Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem

This paper explores 3 problems, the Sieve of Eratosthenes, Discrete Logarithm Problem (DLP) and Integer Factorization. Prime numbers are essential to all cryptographic systems today. The Sieve of Eratosthenes is an algorithm used to find prime numbers up to a given limit. Using the basic version of...

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Main Author: Loo, Jia Ein
Other Authors: Tay Kian Boon
Format: Final Year Project
Language:English
Published: Nanyang Technological University 2023
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Online Access:https://hdl.handle.net/10356/166054
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spelling sg-ntu-dr.10356-1660542023-04-21T15:39:12Z Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem Loo, Jia Ein Tay Kian Boon School of Computer Science and Engineering kianboon.tay@ntu.edu.sg Engineering::Computer science and engineering This paper explores 3 problems, the Sieve of Eratosthenes, Discrete Logarithm Problem (DLP) and Integer Factorization. Prime numbers are essential to all cryptographic systems today. The Sieve of Eratosthenes is an algorithm used to find prime numbers up to a given limit. Using the basic version of the algorithm will work but take a long time just to produce prime numbers below the limit of one billion. Here we explore some improvements on the algorithm and implemented them. The DLP is defined as: given a group Fp, a generator g of the group and an element k of p, find the discrete logarithm to the base g of k in the group p. k = gα mod p From the equation above, we want to find a, given k, g and p. The DLP is known to be a computationally hard problem and is hence is used to construct a secure cryptosystem. However, the DLP is not always hard – it depends on the groups. In this paper, we explore algorithms to exploit weak groups and solve their discrete logarithm efficiently. Integer Factorization is necessary for many other algorithms, such as the Pohlig-Hellman algorithm (which will be explained later in this paper), where factors of a number are required for the algorithm. Integer Factorization is a known NP problem, and there is no algorithm yet that can solve it in polynomial time. The goal of integer factorization is to decompose a number into smaller numbers, also known as prime factors. In this paper, we evaluate different integer factorization algorithms, understanding the constraints of each algorithm. Bachelor of Engineering (Computer Science) 2023-04-19T02:40:27Z 2023-04-19T02:40:27Z 2023 Final Year Project (FYP) Loo, J. E. (2023). Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem. Final Year Project (FYP), Nanyang Technological University, Singapore. https://hdl.handle.net/10356/166054 https://hdl.handle.net/10356/166054 en SCSE22-0329 application/pdf Nanyang Technological University
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Engineering::Computer science and engineering
spellingShingle Engineering::Computer science and engineering
Loo, Jia Ein
Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
description This paper explores 3 problems, the Sieve of Eratosthenes, Discrete Logarithm Problem (DLP) and Integer Factorization. Prime numbers are essential to all cryptographic systems today. The Sieve of Eratosthenes is an algorithm used to find prime numbers up to a given limit. Using the basic version of the algorithm will work but take a long time just to produce prime numbers below the limit of one billion. Here we explore some improvements on the algorithm and implemented them. The DLP is defined as: given a group Fp, a generator g of the group and an element k of p, find the discrete logarithm to the base g of k in the group p. k = gα mod p From the equation above, we want to find a, given k, g and p. The DLP is known to be a computationally hard problem and is hence is used to construct a secure cryptosystem. However, the DLP is not always hard – it depends on the groups. In this paper, we explore algorithms to exploit weak groups and solve their discrete logarithm efficiently. Integer Factorization is necessary for many other algorithms, such as the Pohlig-Hellman algorithm (which will be explained later in this paper), where factors of a number are required for the algorithm. Integer Factorization is a known NP problem, and there is no algorithm yet that can solve it in polynomial time. The goal of integer factorization is to decompose a number into smaller numbers, also known as prime factors. In this paper, we evaluate different integer factorization algorithms, understanding the constraints of each algorithm.
author2 Tay Kian Boon
author_facet Tay Kian Boon
Loo, Jia Ein
format Final Year Project
author Loo, Jia Ein
author_sort Loo, Jia Ein
title Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
title_short Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
title_full Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
title_fullStr Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
title_full_unstemmed Analysis on Sieve of Eratosthenes algorithm and algorithms for solving the discrete logarithm problem
title_sort analysis on sieve of eratosthenes algorithm and algorithms for solving the discrete logarithm problem
publisher Nanyang Technological University
publishDate 2023
url https://hdl.handle.net/10356/166054
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