Number Theory and the RSA Public Key Cryptosystem

Save this PDF as:

Size: px
Start display at page:

Transcription

2 sage: prime[tab] prime_divisors prime_pi prime_range primes prime_factors prime_powers prime_to_m_part primes_first_n If we know the exact name of a command, we can use the help function to obtain further information on that command. For example, the command help(primes) provides documentation on the built-in function primes, as illustrated in the following help session: Help on function primes in module sage.rings.arith: primes(start, stop=none) Returns an iterator over all primes between start and stop-1, inclusive. This is much slower than \code{prime_range}, but potentially uses less memory. This command is like the xrange command, except it only iterates over primes. In some cases it is better to use primes than prime_range, because primes does not build a list of all primes in the range in memory all at once. However it is potentially much slower since it simply calls the \code{next_prime} function repeatedly, and \code{next_prime} is slow, partly because it proves correctness. EXAMPLES: sage: for p in primes(5,10):... print p sage: list(primes(11)) [2, 3, 5, 7] sage: list(primes( , )) [ , , , , ] To quit the above help session, we simply press q on our keyboard. For further assistance on specific problems, we can also search the archive of the sage-support mailing list at or send an to 2 Elementary number theory We first review basic concepts from elementary number theory, including the notion of primes, greatest common divisors, congruences and Euler s phi function. The number theoretic concepts and commands introduced will be referred to in later sections when we present the RSA algorithm. 2

3 Prime numbers Public key cryptography uses many fundamental concepts from number theory, such as prime numbers and greatest common divisors. A positive integer n > 1 is said to be prime if its factors are exclusively 1 and itself. In, we can obtain the first 20 prime numbers using the command primes first n: sage: primes_first_n(20) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] Greatest common divisors Let Z be the set of integers. If a, b Z, then the greatest common divisor (GCD) of a and b is the largest positive factor of both a and b. We use gcd(a, b) to denote this largest positive factor. The GCD of any two distinct primes is 1, and the GCD of 18 and 27 is 9. We can use the command gcd to compute the GCD of any two integers: sage: gcd(3, 59) 1 sage: gcd(18, 27) 9 If gcd(a, b) = 1, we say that a is coprime (or relatively prime) to b. In particular, gcd(3, 59) = 1 so 3 is coprime to 59 and vice versa. Congruences When one integer is divided by a non-zero integer, we usually get a remainder. For example, upon dividing 23 by 5, we get a remainder of 3; when 8 is divided by 5, the remainder is again 3. The notion of congruence helps us to describe the situation in which two integers have the same remainder upon division by a non-zero integer. Let a, b, n Z such that n 0. If a and b have the same remainder upon division by n, then we say that a is congruent to b modulo n and denote this relationship by a b (mod n) This definition is equivalent to saying that n divides the difference of a and b, that is, n (a b). Thus 23 8 (mod 5) because when both 23 and 8 are divided by 5, we end up with a remainder of 3. The command mod allows us to compute such a remainder: sage: mod(23, 5) 3 sage: mod(8, 5) 3 3

4 Euler s phi function Consider all the integers from 1 to 20, inclusive. List all those integers that are coprime to 20. In other words, we want to find those integers n, where 1 n 20, such that gcd(n, 20) = 1. The latter task can be easily accomplished with a little bit of programming: sage: for n in xrange(1, 21):...: if gcd(n, 20) == 1:...: print n,...: The above programming statements can be saved to a text file called, say, /home/ mvngu/totient.sage, organizing it as in Listing 1 to enhance readability. We refer to totient.sage as a script, just as one would refer to a file containing Python code as a Python script. We use 4 space indentations, which is a coding convention in as well as Python programming, instead of tabs. 1 for n in xrange(1, 21): 2 if gcd(n, 20) == 1: 3 print n, Listing 1: Finding all integers 1 n 20 such that gcd(n, 20) = 1. The command load can be used to read the file containing our programming statements into and, upon loading the content of the file, have execute those statements: sage: load "/home/mvngu/totient.sage" From the latter list, there are 8 integers in the closed interval [1, 20] that are coprime to 20. Without explicitly generating the list (1) how can we compute the number of integers in [1, 20] that are coprime to 20? This is where Euler s phi function comes in handy. Let n Z be positive. Then Euler s phi function counts the number of integers a, with 1 a n, such that gcd(a, n) = 1. This number is denoted by ϕ(n). Euler s phi function is sometimes referred to as Euler s totient function, hence the name totient.sage for the above script. The command euler phi implements Euler s phi function. To compute ϕ(20) without explicitly generating the list (1), we proceed as follows: sage: euler_phi(20) 8 4

5 3 How to keep a secret? Cryptography is the science (some might say art) of concealing data. Imagine that we are composing a confidential to someone. Having written the , we can send it in one of two ways. The first, and usually convenient, way is to simply press the send button and not care about how our will be delivered. Sending an in this manner is similar to writing our confidential message on a postcard and post it without enclosing our postcard inside an envelope. Anyone who can access our postcard can see our message. On the other hand, before sending our , we can scramble the confidential message and then press the send button. Scrambling our message is similar to enclosing our postcard inside an envelope. While not 100% secure, at least we know that anyone wanting to read our postcard has to open the envelope. In cryptography parlance, our message is called plaintext. The process of scrambling our message is referred to as encryption. After encrypting our message, the scrambled version is called ciphertext. From the ciphertext, we can recover our original unscrambled message via decryption. Figure 1 illustrates the processes of encryption and decryption. A cryptosystem is comprised of a pair of related encryption and decryption processes. encrypt decrypt plaintext ciphertext plaintext Figure 1: A schema for encryption and decryption. Table 1 provides a very simple method of scrambling a message written in English and using only upper case letters, excluding punctuation characters. Formally, let Σ = {A,B,C,...,Z} be the set of capital letters of the English alphabet. Furthermore, let Φ = {65, 66, 67,..., 90} be the American Standard Code for Information Interchange (ASCII) encodings of the upper case English letters. Then Table 1 explicitly describes the mapping f : Σ Φ. (For those familiar with ASCII, f is actually a common process for encoding elements of Σ, rather than a cryptographic scrambling process per se.) To scramble a message written using the alphabet Σ, we simply replace each capital letter of the message with its corresponding ASCII encoding. However, the scrambling process described in Table 1 provides, cryptographically speaking, very little to no security at all and we strongly discourage its use in practice. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Table 1: ASCII encodings of English capital letters. 5

6 4 Keeping a secret with two keys The Rivest, Shamir, Adleman (RSA) cryptosystem is an example of a public key cryptosystem. RSA uses a public key to encrypt messages and decryption is performed using a corresponding private key. We can distribute our public keys, but for security reasons we should keep our private keys to ourselves. The encryption and decryption processes draw upon techniques from elementary number theory. Algorithm 1 [5, p.165] outlines the RSA procedure for encryption and decryption Choose two primes p and q and let n = pq. Let e Z be positive such that gcd ( e, ϕ(n) ) = 1. Compute a value for d Z such that de 1 (mod ϕ(n)). Our public key is the pair (n, e) and our private key is the triple (p, q, d). For any non-zero integer m < n, encrypt m using c m e (mod n). Decrypt c using m c d (mod n). Algorithm 1: The RSA algorithm for encryption and decryption. The next two sections will step through the RSA algorithm, using to generate public and private keys, and perform encryption and decryption based on those keys. 5 Generating public and private keys Positive integers of the form M m = 2 m 1 are called Mersenne numbers. If p is prime and M p = 2 p 1 is also prime, then M p is called a Mersenne prime. For example, 31 is prime and M 31 = is a Mersenne prime, as can be verified using the command is prime(p). This command returns True if its argument p is precisely a prime number; otherwise it returns False. By definition, a prime must be a positive integer, hence is prime(-2) returns False although we know that 2 is prime. Indeed, the number M 61 = is also a Mersenne prime. We can use M 31 and M 61 to work through step 1 in Algorithm 1: sage: p = (2^31) - 1 sage: is_prime(p) True sage: q = (2^61) - 1 sage: is_prime(q) True sage: n = p * q ; n A word of warning is in order here. In the above code example, the choice of p and q as Mersenne primes, and with so many digits far apart from each other, is a very bad choice in terms of cryptographic security. However, we shall use the above chosen numeric values for p and q for the remainder of this tutorial, always bearing in mind that they have been chosen for pedagogy purposes only. Please refer to [2, 4, 5] for in-depth discussions on the security of RSA, or consult other specialized texts. 6

7 For step 2, we need to find a positive integer that is coprime to ϕ(n). The set of integers is implemented within the module sage.rings.integer ring. Various operations on integers can be accessed via the ZZ.* family of functions. For instance, the command ZZ.random element(n) returns a pseudo-random integer uniformly distributed within the closed interval [0, n 1]. Using a simple programming loop, we can compute the required value of e as follows: sage: e = ZZ.random_element(euler_phi(n)) sage: while gcd(e, euler_phi(n))!= 1:...: e = ZZ.random_element(euler_phi(n))...: sage: e sage: e < n True As e is a pseudo-random integer, its numeric value differs after each execution of e = ZZ.random element(euler phi(n)). To calculate a value for d in step 3 of Algorithm 1, we use the extended Euclidean algorithm. By definition of congruence, de 1 (mod ϕ(n)) is equivalent to de k ϕ(n) = 1 where k Z. From steps 1 and 2, we already know the numeric values of e and ϕ(n). The extended Euclidean algorithm allows us to compute d and k. In, this can be accomplished via the command xgcd. Given two integers x and y, xgcd(x, y) returns a tuple (g, s, t) that satisfies the Bézout identity g = gcd(x, y) = sx + ty. Having computed a value for d, we then use the command mod(d*e, euler phi(n)) to check that d*e is indeed congruent to 1 modulo euler phi(n): sage: bezout = xgcd(e, euler_phi(n)) ; bezout (1, , ) sage: d = Integer(mod(bezout[1], euler_phi(n))) ; d sage: mod(d * e, euler_phi(n)) 1 Thus, our RSA public key is (n, e) = ( , ) (2) and our corresponding private key is (p, q, d) = ( , , ) 7

8 H E L L O W O R L D Table 2: ASCII encodings of HELLOWORLD. 6 Encryption and decryption Suppose we want to scramble the message HELLOWORLD using RSA encryption. From Table 1, our message maps to integers of the ASCII encodings as given in Table 2. Concatenating all the integers in Table 2, our message can be represented by the integer m = (3) There are other more cryptographically secure means for representing our message as an integer. The above process is used for demonstration purposes only and we strongly discourage its use in practice. In, we can obtain an integer representation of our message as follows: sage: m = "HELLOWORLD" sage: m = [ord(c) for c in m] ; m [72, 69, 76, 76, 79, 87, 79, 82, 76, 68] sage: m = ZZ(list(reversed(m)), 100) ; m To encrypt our message, we raise m to the power of e and reduce the result modulo n. The command mod(a^b, n) first computes a^b and then reduces the result modulo n. If the exponent b is a large integer, say with more than 20 digits, then performing modular exponentiation in this naive manner takes quite some time. Brute force (or naive) modular exponentiation is inefficient and, when performed using a computer, can quickly consume a huge quantity of the computer s memory or result in overflow messages. For instance, if we perform naive modular exponentiation using the command mod(m^e, n), where m is per (3), and n and e are as in (2), we would get an error message similar to the following: sage: mod(m^e, n) RuntimeError Traceback (most recent call last) /home/mvngu/<ipython console> in <module>() /home/mvngu/usr/bin/sage-3.1.4/local/lib/python2.5/site-packages/sage/rings/integer.so in sage.rings.integer.integer. pow (sage/rings/integer.c:9650)() RuntimeError: exponent must be at most There is a trick to efficiently perform modular exponentiation, called the method of repeated squaring [1, p.879]. Suppose we want to compute a b mod n. First, let d := 1 8

9 and obtain the binary representation of b, say (b 1, b 2,...,b k ) where each b i Z 2. For i := 1,..., k, let d := d 2 mod n and if b i = 1 then let d := da mod n. This algorithm is implemented in Listing 2, which defines a function called modexp to perform modular exponentiation using repeated squaring. 1 def modexp(a, b, n): 2 """ 3 Modular exponentiation using repeated squaring. 4 That is, we want to compute a^b mod n. 5 6 INPUT 7 a -- a positive integer 8 b -- a positive integer 9 n -- a positive integer OUTPUT 12 a^b mod n 13 """ 14 d = 1 15 for i in list(integer.binary(b)): 16 d = mod(d * d, n) 17 if Integer(i) == 1: 18 d = mod(d * a, n) 19 return Integer(d) Listing 2: function for modular exponentiation via repeated squaring. The code in Listing 2 can be saved to a file called, say, /home/mvngu/modexp.sage. The file can then be read into using the command load. Let us read the above file into and encrypt our message: sage: load "/home/mvngu/modexp.sage" sage: c = modexp(m, e, n) ; c Thus c = is the ciphertext. To recover our plaintext, we raise c to the power of d and reduce the result modulo n. Again, we use modular exponentiation via repeated squaring in the decryption process: sage: modexp(c, d, n) sage: m Notice in the last output that the value is the same as the integer that represents our original message. Hence we have recovered our plaintext. 9

10 7 Further discussion This tutorial has introduced the reader to some basic programming from within a session, in particular the code Listings 1 and 2. The official tutorial on can be found online at or in your local installation of at the directory SAGE_ROOT/devel/doc-main/html/tut/index.html. In order to make effective use of and extend its functionalities, the reader is encouraged to consult the official Python documentation at Some code examples presented in this tutorial are said to be un-pythonic, in the sense that they do not make effective use of various features of the Python programming language. For instance, the function modexp in Listing 2 can also be implemented as follows: 1 def modexp(a, b, n): 2 d = 1 3 for i in reversed(b.digits()): 4 d = mod(d * d, n) 5 if i == 1: 6 d = mod(d * a, n) 7 return Integer(d) Listing 3: A Pythonic version of modexp. modules in the directory SAGE_ROOT/devel/sage-main/sage/crypto can be used to complement a course in cryptography. David Kohel s notes at uses to teach a course in cryptography. Now it is over to you to explore what you can do using. References [1] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. The MIT Press, USA, 2nd edition, [2] A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. CRC Press, Boca Raton, FL, USA, [3] W. Stein. : Open Source Mathematics Software (version 3.1.4). The Group, 03 November [4] D. R. Stinson. Cryptography: Theory and Practice. Chapman & Hall/CRC, Boca Raton, USA, 3rd edition, [5] W. Trappe and L. C. Washington. Introduction to Cryptography with Coding Theory. Pearson Prentice Hall, Upper Saddle River, New Jersey, USA, 2nd edition,

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

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

3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)

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

An Introduction to the RSA Encryption Method

April 17, 2012 Outline 1 History 2 3 4 5 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

Public-Key Cryptography. Oregon State University

Public-Key Cryptography Çetin Kaya Koç Oregon State University 1 Sender M Receiver Adversary Objective: Secure communication over an insecure channel 2 Solution: Secret-key cryptography Exchange the key

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

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

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

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

Chapter 10 Asymmetric-Key Cryptography

Chapter 10 Asymmetric-Key Cryptography Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10.1 Chapter 10 Objectives Present asymmetric-key cryptography. Distinguish

A SOFTWARE COMPARISON OF RSA AND ECC

International Journal Of Computer Science And Applications Vol. 2, No. 1, April / May 29 ISSN: 974-13 A SOFTWARE COMPARISON OF RSA AND ECC Vivek B. Kute Lecturer. CSE Department, SVPCET, Nagpur 9975549138

9 Modular Exponentiation and Cryptography

9 Modular Exponentiation and Cryptography 9.1 Modular Exponentiation Modular arithmetic is used in cryptography. In particular, modular exponentiation is the cornerstone of what is called the RSA system.

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±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

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

Cryptography. Helmer Aslaksen Department of Mathematics National University of Singapore

Cryptography Helmer Aslaksen Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/sfm/ 1 Basic Concepts There are many situations in life where

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 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

Chapter 10 Asymmetric-Key Cryptography

Chapter 10 Asymmetric-Key Cryptography Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10.1 Chapter 10 Objectives To distinguish between two cryptosystems: symmetric-key

Chapter 9 Public Key Cryptography and RSA

Chapter 9 Public Key Cryptography and RSA Cryptography and Network Security: Principles and Practices (3rd Ed.) 2004/1/15 1 9.1 Principles of Public Key Private-Key Cryptography traditional private/secret/single

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,

PUBLIC KEY ENCRYPTION

PUBLIC KEY ENCRYPTION http://www.tutorialspoint.com/cryptography/public_key_encryption.htm Copyright tutorialspoint.com Public Key Cryptography Unlike symmetric key cryptography, we do not find historical

ΕΠΛ 674: Εργαστήριο 3

ΕΠΛ 674: Εργαστήριο 3 Ο αλγόριθμος ασύμμετρης κρυπτογράφησης RSA Παύλος Αντωνίου Department of Computer Science Private-Key Cryptography traditional private/secret/single key cryptography uses one key

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

MTH 382 Number Theory Spring 2003 Practice Problems for the Final

MTH 382 Number Theory Spring 2003 Practice Problems for the Final (1) Find the quotient and remainder in the Division Algorithm, (a) with divisor 16 and dividend 95, (b) with divisor 16 and dividend -95,

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

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,

The Mathematics of RSA

The Mathematics of RSA Dimitri Papaioannou May 24, 2007 1 Introduction Cryptographic systems come in two flavors. Symmetric or Private key encryption and Asymmetric or Public key encryption. Strictly speaking,

Notes for Recitation 5

6.042/18.062J Mathematics for Computer Science September 24, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 5 1 Exponentiation and Modular Arithmetic Recall that RSA encryption and decryption

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

RSA Encryption. Grade Levels. Objectives and Topics. Introduction and Outline (54)(2)(2) (27)(2)(2)(2) (9)(3)(2)(2)(2) (3)(3)(3)(2)(2)(2)

RSA Encryption Grade Levels This activity is intended for high schools students, grades 10 12. Objectives and Topics One of the classical examples of applied mathematics is encryption. Through this activity,

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

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

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.

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.

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

1) A very simple example of RSA encryption

Solved Examples 1) A very simple example of RSA encryption This is an extremely simple example using numbers you can work out on a pocket calculator (those of you over the age of 35 45 can probably even

Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs

Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs Enes Pasalic University of Primorska Koper, 2014 Contents 1 Preface 3 2 Problems 4 2 1 Preface This is a

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

The RSA Algorithm. Evgeny Milanov. 3 June 2009

The RSA Algorithm Evgeny Milanov 3 June 2009 In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman introduced a cryptographic algorithm, which was essentially to replace the less secure National Bureau

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

Symmetric Key cryptosystem

SFWR C03: Computer Networks and Computer Security Mar 8-11 200 Lecturer: Kartik Krishnan Lectures 22-2 Symmetric Key cryptosystem Symmetric encryption, also referred to as conventional encryption or single

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,

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

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.

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

UOSEC Week 2: Asymmetric Cryptography. Frank IRC kee Adam IRC xe0 IRC: irc.freenode.net #0x4f

UOSEC Week 2: Asymmetric Cryptography Frank farana@uoregon.edu IRC kee Adam pond2@uoregon.edu IRC xe0 IRC: irc.freenode.net #0x4f Agenda HackIM CTF Results GITSC CTF this Saturday 10:00am Basics of Asymmetric

A Study on Asymmetric Key Cryptography Algorithms

A Study on Asymmetric Key Cryptography Algorithms ASAITHAMBI.N School of Computer Science and Engineering, Bharathidasan University, Trichy, asaicarrier@gmail.com Abstract Asymmetric key algorithms use

CS Computer and Network Security: Applied Cryptography

CS 5410 - Computer and Network Security: Applied Cryptography Professor Patrick Traynor Spring 2016 Reminders Project Ideas are due on Tuesday. Where are we with these? Assignment #2 is posted. Let s get

NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES

NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES Ounasser Abid 1, Jaouad Ettanfouhi 2 and Omar Khadir 3 1,2,3 Laboratory of Mathematics, Cryptography and Mechanics, Department of Mathematics, Fstm,

Data Encryption A B C D E F G H I J K L M N O P Q R S T U V W X Y Z. we would encrypt the string IDESOFMARCH as follows:

Data Encryption Encryption refers to the coding of information in order to keep it secret. Encryption is accomplished by transforming the string of characters comprising the information to produce a new

CRYPTOGRAPHIC ALGORITHMS (AES, RSA)

CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA CRYPTOGRAPHIC ALGORITHMS (AES, RSA) A PAPER SUBMITTED TO PROFESSOR GILBERT S. YOUNG IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE COURSE CS530 : ADVANCED

The RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm

The RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm Maria D. Kelly December 7, 2009 Abstract The RSA algorithm, developed in 1977 by Rivest, Shamir, and Adlemen, is an algorithm

Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

Module 5: Basic Number Theory

Module 5: Basic Number Theory Theme 1: Division Given two integers, say a and b, the quotient b=a may or may not be an integer (e.g., 16=4 =4but 12=5 = 2:4). Number theory concerns the former case, and

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

The Laws of Cryptography Cryptographers Favorite Algorithms

2 The Laws of Cryptography Cryptographers Favorite Algorithms 2.1 The Extended Euclidean Algorithm. The previous section introduced the field known as the integers mod p, denoted or. Most of the field

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

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

Today ENCRYPTION. Cryptography example. Basic principles of cryptography

Today ENCRYPTION The last class described a number of problems in ensuring your security and privacy when using a computer on-line. This lecture discusses one of the main technological solutions. The use

Computer and Network Security

MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest

An Approach to Shorten Digital Signature Length

Computer Science Journal of Moldova, vol.14, no.342, 2006 An Approach to Shorten Digital Signature Length Nikolay A. Moldovyan Abstract A new method is proposed to design short signature schemes based

Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES

www.arpapress.com/volumes/vol8issue1/ijrras_8_1_10.pdf SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES Malek Jakob Kakish Amman Arab University, Department of Computer Information Systems, P.O.Box 2234,

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

Solutions to Practice Problems

Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

= 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

A chosen text attack on the RSA cryptosystem and some discrete logarithm schemes

A chosen text attack on the RSA cryptosystem and some discrete logarithm schemes Y. Desmedt Aangesteld Navorser NFWO Katholieke Universiteit Leuven Laboratorium ESAT B-3030 Heverlee, Belgium A. M. Odlyzko

Factoring Algorithms

Institutionen för Informationsteknologi Lunds Tekniska Högskola Department of Information Technology Lund University Cryptology - Project 1 Factoring Algorithms The purpose of this project is to understand

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

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

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.

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

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering. Introduction to Cryptography ECE 597XX/697XX

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Introduction to Cryptography ECE 597XX/697XX Part 6 Introduction to Public-Key Cryptography Israel Koren ECE597/697 Koren Part.6.1

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 Cryptosystem. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong. RSA Cryptosystem

Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong In this lecture, we will discuss the RSA cryptosystem, which is widely adopted as a way to encrypt a message, or

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

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

Chapter 9. Computational Number Theory. 9.1 The basic groups Integers mod N Groups

Chapter 9 Computational Number Theory 9.1 The basic groups We let Z = {..., 2, 1,0,1,2,...} denote the set of integers. We let Z + = {1,2,...} denote the set of positive integers and N = {0,1,2,...} the

Primality - Factorization

Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

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

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

Network Security. Security Attacks. Normal flow: Interruption: 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室

Network Security 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室 Security Attacks Normal flow: sender receiver Interruption: Information source Information destination

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

Comparative Analysis for Performance acceleration of Modern Asymmetric Crypto Systems

J. of Comp. and I.T. Vol. 3(1&2), 1-6 (2012). Comparative Analysis for Performance acceleration of Modern Asymmetric Crypto Systems RAJ KUMAR 1 and V.K. SARASWAT 2 1,2 Department of Computer Science, ICIS

On Generalized Fermat Numbers 3 2n +1

Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic

Lukasz Pater CMMS Administrator and Developer

Lukasz Pater CMMS Administrator and Developer EDMS 1373428 Agenda Introduction Why do we need asymmetric ciphers? One-way functions RSA Cipher Message Integrity Examples Secure Socket Layer Single Sign

SECURITY IN NETWORKS

SECURITY IN NETWORKS GOALS Understand principles of network security: Cryptography and its many uses beyond confidentiality Authentication Message integrity Security in practice: Security in application,

MODULAR ARITHMETIC KEITH CONRAD. Introduction We will define the notion of congruent integers (with respect to a modulus) and develop some basic ideas of modular arithmetic. Applications of modular arithmetic

ANALYSIS OF RSA ALGORITHM USING GPU PROGRAMMING

ANALYSIS OF RSA ALGORITHM USING GPU PROGRAMMING Sonam Mahajan 1 and Maninder Singh 2 1 Department of Computer Science Engineering, Thapar University, Patiala, India 2 Department of Computer Science Engineering,

NUMBER THEORY AND CRYPTOGRAPHY

NUMBER THEORY AND CRYPTOGRAPHY KEITH CONRAD 1. Introduction Cryptography is the study of secret messages. For most of human history, cryptography was important primarily for military or diplomatic purposes

EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

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

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.

CRC Press has granted the following specific permissions for the electronic version of this book:

This is a Chapter from the Handbook of Applied Cryptography, by A. Menezes, P. van Oorschot, and S. Vanstone, CRC Press, 1996. For further information, see www.cacr.math.uwaterloo.ca/hac CRC Press has

Outline. Cryptography. Bret Benesh. Math 331

Outline 1 College of St. Benedict/St. John s University Department of Mathematics Math 331 2 3 The internet is a lawless place, and people have access to all sorts of information. What is keeping people

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Karagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Karagpur Lecture No. #06 Cryptanalysis of Classical Ciphers (Refer