An Introduction to the RSA Encryption Method
|
|
|
- Eunice Lynch
- 7 years ago
- Views:
Transcription
1 April 17, 2012
2 Outline 1 History
3 History RSA stands for Rivest, Shamir, and Adelman, the last names of the designers It was first published in 1978 as one of the first public-key crytographic systems A public-key system means the algorithm for encrypting a message is publicly known but the algorithm to decrypt the message is only privately known (by the person who set up the system)
4 Review Definition a b (mod c) a = b + kc for some integer k.
5 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1)8
6 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1) (mod 8)
7 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1) (mod 8) 7 13 ( 1) 13
8 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1) (mod 8) 7 13 ( 1) 13 1
9 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1) (mod 8) 7 13 ( 1) (mod 8)
10 Review Definition a b (mod c) a = b + kc for some integer k (mod 4) because 21 = 1 + (5) (mod 11) because 25 = 3 + (2) (mod 8) because 1 = 7 + ( 1) (mod 8) 7 13 ( 1) (mod 8)
11 Necessary Theorems for RSA - φ(n) Definition If n is a positive integer, then Euler s phi function, φ(n), returns the number of integers k in the range 1 k n for which gcd(n, k) = 1.
12 Necessary Theorems for RSA - φ(n) Definition If n is a positive integer, then Euler s phi function, φ(n), returns the number of integers k in the range 1 k n for which gcd(n, k) = 1. Theorem (Euler s Theorem) If n > 0 and a are relatively prime integers, then a φ(n) 1 (mod n).
13 Necessary Theorems for RSA - φ(n) Definition If n is a positive integer, then Euler s phi function, φ(n), returns the number of integers k in the range 1 k n for which gcd(n, k) = 1. Theorem (Euler s Theorem) If n > 0 and a are relatively prime integers, then a φ(n) 1 (mod n). Corollary If b 1 b 2 (mod φ(n)), then a b 1 a b 2 (mod n).
14 Setting up your own RSA system Pick p and q to be large prime numbers, and let n = pq. Then pick an e such that gcd(e, φ(n)) = 1. e is your encryption exponent.
15 Setting up your own RSA system Pick p and q to be large prime numbers, and let n = pq. Then pick an e such that gcd(e, φ(n)) = 1. e is your encryption exponent. Now, solve for d where ed 1 (mod φ(n)). This can be done with something called the Extended Euclidean Algorithm, or by solving the Linear Diophantine Equation: ed = 1 + kφ(n). d is your decryption exponent.
16 Setting up your own RSA system Pick p and q to be large prime numbers, and let n = pq. Then pick an e such that gcd(e, φ(n)) = 1. e is your encryption exponent. Now, solve for d where ed 1 (mod φ(n)). This can be done with something called the Extended Euclidean Algorithm, or by solving the Linear Diophantine Equation: ed = 1 + kφ(n). d is your decryption exponent. You now have your own RSA system! Public Key - (n, e) Private Key - (d)
17 Using your RSA system When someone wants to send you a message they: 1 Convert their message into a number in a simple agreed upon way such as a=01, b=02, c= Compute the ciphertext c m e (mod n) 3 Send you c
18 Using your RSA system When someone wants to send you a message they: 1 Convert their message into a number in a simple agreed upon way such as a=01, b=02, c= Compute the ciphertext c m e (mod n) 3 Send you c To decrypt their message you: 1 Compute m c d (mod n) 2 Convert their message back into letters and words
19 (Set-Up and Encryption) First, set up your RSA system. Pick p = 5, q =11. Let n = pq = 55. Now pick e = 3. Then ed 1 (mod φ(n)) = d = 27. Since (mod 40). Your RSA system is now set up. Make n and e public.
20 (Set-Up and Encryption) First, set up your RSA system. Pick p = 5, q =11. Let n = pq = 55. Now pick e = 3. Then ed 1 (mod φ(n)) = d = 27. Since (mod 40). Your RSA system is now set up. Make n and e public. Let s say that your friend wants to send you the message m=18. They will compute c where c m e (mod n). c m e (mod 55) because 18 3 = 5832 = 2 + (106)55. Your friend will send you the ciphertext c = 2.
21 (Decryption) You just recieved c = 2 from your friend. Use your private key, d = 27, to compute their message m. m c d (mod 55) because 2 27 = = 18 + ( )55. So your friend sent you the message m = 18.
22 RSA Why does m c d message m? (mod n) work to get you back the original. Let p and q be prime, n = pq, ed 1 (mod φ(n)). Then k Z such that ed = 1 + kφ(n). Also let m < n be a message and let c m e (mod n). Then, c d (m e ) d m ed m 1+kφ(n) m (mod n).
The application of prime numbers to RSA encryption
The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered
Public Key Cryptography and RSA. Review: Number Theory Basics
Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and
Public Key Cryptography: RSA and Lots of Number Theory
Public Key Cryptography: RSA and Lots of Number Theory Public vs. Private-Key Cryptography We have just discussed traditional symmetric cryptography: Uses a single key shared between sender and receiver
RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true?
RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e -1
Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
Some practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod
Mathematics of Internet Security. Keeping Eve The Eavesdropper Away From Your Credit Card Information
The : Keeping Eve The Eavesdropper Away From Your Credit Card Information Department of Mathematics North Dakota State University 16 September 2010 Science Cafe Introduction Disclaimer: is not an internet
The Mathematical Cryptography of the RSA Cryptosystem
The Mathematical Cryptography of the RSA Cryptosystem Abderrahmane Nitaj Laboratoire de Mathématiques Nicolas Oresme Université de Caen, France abderrahmanenitaj@unicaenfr http://wwwmathunicaenfr/~nitaj
RSA Encryption. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003
RSA Encryption Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003 1 Public Key Cryptography One of the biggest problems in cryptography is the distribution of keys.
Applied Cryptography Public Key Algorithms
Applied Cryptography Public Key Algorithms Sape J. Mullender Huygens Systems Research Laboratory Universiteit Twente Enschede 1 Public Key Cryptography Independently invented by Whitfield Diffie & Martin
Discrete Mathematics, Chapter 4: Number Theory and Cryptography
Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility
Primes, Factoring, and RSA A Return to Cryptography. Table of contents
Primes, Factoring, and RSA A Return to Cryptography Foundations of Cryptography Computer Science Department Wellesley College Table of contents Introduction Generating Primes RSA Assumption A classic hard
Number Theory and Cryptography using PARI/GP
Number Theory and Cryptography using Minh Van Nguyen nguyenminh2@gmail.com 25 November 2008 This article uses to study elementary number theory and the RSA public key cryptosystem. Various commands will
MATH 13150: Freshman Seminar Unit 10
MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers
Advanced Cryptography
Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.
Primality Testing and Factorization Methods
Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,
Number Theory and the RSA Public Key Cryptosystem
Number Theory and the RSA Public Key Cryptosystem Minh Van Nguyen nguyenminh2@gmail.com 05 November 2008 This tutorial uses to study elementary number theory and the RSA public key cryptosystem. A number
Software Tool for Implementing RSA Algorithm
Software Tool for Implementing RSA Algorithm Adriana Borodzhieva, Plamen Manoilov Rousse University Angel Kanchev, Rousse, Bulgaria Abstract: RSA is one of the most-common used algorithms for public-key
Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms
Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport
Lecture 3: One-Way Encryption, RSA Example
ICS 180: Introduction to Cryptography April 13, 2004 Lecturer: Stanislaw Jarecki Lecture 3: One-Way Encryption, RSA Example 1 LECTURE SUMMARY We look at a different security property one might require
Lecture 6 - Cryptography
Lecture 6 - Cryptography CSE497b - Spring 2007 Introduction Computer and Network Security Professor Jaeger www.cse.psu.edu/~tjaeger/cse497b-s07 Question 2 Setup: Assume you and I don t know anything about
Outline. CSc 466/566. Computer Security. 8 : Cryptography Digital Signatures. Digital Signatures. Digital Signatures... Christian Collberg
Outline CSc 466/566 Computer Security 8 : Cryptography Digital Signatures Version: 2012/02/27 16:07:05 Department of Computer Science University of Arizona collberg@gmail.com Copyright c 2012 Christian
Computing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
RSA Attacks. By Abdulaziz Alrasheed and Fatima
RSA Attacks By Abdulaziz Alrasheed and Fatima 1 Introduction Invented by Ron Rivest, Adi Shamir, and Len Adleman [1], the RSA cryptosystem was first revealed in the August 1977 issue of Scientific American.
GREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
Cryptography and Network Security Chapter 9
Cryptography and Network Security Chapter 9 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 9 Public Key Cryptography and RSA Every Egyptian received two names,
Elements of Applied Cryptography Public key encryption
Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let
CIS 5371 Cryptography. 8. Encryption --
CIS 5371 Cryptography p y 8. Encryption -- Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: All-or-nothing secrecy.
Introduction to Hill cipher
Introduction to Hill cipher We have explored three simple substitution ciphers that generated ciphertext C from plaintext p by means of an arithmetic operation modulo 26. Caesar cipher: The Caesar cipher
Index Calculation Attacks on RSA Signature and Encryption
Index Calculation Attacks on RSA Signature and Encryption Jean-Sébastien Coron 1, Yvo Desmedt 2, David Naccache 1, Andrew Odlyzko 3, and Julien P. Stern 4 1 Gemplus Card International {jean-sebastien.coron,david.naccache}@gemplus.com
1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works.
MATH 13150: Freshman Seminar Unit 18 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. 1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the internet
CHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
The Mathematics of the RSA Public-Key Cryptosystem
The Mathematics of the RSA Public-Key Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through
Separable & Secure Data Hiding & Image Encryption Using Hybrid Cryptography
502 Separable & Secure Data Hiding & Image Encryption Using Hybrid Cryptography 1 Vinay Wadekar, 2 Ajinkya Jadhavrao, 3 Sharad Ghule, 4 Akshay Kapse 1,2,3,4 Computer Engineering, University Of Pune, Pune,
Timing Attacks on software implementation of RSA
Timing Attacks on software implementation of RSA Project Report Harshman Singh 903-40-5260 singhha@cs.orst.edu June 07, 2004 Abstract Timing attacks enable an attacker to extract secret information from
Public Key (asymmetric) Cryptography
Public-Key Cryptography UNIVERSITA DEGLI STUDI DI PARMA Dipartimento di Ingegneria dell Informazione Public Key (asymmetric) Cryptography Luca Veltri (mail.to: luca.veltri@unipr.it) Course of Network Security,
RSA and Primality Testing
and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2
Digital Signatures. Murat Kantarcioglu. Based on Prof. Li s Slides. Digital Signatures: The Problem
Digital Signatures Murat Kantarcioglu Based on Prof. Li s Slides Digital Signatures: The Problem Consider the real-life example where a person pays by credit card and signs a bill; the seller verifies
Notes on Network Security Prof. Hemant K. Soni
Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications
Overview of Public-Key Cryptography
CS 361S Overview of Public-Key Cryptography Vitaly Shmatikov slide 1 Reading Assignment Kaufman 6.1-6 slide 2 Public-Key Cryptography public key public key? private key Alice Bob Given: Everybody knows
CIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives
CIS 6930 Emerging Topics in Network Security Topic 2. Network Security Primitives 1 Outline Absolute basics Encryption/Decryption; Digital signatures; D-H key exchange; Hash functions; Application of hash
LUC: A New Public Key System
LUC: A New Public Key System Peter J. Smith a and Michael J. J. Lennon b a LUC Partners, Auckland UniServices Ltd, The University of Auckland, Private Bag 92019, Auckland, New Zealand. b Department of
MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS
MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:00-3:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:
Every Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
Module: Applied Cryptography. Professor Patrick McDaniel Fall 2010. CSE543 - Introduction to Computer and Network Security
CSE543 - Introduction to Computer and Network Security Module: Applied Cryptography Professor Patrick McDaniel Fall 2010 Page 1 Key Distribution/Agreement Key Distribution is the process where we assign
CSCE 465 Computer & Network Security
CSCE 465 Computer & Network Security Instructor: Dr. Guofei Gu http://courses.cse.tamu.edu/guofei/csce465/ Public Key Cryptogrophy 1 Roadmap Introduction RSA Diffie-Hellman Key Exchange Public key and
Homework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
Cryptography and Network Security
Cryptography and Network Security Fifth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared
Outline. CSc 466/566. Computer Security. 7 : Cryptography Public Key. History of Public Key Cryptography. Public-key Algorithms. Christian Collberg
CSc 466/566 Computer 7 : Cryptography Public Key Version: 2012/02/15 16:15:24 Department of Computer Science University of Arizona collberg@gmail.com Copyright c 2012 Christian Collberg Christian Collberg
MATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction
MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key
Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm
MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following
Secure Network Communication Part II II Public Key Cryptography. Public Key Cryptography
Kommunikationssysteme (KSy) - Block 8 Secure Network Communication Part II II Public Key Cryptography Dr. Andreas Steffen 2000-2001 A. Steffen, 28.03.2001, KSy_RSA.ppt 1 Secure Key Distribution Problem
Number Theory: A Mathemythical Approach. Student Resources. Printed Version
Number Theory: A Mathemythical Approach Student Resources Printed Version ii Contents 1 Appendix 1 2 Hints to Problems 3 Chapter 1 Hints......................................... 3 Chapter 2 Hints.........................................
An Efficient Data Security in Cloud Computing Using the RSA Encryption Process Algorithm
An Efficient Data Security in Cloud Computing Using the RSA Encryption Process Algorithm V.Masthanamma 1,G.Lakshmi Preya 2 UG Scholar, Department of Information Technology, Saveetha School of Engineering
EXAM questions for the course TTM4135 - Information Security June 2010. Part 1
EXAM questions for the course TTM4135 - Information Security June 2010 Part 1 This part consists of 6 questions all from one common topic. The number of maximal points for every correctly answered question
Solutions to Problem Set 1
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467b: Cryptography and Computer Security Handout #8 Zheng Ma February 21, 2005 Solutions to Problem Set 1 Problem 1: Cracking the Hill cipher Suppose
= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
Lecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
RSA Keys with Common Factors
RSA Keys with Common Factors Joppe W. Bos Cryptography group extreme Computing Group, Microsoft Research 1 / 19 Outline 2 / 19 Public-Key Cryptography 3 / 19 Cryptanalysis of Public-Key Cryptography Popular
Network Security. Computer Networking Lecture 08. March 19, 2012. HKU SPACE Community College. HKU SPACE CC CN Lecture 08 1/23
Network Security Computer Networking Lecture 08 HKU SPACE Community College March 19, 2012 HKU SPACE CC CN Lecture 08 1/23 Outline Introduction Cryptography Algorithms Secret Key Algorithm Message Digest
Factoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
Data Grid Privacy and Secure Storage Service in Cloud Computing
Data Grid Privacy and Secure Storage Service in Cloud Computing L.Revathi 1, S.Karthikeyan 2 1 Research Scholar, Department of Computer Applications, Dr. M.G.R. Educational and Research Institute University,
Computer Security: Principles and Practice
Computer Security: Principles and Practice Chapter 20 Public-Key Cryptography and Message Authentication First Edition by William Stallings and Lawrie Brown Lecture slides by Lawrie Brown Public-Key Cryptography
CS 758: Cryptography / Network Security
CS 758: Cryptography / Network Security offered in the Fall Semester, 2003, by Doug Stinson my office: DC 3122 my email address: dstinson@uwaterloo.ca my web page: http://cacr.math.uwaterloo.ca/~dstinson/index.html
Math 453: Elementary Number Theory Definitions and Theorems
Math 453: Elementary Number Theory Definitions and Theorems (Class Notes, Spring 2011 A.J. Hildebrand) Version 5-4-2011 Contents About these notes 3 1 Divisibility and Factorization 4 1.1 Divisibility.......................................
Chapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
Public-key cryptography RSA
Public-key cryptography RSA NGUYEN Tuong Lan LIU Yi Master Informatique University Lyon 1 Objective: Our goal in the study is to understand the algorithm RSA, some existence attacks and implement in Java.
Cloud and Mobile Computing
Cloud and Mobile Computing Protect Privacy in Offloading Yung-Hsiang Lu Electrical and Computer Engineering Purdue University Technological Trends Mobile systems become primary computing platforms for
Introduction to Cryptography CS 355
Introduction to Cryptography CS 355 Lecture 30 Digital Signatures CS 355 Fall 2005 / Lecture 30 1 Announcements Wednesday s lecture cancelled Friday will be guest lecture by Prof. Cristina Nita- Rotaru
1 Digital Signatures. 1.1 The RSA Function: The eth Power Map on Z n. Crypto: Primitives and Protocols Lecture 6.
1 Digital Signatures A digital signature is a fundamental cryptographic primitive, technologically equivalent to a handwritten signature. In many applications, digital signatures are used as building blocks
Embedding more security in digital signature system by using combination of public key cryptography and secret sharing scheme
International Journal of Computer Sciences and Engineering Open Access Research Paper Volume-4, Issue-3 E-ISSN: 2347-2693 Embedding more security in digital signature system by using combination of public
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational
How To Use Amazon Cloud 2 On Linux And Windows 2 On A Pc Or Mac Or Ipad (For Pc) On A Microsoft Mac Or Macbook Or Ipa (For Mac) On An Ubuntu Or Ipro (For Windows
Amazon AWS Tutorial II: Windows and Linux on EC2 Shuang Luan Department of Computer Science Department of Radiology University of New Mexico Amazon EC2 Getting Started Official Website: http://aws.amazon.com/ec2
Continued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
159.334 Computer Networks. Network Security 1. Professor Richard Harris School of Engineering and Advanced Technology
Network Security 1 Professor Richard Harris School of Engineering and Advanced Technology Presentation Outline Overview of Identification and Authentication The importance of identification and Authentication
CIS 433/533 - Computer and Network Security Public Key Crypto/ Cryptographic Protocols
CIS 433/533 - Computer and Network Security Public Key Crypto/ Cryptographic Protocols Professor Kevin Butler Winter 2010 Computer and Information Science Key Distribution/Agreement Key Distribution is
Math 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
DIVISIBILITY AND GREATEST COMMON DIVISORS
DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will
A Comparison Of Integer Factoring Algorithms. Keyur Anilkumar Kanabar
A Comparison Of Integer Factoring Algorithms Keyur Anilkumar Kanabar Batchelor of Science in Computer Science with Honours The University of Bath May 2007 This dissertation may be made available for consultation
Digital Signatures. Meka N.L.Sneha. Indiana State University. nmeka@sycamores.indstate.edu. October 2015
Digital Signatures Meka N.L.Sneha Indiana State University nmeka@sycamores.indstate.edu October 2015 1 Introduction Digital Signatures are the most trusted way to get documents signed online. A digital
Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay
Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security - MA61027 Modern/Public-key cryptography started in 1976 with the publication of the following paper. W. Diffie
Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
Shor s algorithm and secret sharing
Shor s algorithm and secret sharing Libor Nentvich: QC 23 April 2007: Shor s algorithm and secret sharing 1/41 Goals: 1 To explain why the factoring is important. 2 To describe the oldest and most successful
An Overview of Integer Factoring Algorithms. The Problem
An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm
Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
I. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
Network Security. Gaurav Naik Gus Anderson. College of Engineering. Drexel University, Philadelphia, PA. Drexel University. College of Engineering
Network Security Gaurav Naik Gus Anderson, Philadelphia, PA Lectures on Network Security Feb 12 (Today!): Public Key Crypto, Hash Functions, Digital Signatures, and the Public Key Infrastructure Feb 14:
Keywords- Cloud Computing, Android Platform, Encryption, Decryption, NTRU, RSA, DES, throughput.
Volume 3, Issue 11, November 2013 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Implementation
Java Coding Ground With Security Editor
IJCST Vo l 6, Is s u e 1, Ja n - Ma r c h 2015 ISSN : 0976-8491 (Online) ISSN : 2229-4333 (Print) Java Coding Ground With Security Editor 1 ISurya Prabha, 2 GPrabhakar Reddy, 3 V Ramakrishna, 4 YDeepika
Factoring integers, Producing primes and the RSA cryptosystem Harish-Chandra Research Institute
RSA cryptosystem HRI, Allahabad, February, 2005 0 Factoring integers, Producing primes and the RSA cryptosystem Harish-Chandra Research Institute Allahabad (UP), INDIA February, 2005 RSA cryptosystem HRI,
CSE 30331 Final Project: crypchat Encrypted Chat Program Christopher Durr, Matthew Mooney, Paul Zakas
CSE 30331 Final Project: crypchat Encrypted Chat Program Christopher Durr, Matthew Mooney, Paul Zakas Abstract: Secure communication over the internet is critical in today s world. Our project is an encrypted
Basic Algorithms In Computer Algebra
Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,
Mathematical Model Based Total Security System with Qualitative and Quantitative Data of Human
Int Jr of Mathematics Sciences & Applications Vol3, No1, January-June 2013 Copyright Mind Reader Publications ISSN No: 2230-9888 wwwjournalshubcom Mathematical Model Based Total Security System with Qualitative
Today s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
Network Security. Abusayeed Saifullah. CS 5600 Computer Networks. These slides are adapted from Kurose and Ross 8-1
Network Security Abusayeed Saifullah CS 5600 Computer Networks These slides are adapted from Kurose and Ross 8-1 Public Key Cryptography symmetric key crypto v requires sender, receiver know shared secret
Lectures on Number Theory. Lars-Åke Lindahl
Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder
Evaluation of Digital Signature Process
Evaluation of Digital Signature Process Emil SIMION, Ph. D. email: esimion@fmi.unibuc.ro Agenda Evaluation of digital signatures schemes: evaluation criteria; security evaluation; security of hash functions;
The Euclidean Algorithm
The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have
8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.