Prime Numbers Comparison using Sieve of Eratosthenes and Sieve of Sundaram Algorithm

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Prime Numbers Comparison using Sieve of Eratosthenes and Sieve of Sundaram Algorithm To cite this article: D Abdullah et al 2018 J. Phys.: Conf. Ser View the article online for updates and enhancements. This content was downloaded from IP address on 13/07/2018 at 09:13

2 Prime Numbers Comparison using Sieve of Eratosthenes and Sieve of Sundaram Algorithm D Abdullah 1, R Rahim 2, D Apdilah 3, S Efendi 4, T Tulus 5 and S Suwilo 5 1 Department of Informatics, Universitas Malikussaleh, Aceh, Indonesia 2 School of Computer and Communication Engineering, Universiti Malaysia Perlis, Malaysia 3 Department of Informatics Engineering, AMIK Intel Com Global Inda, Indonesia 4 Faculty of Computer Science and Information Technology, Universitas Sumatera Utara, Indonesia 5 Department of Mathematics, Universitas Sumatera Utara, Indonesia dahlan@unimal.ac.id Abstract. Prime numbers are numbers that have their appeal to researchers due to the complexity of these numbers, many algorithms that can be used to generate prime numbers ranging from simple to complex computations, Sieve of Eratosthenes and Sieve of Sundaram are two algorithm that can be used to generate Prime numbers of randomly generated or sequential numbered random numbers, testing in this study to find out which algorithm is better used for large primes in terms of time complexity, the test also assisted with applications designed using Java language with code optimization and Maximum memory usage so that the testing process can be simultaneously and the results obtained can be objective 1. Introduction Prime numbers are the unique numbers because this number cannot be divided by other positive integers, except 1 and itself [1]. In fact, it is also mentioned that prime numbers are universal languages that can understand by all beings with a high level of intelligence [1] [2] [3]. Prime numbers have several benefits, one of which is concerning disguise messages or often called cryptograph [2]. Determine the prime numbers can be determined manually of course by counting one by one and this takes a very long time or generated by using certain algorithms, such as Sieve of Eratosthenes and Sieve of Sundaram algorithms. Algorithm Sieve of Eratosthenes and Sieve of Sundaram is an algorithm used to eliminate numbers that do not prime from a collection of numbers that exist [4] [5], this process is quite good as an algorithm that could be applied to various processes of cryptographic algorithms [5], the research will be tested to compare the two algorithms In determining the prime of a large number and knowing which algorithm is safe for use on small and large numbers. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 2. Theory The basis of number theory is the number itself, and the core block of numbers is a prime number [6] [7], so prime numbers are often said to be central to number theory [6]. Prime numbers are numbers that have only two factors, namely the number one and the number itself. Numbers that have more than two factors are referred to as composite numbers. Numbers one has only one factor and is considered not prime and composite numbers [8]. When a composite number is written as the product of all the prime factors, it is said to be the main factorization of that number. Factorization of a number into its basic constituent element also called prime decomposition. Typically, factors are listed from the smallest to the largest [9] [10]. Sieve of Eratosthenes is a way to find all prime numbers between 1 and a number n. The algorithm discovered by Eratosthenes, an ancient Greek scientist. Sieve of Eratosthenes is the simplest way to find prime numbers [4]. Here is the step of the Sieve of Eratosthenes algorithm process [4] [5] [9]: 1. Determine the number to be tested it is primarily, from 2 to the largest desired number, call it this list as A list. 2. Mark the number two from list A and move it into another list that holds the primes to be searched. 3. Delete all numbers that are multiples of 2 in list A. 4. The next number found in list A is a prime number. Mark and move to list B. 5. Delete all numbers that are multiples of primes in list An earlier, numbers that have been removed from the list do not need to be in the process 6. Repeat process 4 and five until no numbers are left in list A. Sieve of Sundaram algorithm is an algorithm generating prime numbers that begin with the value after and conducted by n2 + 1, the function of determining prime numbers using the following formula: i. j N, 1 i j The above formula is used to process all numbers that are not primed in a set of values, to determine prime numbers using the following equation: i. + j + 2ij n 3. Result and Discussion The method of testing prime number generator using the Sieve of Eratosthenes algorithm based on the steps previously described, here is the test to determine the primes of the numbers First step show numbers 1 to 100 Table 1. Number 1 to 100 2

4 2. The second step, Removing the number 1 and multiples of 2 except the number 2 itself, the result show below Table 2. Remove number multiple of 2 3. Third step, removing multiples of 3 except the number 3 itself Table 3. Remove number multiple of 3 4. Fourth step, removing multiples of 5 except the number 5 itself Table 4. Remove number multiple of 5 3

5 5. Fifth step, removing multiples of 7 except the number 7 itself Table 5. Remove number multiple of 7 The end result of the Sieve of Eratosthenes algorithm process is as follows: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97, for experiments using sieve of sundaram perform in a way that is similar to the sieve of Eratosthenes. The above test is performed to show the process of determining the primes of each algorithm, to find out which algorithm is faster in determining the prime number from small numbers to large is by testing using application that develops by using Java, see the figure below Figure 1. Prime Number Generator Java The above application is made to perform simultaneous testing to generate random numbers and determining the prime numbers from a random number produced by the algorithm Sieve of Eratosthenes and Sieve of Sundaram, for example, as follows: 4

6 Figure 2. Prime Number Result with Application The above test appears to display the primes of the numbers with the time displayed simultaneously, for a detailed test can be seen from the table below: Table 6. List Numbers No Number Sieve of Eratosthenes (Second) Sieve of Sundaram (Second) :0: :0: :0: :0: :0: :0: :0: :0: :0: :0: :0: :0: :2: :3: :5: :8: :11: :16: :23: :31: From the table above obtained graph of testing for the number of primes produced by using Sieve of Eratosthenes and Sieve of Sundaram algorithm as follows: Time Comparison 36:00,0 28:48,0 21:36,0 14:24,0 07:12,0 00:00, Sieve of Eratosthenes Sieve of Sundaram Figure 3. Time Graph Comparison 5

7 The graph (Figure 3) explains the greater the number of primes produced then the sieve of Eratosthenes is better than the Sieve of Sundaram algorithm. 4. Conclusion The experiment to compare prime generator with Sieve of Eratosthenes and Sieve of Sundaram algorithms has found that for small prime numbers Sieve of Sundaram is better than Sieve of Eratosthenes; but to display a large prime number Sieve of Eratosthenes is better, experiment do with the help of applications built using Java with the optimization of code and use buffer memory to optimize process prime generator. References [1] Kim YS, Shrestha S R and Jang JW 2011 Refined Algorithm for Prime Number Generation in Embedded Security Systems IEEE Asia-Pacific Services Computing Conference (APSCC) (South Korea: Jeju Island). [2] Qing Z and Zhihua H 2011 The large prime numbers generation of RSA algorithm based on genetic algorithm Int. Conf. on Intelligence Sc. and Information Eng. (China: Wuhan). [3] Igumnov V S 2004 Generation of the Large Random Prime Numbers Proc. 5th Annual Int. Siberian Workshop (Russia: Erlagol, Altai). [4] O'Neill M E 2009 The Genuine Sieve of Eratosthenes J. of Functional Programming 19 pp [5] Lambert M 2004 Calculating the Sieve of Eratosthenes J. of Functional Programming 14 pp [6] Sarukkai R R 1995 Prime Numbers and Output Codes IEEE Int. Conf. on Neural Networks (Australia: Perth, WA). [7] Dey S, Abraham A and Sanyal S 2007 An LSB Data Hiding Technique Using Prime Numbers Third Int. Symposium on Information Assurance and Security (UK: Manchester). [8] Liu D 2011 Iteration Distribution of Prime Numbers J. of Mathematics Research 3 pp [9] Jullien G A 1980 Implementation of Multiplication, Modulo a Prime Number, with Applications to Number Theoretic Transforms IEEE Transactions on Computers 29 pp [10] Fu Y and Xu S 2010 Generalized Fast Coding Steganographic Algorithm Based on Prime Number IEEE Int. Conf. on Information Theory and Information Security (ICITIS) (China: Beijing). 6