new method of integer factorization. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or and its prime factors [1]. This fact is the basis behind many of the cryptosystems currently in use.