4 PKI Public Key Infrastructure
|
|
|
- Herbert Hart
- 6 years ago
- Views:
Transcription
1 67 PKI 4.1 PKI history 4 PKI Public Key Infrastructure 4.1 PKI history Classical cryptography Example form II WW: Enigma dates back thousands of years symmetric key
2 68 PKI 4.1 PKI history Symmetric key cryptography has a good property: possible to devise secure but at the same time fast algorithms! Public key cryptography: non-symmetric key Clifford Cocks ca 1973 secretly (until 1997, (The Government Communications Headquarters (GCHQ) British intelligence agency responsible for providing signals intelligence (SIGINT) and information assurance)); in parallel Rivest, Shamir and Adleman 1976 (their picture) ( photos/len-adi-ron.jpg) * RSA algoritm or * Cocks/RSA)
3 69 PKI 4.1 PKI history One-way functions Definition. Function f : X Y is a one-way function if it is easy to calculate y=f(x), but knowing only y it is computationally impossible to find efficiently x such that f(x)=y. Function f has a backdoor C, if knowing the information C it is possible to invert the function f efficiently. Existence of one-way functions not proven mathematically but suitable candidates exist one of the best-known candidates: calculation of number x b modulo n (1977) It is easy to find solution to the equation x b ymod n explicitly trying all the values of x but Howitto is make computationally the task more inefficient difficult? n is large
4 70 PKI 4.2 RSA 4.2 RSA Definition. For a given positive integer n, two integers a and b are called congruent modulo n, written a bmodn if a b is divisible by n (or equivalently if a and b have the same remainder when divided by n) Example. 7 2 mod 5
5 71 PKI 4.2 RSA RSA algoritm main idea based on an example: 2 1 2mod mod mod mod mod mod mod mod mod mod mod mod mod mod mod mod mod mod19
6 72 PKI 4.2 RSA RSA is a Block-algorithm: both the original text M as well as the encrypted text C consist of blocks of length 0 to n Choose 2 large random distinct prime numbers p q. Calculate n = pq 2. Calculate ϕ = (p 1)(q 1) (Euler s totient function) 3. Choose integer e (1 < e < ϕ), which is coprime to ϕ, i.e. gcd(e,ϕ) = 1 (Definition. Integer and its coprime have their greatest common divisor 1. e.g., 6 and 35 are coprimes, 6 and 27 are not coprimes (divisible by 3) (How to determine GCD? Euclidean algorithm (Python code:) def gcd(a, b): if b == 0: return a else: return gcd(b,a%b) ) 4. Find d, for which d e 1modϕ. (i.e. (d e 1)modϕ = 0 ); Here: n and e public key, n and d private key Encrypting: M transformed into the ciphertext: C = M e modn Decrypting: M = C d modn.
7 73 PKI 4.3 RSA help algorithms 4.3 RSA help algorithms Example: Euclidean algorithm and its extension Euclidean algorithm: 2107=2* =2* =1* =5* =2*21+7 =>GCD(2107,896)=7 Perform back-substitution: 7=49-2*21 =49-2*(266-5*49) =-2*266+11*49 =-2*266+11*( ) =11*315-13*266 =11*315-13*(896-2*315) =-13*896+37*315 =-13*896+37*(2107-2*986) =(37)*2107+(-87)*896 Therefore, for implementing RSA algorithm steps 3. and 4. extended Euclidean algorithm can be used:
8 74 PKI 4.3 RSA help algorithms def exeu(a, b): # (Extended Euclidean algorithm) q=0l; r=0l; x = [0L,0L,0L] y = [0L,0L,0L] if not b: return [1,0] else: x[2] = 1; x[1] = 0 y[2] = 0; y[1] = 1 while (b>0): q=a/b r=a-q*b x[0]=x[2]-q*x[1]; y[0]=y[2]-q*y[1] a,b=b,r x[2]=x[1];x[1]=x[0]; y[2]=y[1];y[1]=y[0]; return [x[2],y[2]]
9 75 PKI 4.3 RSA help algorithms Extended Euclidean algorithm gives: GCD(a, b) = x a + y b Example 1: Let p = 3 and q = 11 => n = 3 11 = 33 ϕ(n) = (3 1) (11 1) = 20 Choose randomly e such that GCD(e,20)=1. Let e = 7, GCD(7,20)=1 Using the function above: exeu(20,7) gives: [-1,3], i.e. 1=-1*20+3*7 => d = 3 (We can say that d e = 1modϕ. (i.e. (d e 1)modϕ = 0) Public key is: (n = 33, e = 7) Secret key: (n = 33, d = 3) Let M = 2. Then C = 2 7 mod33 = 29 and M = 29 3 mod33 = 2.
10 76 PKI 4.3 RSA help algorithms Example 2: Let p = 47 and q = 71 => n = p q = 3337 ϕ(n) = (p 1) (q 1) = 3220 Choose randomly e such that GCD(e,3220)=1. Let e = 79, GCD(79,3220)=1 Using the extended Euclidean algorithm exeu(3220,79) gives: [-25,1019], i.e. 1=-25* *79 => d = 1019 (We can say that d e = 1modϕ. (i.e. (d e 1)modϕ = 0) Public key is: (n = 3337,e = 79) Secret key is: (n = 3337,d = 1019) Let M = 582. Then C = mod3337 = 776 and M = mod3337 = 582
11 77 PKI 4.3 RSA help algorithms Helping tools for implementing RSA algorithm Fermat s primality test fast, gives right answer for most of composite numbers with proability greater than 1/2 Miller-Rabin s test slower but much more accurate identifying some extremely bad composite numbers Carmichael s numbers
12 78 PKI 4.3 RSA help algorithms Fermat s prime number test Fermat s Little Theorem: If p is a prime number then for any integer a a p a will be evenly divisible by p. i.e. if p is a prime and a is an integer coprime to p, then a p 1 1 will be evenly divisible by p: a p 1 1(mod p) For testing: choose random integer a < p and perform above check Let n be composite number. There exist such values of a for which a n 1 1(mod n) called Fermat s liars (in this case n is called Fermat pseudoprime to base a), rest are Fermat s witnesses to number n Carmichael numbers composite numbers n, for which Fermat s liars are all numbers a for which GCD(n,a)=1.
13 79 PKI 4.3 RSA help algorithms Carmichael numbers:
14 80 PKI 4.3 RSA help algorithms Carmichael number multipliers: 561 : [3, 11, 17] 1105 : [5, 13, 17] 1729 : [7, 13, 19] 2465 : [5, 17, 29] 2821 : [7, 13, 31] 6601 : [7, 23, 41] 8911 : [7, 19, 67] : [5, 29, 73] : [7, 31, 73] : [13, 37, 61] : [7, 11, 13, 41] : [13, 37, 97] : [7, 73, 103] : [3, 5, 47, 89] : [7, 13, 19, 37] : [11, 13, 17, 31] : [7, 11, 13, 101] : [13, 37, 241] : [7, 13, 19, 73] : [17, 41, 233] : [7, 13, 31, 61] : [7, 13, 19, 109] : [41, 61, 101] : [5, 17, 29, 113] : [37, 73, 109] : [13, 61, 397] : [19, 43, 409] : [13, 17, 23, 67] : [31, 61, 211] : [41, 73, 137] : [5, 19, 29, 163] : [37, 73, 181] : [31, 61, 271]
15 81 PKI 4.3 RSA help algorithms Miller-Rabin s test 1. Choose odd n => n 1 = 2 s d, where s -- integer, d -- odd. 2. Choose random integer a < n. 3. If a d modn = 1: return probably non-prime! 4. for j in [0 : s 1]: if (a 2 jd modn = 1: return probably prime! 5. return non-prime! If the test returns probably prime for t tests, then probability that n is prime is 1 4 t. Example, for t = 10 we have n as prime with probability of Usually, the choice for t 40.
16 82 PKI 4.4 Other algorithms 4.4 Other algorithms Other public key algorithms ElGamal DSS Diffie-Hellmann LUC XTR
17 83 PKI 4.5 Public Key Publishing 4.5 Public Key Publishing Distributing public keys How to make authentic copy of a public key to all potential checkers of the signature? Draw-back: Modification of a key can be used for man-in-the-middle attacks Idea: Public keys in a phonebook Idea by Diffie and Hellman, in 1976 Phonebook available in the Internet as on-line service
18 84 PKI 4.5 Public Key Publishing Certificate idea 1978 Kohnfelder: global online service for public key distribution not feasible due to high need for communication idea of certificates: each phone-book entry (ceritficate) separately signed by trusted service provider Certificate: document connecting a person with his/her public key Owner s ID Owner s public key Certificate issuer Validity period
19 85 PKI 4.5 Public Key Publishing etc. Benefits with certificates: can be distributed independently no need for phonebook -checks each time => reduced communication needs; communication channels load more even: no bottle-neck near the phonebook
20 86 PKI 4.5 Public Key Publishing Terms and abbreviations associated with certificates Certification Authority (CA) Certificate policy (CP) Certification path Certification Practice Statement (CPS) Certificate revocation list (CRL) Issuing certification authority (issuing CA) Public Key Certificate (PKC) data structure containing end user [/device/computer] public key together with some other useful information digitally signed with the secret key of the issuing CA Registration authority (RA) enterprise performing certificate identification, authorisation, but does not issue or sign certificates. RA fullfills tasks delegated by CA.
21 87 PKI 4.5 Public Key Publishing How are certificates issued? Certificates are issued by trusted party CA (Certification Authority) CA has CP (Certification Policy, documented set of rules) regulating whom to, how and for what purposes certificates are issued CA has to reliably protect its secret key CA has to perform regular audits for its actual security situation Using a certificate: It is enough if all know the CA public key! Merkle s tree structure is used
22 88 PKI 4.5 Public Key Publishing Merkle s tree (also, the Tree Hash Exchange (THEX) format) ROOT=IC(E+F) / \ / \ E=IC(A+B) F=IC(C+D) / \ / \ / \ / \ A=EC(S1) B=EC(S2) C=EC(S3) D=EC(S4) EC - end certificate IC - interim certificate
23 89 PKI 4.5 Public Key Publishing Storing the certificates Public key is made available to everybody Secret keys are kept in a CA typically on a separate storage device: online (ID-card (or similar), usually special FIPS 3 cryptodevice) in an isolated storage (e.g. USB flash memory kept away in a safe) Usually CA activities on a separate machine isolated from the network
24 90 PKI 4.5 Public Key Publishing Eestonian Grid policy: key protected with over 15 character password CA is backed up (CA database as well as public-secret key) All activities are getting logged CRL is published at webpage Baltic Grid webpage ( balticgrid.org) (Estonian Grid webpage ( ee/ca)) Additional information on CA and its activities: EuroPKI (
Dr. Jinyuan (Stella) Sun Dept. of Electrical Engineering and Computer Science University of Tennessee Fall 2010
CS 494/594 Computer and Network Security Dr. Jinyuan (Stella) Sun Dept. of Electrical Engineering and Computer Science University of Tennessee Fall 2010 1 Public Key Cryptography Modular Arithmetic RSA
Public Key Algorithms
Public Key Algorithms CS 472 Spring 13 Lecture 6 Mohammad Almalag 2/19/2013 Public Key Algorithms - Introduction Public key algorithms are a motley crew, how? All hash algorithms do the same thing: Take
Public Key Cryptography
Public Key Cryptography Giuseppe F. Italiano Universita` di Roma Tor Vergata italiano@disp.uniroma2.it Motivation Until early 70s, cryptography was mostly owned by government and military Symmetric cryptography
Lecture 2 Applied Cryptography (Part 2)
Lecture 2 Applied Cryptography (Part 2) Patrick P. C. Lee Tsinghua Summer Course 2010 2-1 Roadmap Number theory Public key cryptography RSA Diffie-Hellman DSA Certificates Tsinghua Summer Course 2010 2-2
Public Key Algorithms
Public Key Algorithms 1 Public Key Algorithms It is necessary to know some number theory to really understand how and why public key algorithms work Most of the public key algorithms are based on modular
Public Key Cryptography
graphy CSS322: Security and Cryptography Sirindhorn International Institute of Technology Thammasat University Prepared by Steven Gordon on 29 December 2011 CSS322Y11S2L07, Steve/Courses/2011/S2/CSS322/Lectures/rsa.tex,
CS669 Network Security
UNIT II PUBLIC KEY ENCRYPTION Uniqueness Number Theory concepts Primality Modular Arithmetic Fermet & Euler Theorem Euclid Algorithm RSA Elliptic Curve Cryptography Diffie Hellman Key Exchange Uniqueness
Applied Cryptography and Computer Security CSE 664 Spring 2018
Applied Cryptography and Computer Security Lecture 13: Public-Key Cryptography and RSA Department of Computer Science and Engineering University at Buffalo 1 Public-Key Cryptography What we already know
Chapter 3 Public Key Cryptography
Cryptography and Network Security Chapter 3 Public Key Cryptography Lectured by Nguyễn Đức Thái Outline Number theory overview Public key cryptography RSA algorithm 2 Prime Numbers A prime number is an
Lecture 6: Overview of Public-Key Cryptography and RSA
1 Lecture 6: Overview of Public-Key Cryptography and RSA Yuan Xue In this lecture, we give an overview to the public-key cryptography, which is also referred to as asymmetric cryptography. We will first
CSCI 454/554 Computer and Network Security. Topic 5.2 Public Key Cryptography
CSCI 454/554 Computer and Network Security Topic 5.2 Public Key Cryptography Outline 1. Introduction 2. RSA 3. Diffie-Hellman Key Exchange 4. Digital Signature Standard 2 Introduction Public Key Cryptography
Chapter 9 Public Key Cryptography. WANG YANG
Chapter 9 Public Key Cryptography WANG YANG wyang@njnet.edu.cn Content Introduction RSA Diffie-Hellman Key Exchange Introduction Public Key Cryptography plaintext encryption ciphertext decryption plaintext
Outline. CSCI 454/554 Computer and Network Security. Introduction. Topic 5.2 Public Key Cryptography. 1. Introduction 2. RSA
CSCI 454/554 Computer and Network Security Topic 5.2 Public Key Cryptography 1. Introduction 2. RSA Outline 3. Diffie-Hellman Key Exchange 4. Digital Signature Standard 2 Introduction Public Key Cryptography
Outline. Public Key Cryptography. Applications of Public Key Crypto. Applications (Cont d)
Outline AIT 682: Network and Systems Security 1. Introduction 2. RSA 3. Diffie-Hellman Key Exchange 4. Digital Signature Standard Topic 5.2 Public Key Cryptography Instructor: Dr. Kun Sun 2 Public Key
Applied Cryptography and Network Security
Applied Cryptography and Network Security William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #8: RSA Didn t we learn about RSA last time? During the last lecture, we saw what RSA does and learned
RSA (material drawn from Avi Kak Lecture 12, Lecture Notes on "Computer and Network Security" Used in asymmetric crypto.
RSA (material drawn from Avi Kak (kak@purdue.edu) Lecture 12, Lecture Notes on "Computer and Network Security" Used in asymmetric crypto. protocols The RSA algorithm is based on the following property
Public Key Algorithms
CSE597B: Special Topics in Network and Systems Security Public Key Cryptography Instructor: Sencun Zhu The Pennsylvania State University Public Key Algorithms Public key algorithms RSA: encryption and
ASYMMETRIC CRYPTOGRAPHY
ASYMMETRIC CRYPTOGRAPHY CONTENT: 1. Number Theory 2. One Way Function 3. Hash Function 4. Digital Signature 5. RSA (Rivest-Shamir Adleman) References: 1. Applied Cryptography, Bruce Schneier 2. Cryptography
Chapter 9. Public Key Cryptography, RSA And Key Management
Chapter 9 Public Key Cryptography, RSA And Key Management RSA by Rivest, Shamir & Adleman of MIT in 1977 The most widely used public-key cryptosystem is RSA. The difficulty of attacking RSA is based on
RSA (algorithm) History
RSA (algorithm) RSA is an algorithm for public-key cryptography that is based on the presumed difficulty of factoring large integers, the factoring problem. RSA stands for Ron Rivest, Adi Shamir and Leonard
ECE 646 Fall 2009 Final Exam December 15, Multiple-choice test
ECE 646 Fall 2009 Final Exam December 15, 2009 Multiple-choice test 1. (1 pt) Parallel processing can be used to speed up the following cryptographic transformations (please note that multiple answers
CS Network Security. Nasir Memon Polytechnic University Module 7 Public Key Cryptography. RSA.
CS 393 - Network Security Nasir Memon Polytechnic University Module 7 Public Key Cryptography. RSA. Course Logistics Homework 2 revised. Due next Tuesday midnight. 2/26,28/02 Module 7 - Pubic Key Crypto
Channel Coding and Cryptography Part II: Introduction to Cryptography
Channel Coding and Cryptography Part II: Introduction to Cryptography Prof. Dr.-Ing. habil. Andreas Ahrens Communications Signal Processing Group, University of Technology, Business and Design Email: andreas.ahrens@hs-wismar.de
Public Key Encryption. Modified by: Dr. Ramzi Saifan
Public Key Encryption Modified by: Dr. Ramzi Saifan Prime Numbers Prime numbers only have divisors of 1 and itself They cannot be written as a product of other numbers Prime numbers are central to number
Cryptography Symmetric Cryptography Asymmetric Cryptography Internet Communication. Telling Secrets. Secret Writing Through the Ages.
Telling Secrets Secret Writing Through the Ages William Turner Department of Mathematics & Computer Science Wabash College Crawfordsville, IN 47933 Tuesday 4 February 2014 W. J. Turner Telling Secrets
Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl. Chapter 6 Introduction to Public-Key Cryptography
Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl www.crypto-textbook.com Chapter 6 Introduction to Public-Key Cryptography ver. November 18, 2010 These
Overview. Public Key Algorithms I
Public Key Algorithms I Dr. Arjan Durresi Louisiana State University Baton Rouge, LA 70810 Durresi@csc.lsu.Edu These slides are available at: http://www.csc.lsu.edu/~durresi/csc4601-04/ Louisiana State
Total points: 71. Total time: 75 minutes. 9 problems over 7 pages. No book, notes, or calculator
CMSC 414 F08 Exam 1 Page 1 of 10 Name: Total points: 71. Total time: 75 minutes. 9 problems over 7 pages. No book, notes, or calculator 1. [14 points] a. Are n=221 and e=3 valid numbers for RSA. Explain.
Public-Key Cryptography. Professor Yanmin Gong Week 3: Sep. 7
Public-Key Cryptography Professor Yanmin Gong Week 3: Sep. 7 Outline Key exchange and Diffie-Hellman protocol Mathematical backgrounds for modular arithmetic RSA Digital Signatures Key management Problem:
An overview and Cryptographic Challenges of RSA Bhawana
An overview and Cryptographic Challenges of RSA Bhawana Department of CSE, Shanti Devi Institute of Technology & Management, Israna, Haryana India ABSTRACT: With the introduction of the computer, the need
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È.
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È. Let Ò Ô Õ. Pick ¾ ½ ³ Òµ ½ so, that ³ Òµµ ½. Let ½ ÑÓ ³ Òµµ. Public key: Ò µ. Secret key Ò µ.
Lecture 6 - Cryptography
Lecture 6 - Cryptography CMPSC 443 - Spring 2012 Introduction Computer and Network Security Professor Jaeger www.cse.psu.edu/~tjaeger/cse443-s12 Question Setup: Assume you and I donʼt know anything about
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È.
RSA (Rivest Shamir Adleman) public key cryptosystem: Key generation: Pick two large prime Ô Õ ¾ numbers È. Let Ò Ô Õ. Pick ¾ ½ ³ Òµ ½ so, that ³ Òµµ ½. Let ½ ÑÓ ³ Òµµ. Public key: Ò µ. Secret key Ò µ.
Asymmetric Primitives. (public key encryptions and digital signatures)
Asymmetric Primitives (public key encryptions and digital signatures) An informal, yet instructive account of asymmetric primitives Timeline of the invention of public-key cryptography 1970-1974 British
Public Key Cryptography 2. c Eli Biham - December 19, Public Key Cryptography 2
Public Key Cryptography 2 c Eli Biham - December 19, 2012 346 Public Key Cryptography 2 RSA Reference: Rivest, Shamir, Adleman, A Method for Obtaining Digital Signatures and Public Key Cryptosystems, CACM,
Network Security. Chapter 4 Public Key Cryptography. Public Key Cryptography (4) Public Key Cryptography
Chair for Network Architectures and Services Department of Informatics TU München Prof. Carle Encryption/Decryption using Public Key Cryptography Network Security Chapter 4 Public Key Cryptography However,
Public Key Cryptography and RSA
Public Key Cryptography and RSA Major topics Principles of public key cryptosystems The RSA algorithm The Security of RSA Motivations A public key system is asymmetric, there does not have to be an exchange
Public Key Cryptography and the RSA Cryptosystem
Public Key Cryptography and the RSA Cryptosystem Two people, say Alice and Bob, would like to exchange secret messages; however, Eve is eavesdropping: One technique would be to use an encryption technique
Some Stuff About Crypto
Some Stuff About Crypto Adrian Frith Laboratory of Foundational Aspects of Computer Science Department of Mathematics and Applied Mathematics University of Cape Town This work is licensed under a Creative
Admin ENCRYPTION. Admin. Encryption 10/29/15. Assignment 6. 4 more assignments: Midterm next Thursday. What is it and why do we need it?
Admin Assignment 6 4 more assignments:! Assignment 7, due 11/13 5pm! Assignment 8, due 11/20 5pm! Assignments 9 & 10, due 12/9 11:59pm ENCRYPTION David Kauchak CS52 Spring 2015 Admin Midterm next Thursday!
CS 6324: Information Security More Info on Key Establishment: RSA, DH & QKD
ERIK JONSSON SCHOOL OF ENGINEERING & COMPUTER SCIENCE Cyber Security Research and Education Institute CS 6324: Information Security Dr. Junia Valente Department of Computer Science The University of Texas
Part VI. Public-key cryptography
Part VI Public-key cryptography Drawbacks with symmetric-key cryptography Symmetric-key cryptography: Communicating parties a priori share some secret information. Secure Channel Alice Unsecured Channel
CSC 474/574 Information Systems Security
CSC 474/574 Information Systems Security Topic 2.5 Public Key Algorithms CSC 474/574 Dr. Peng Ning 1 Public Key Algorithms Public key algorithms covered in this class RSA: encryption and digital signature
L13. Reviews. Rocky K. C. Chang, April 10, 2015
L13. Reviews Rocky K. C. Chang, April 10, 2015 1 Foci of this course Understand the 3 fundamental cryptographic functions and how they are used in network security. Understand the main elements in securing
Public Key Cryptography, OpenPGP, and Enigmail. 31/5/ Geek Girls Carrffots GVA
Public Key Cryptography, OpenPGP, and Enigmail Cryptography is the art and science of transforming (encrypting) a message so only the intended recipient can read it Symmetric Cryptography shared secret
RSA System setup and test
RSA System setup and test Project 1, EITF55 Security, 2018 Ben Smeets Dept. of Electrical and Information Technology, Lund University, Sweden Last revised by Ben Smeets on 2018 01 12 at 01:06 What you
The evolving storage encryption market
The evolving storage encryption market Alexander (S andy) S tewart S un M icros ys tems 1 S toragetek Drive, Louis ville, CO 80028 P hone:+1-303-673-2775 FAX: +1-303-661-5743 E-mail: alexander.s tewart@
Cryptography and Network Security
Cryptography and Network Security CRYPTOGRAPHY AND NETWORK SECURITY PRAKASH C. GUPTA Former Head Department of Information Technology Maharashtra Institute of Technology Pune Delhi-110092 2015 CRYPTOGRAPHY
RSA. Public Key CryptoSystem
RSA Public Key CryptoSystem DIFFIE AND HELLMAN (76) NEW DIRECTIONS IN CRYPTOGRAPHY Split the Bob s secret key K to two parts: K E, to be used for encrypting messages to Bob. K D, to be used for decrypting
Other Uses of Cryptography. Cryptography Goals. Basic Problem and Terminology. Other Uses of Cryptography. What Can Go Wrong? Why Do We Need a Key?
ryptography Goals Protect private communication in the public world and are shouting messages over a crowded room no one can understand what they are saying 1 Other Uses of ryptography Authentication should
Distributed Systems. 26. Cryptographic Systems: An Introduction. Paul Krzyzanowski. Rutgers University. Fall 2015
Distributed Systems 26. Cryptographic Systems: An Introduction Paul Krzyzanowski Rutgers University Fall 2015 1 Cryptography Security Cryptography may be a component of a secure system Adding cryptography
Programming Techniques in Computer Algebra
Programming Techniques in Computer Algebra Prof. Dr. Wolfram Koepf Universität Kassel http://www.mathematik.uni-kassel.de/~koepf March 18, 2010 Yaounde, Cameroon Abstract Topics of This Talk In this talk
1 Extended Euclidean Algorithm
CS 124 Section #8 RSA, Random Walks, Linear Programming 3/27/17 1 Extended Euclidean Algorithm Given a, b, find x, y such that ax + by = d where d is the GCD of a, b. This will be necessary in implementing
Network Encryption. Dr. Michael Ritter. September 18 th, 2015
Network Encryption Dr. Michael Ritter September 18 th, 2015 What the heck is Elliptic Curve Diffie-Hellman? Dr. Michael Ritter September 18 th, 2015 ADVA Optical Networking at a Glance Value-added connectivity
Diffie-Hellman. Part 1 Cryptography 136
Diffie-Hellman Part 1 Cryptography 136 Diffie-Hellman Invented by Williamson (GCHQ) and, independently, by D and H (Stanford) A key exchange algorithm o Used to establish a shared symmetric key Not for
Encryption Details COMP620
Encryption Details COMP620 Encryption is a powerful defensive weapon for free people. It offers a technical guarantee of privacy, regardless of who is running the government It s hard to think of a more
Number Theory and RSA Public-Key Encryption
Number Theory and RSA Public-Key Encryption Dr. Natarajan Meghanathan Associate Professor of Computer Science Jackson State University E-mail: natarajan.meghanathan@jsums.edu CIA Triad: Three Fundamental
Public-key encipherment concept
Date: onday, October 21, 2002 Prof.: Dr Jean-Yves Chouinard Design of Secure Computer Systems CSI4138/CEG4394 Notes on Public Key Cryptography Public-key encipherment concept Each user in a secure communication
Great Theoretical Ideas in Computer Science. Lecture 27: Cryptography
15-251 Great Theoretical Ideas in Computer Science Lecture 27: Cryptography What is cryptography about? Adversary Eavesdropper I will cut his throat I will cut his throat What is cryptography about? loru23n8uladjkfb!#@
Kurose & Ross, Chapters (5 th ed.)
Kurose & Ross, Chapters 8.2-8.3 (5 th ed.) Slides adapted from: J. Kurose & K. Ross \ Computer Networking: A Top Down Approach (5 th ed.) Addison-Wesley, April 2009. Copyright 1996-2010, J.F Kurose and
Key Exchange. References: Applied Cryptography, Bruce Schneier Cryptography and Network Securiy, Willian Stallings
Key Exchange References: Applied Cryptography, Bruce Schneier Cryptography and Network Securiy, Willian Stallings Outlines Primitives Root Discrete Logarithm Diffie-Hellman ElGamal Shamir s Three Pass
RSA: PUBLIC KEY ALGORITHMS
Fernando Rosendo [fernando.rosendo@iweb.com.br] i.web Labs Brazil Theory and Implementation Public Key Algorithms based on mathematical properties which allow the cryptographic process (encryption) using
Chair for Network Architectures and Services Department of Informatics TU München Prof. Carle. Network Security
Chair for Network Architectures and Services Department of Informatics TU München Prof. Carle Network Security Chapter 2 Basics 2.2 Public Key Cryptography Encryption/Decryption using Public Key Cryptography
Elements of Cryptography and Computer and Networking Security Computer Science 134 (COMPSCI 134) Fall 2016 Instructor: Karim ElDefrawy
Elements of Cryptography and Computer and Networking Security Computer Science 134 (COMPSCI 134) Fall 2016 Instructor: Karim ElDefrawy Homework 2 Due: Friday, 10/28/2016 at 11:55pm PT Will be posted on
Study Guide to Mideterm Exam
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467b: Cryptography and Computer Security Handout #7 Professor M. J. Fischer February 20, 2012 Study Guide to Mideterm Exam For the exam, you are responsible
PUBLIC KEY CRYPTO. Anwitaman DATTA SCSE, NTU Singapore CX4024. CRYPTOGRAPHY & NETWORK SECURITY 2018, Anwitaman DATTA
PUBLIC KEY CRYPTO Anwitaman DATTA SCSE, NTU Singapore Acknowledgement: The following lecture slides are based on, and uses material from the text book Cryptography and Network Security (various eds) by
CSC/ECE 774 Advanced Network Security
Computer Science CSC/ECE 774 Advanced Network Security Topic 2. Network Security Primitives CSC/ECE 774 Dr. Peng Ning 1 Outline Absolute basics Encryption/Decryption; Digital signatures; D-H key exchange;
Uzzah and the Ark of the Covenant
Uzzah and the Ark of the Covenant And when they came to the threshing floor of Chidon, Uzzah put out his hand to take hold of the ark, for the oxen stumbled. 10 And the anger of the LORD was kindled against
! Addition! Multiplication! Bigger Example - RSA cryptography
! Addition! Multiplication! Bigger Example - RSA cryptography Modular Arithmetic Modular Exponentiation Primality Testing (Fermat s little theorem) Probabilistic algorithm Euclid s Algorithm for gcd (greatest
RSA Public Key Encryption 1. Ivor Page 2
RSA Public Key Encryption 1 RSA Public Key Encryption 1 Ivor Page 2 One of the most important methods of providing secrecy and authentication is encryption. Secrecy means the secure transmission of data
Algorithms (III) Yijia Chen Shanghai Jiaotong University
Algorithms (III) Yijia Chen Shanghai Jiaotong University Review of the Previous Lecture Factoring: Given a number N, express it as a product of its prime factors. Many security protocols are based on the
Public Key Encryption
Public Key Encryption A case study THE RSA CRYPTOSYSTEM Public 31/05/14 Key Encryption 2 Rivest Shamir Adleman (1978) Key generation 1. Generate two large, distinct primes p, q (100 200 decimal digits)
Modern cryptography 2. CSCI 470: Web Science Keith Vertanen
Modern cryptography 2 CSCI 470: Web Science Keith Vertanen Modern cryptography Overview Asymmetric cryptography Diffie-Hellman key exchange (last time) Pubic key: RSA Pretty Good Privacy (PGP) Digital
Cryptography and Network Security. Sixth Edition by William Stallings
Cryptography and Network Security Sixth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA Misconceptions Concerning Public-Key Encryption Public-key encryption is more secure from
Math From Scratch Lesson 22: The RSA Encryption Algorithm
Math From Scratch Lesson 22: The RSA Encryption Algorithm W. Blaine Dowler January 2, 2012 Contents 1 What Is Encryption? 1 2 What Is RSA Encryption? 2 2.1 Key Generation............................ 2
Assignment 9 / Cryptography
Assignment 9 / Cryptography Michael Hauser March 2002 Tutor: Mr. Schmidt Course: M.Sc Distributed Systems Engineering Lecturer: Mr. Owens CONTENTS Contents 1 Introduction 3 2 Simple Ciphers 3 2.1 Vignère
Cryptography in Lotus Notes/Domino Pragmatic Introduction for Administrators
Cryptography in Lotus Notes/Domino Pragmatic Introduction for Administrators Belfast, 11-Nov-2010 Innovative Software Solutions. Thomas Bahn - graduated in mathematics, University of Hannover - developing
Module: Cryptographic Protocols. Professor Patrick McDaniel Spring CMPSC443 - Introduction to Computer and Network Security
CMPSC443 - Introduction to Computer and Network Security Module: Cryptographic Protocols Professor Patrick McDaniel Spring 2009 1 Key Distribution/Agreement Key Distribution is the process where we assign
Elliptic Curve Public Key Cryptography
Why? Elliptic Curve Public Key Cryptography ECC offers greater security for a given key size. Why? Elliptic Curve Public Key Cryptography ECC offers greater security for a given key size. The smaller key
Lecture 3 Algorithms with numbers (cont.)
Advanced Algorithms Floriano Zini Free University of Bozen-Bolzano Faculty of Computer Science Academic Year 2013-2014 Lecture 3 Algorithms with numbers (cont.) 1 Modular arithmetic For cryptography it
Hash Functions, Public-Key Encryption CMSC 23200/33250, Autumn 2018, Lecture 6
Hash Functions, Public-Key Encryption CMSC 23200/33250, Autumn 2018, Lecture 6 David Cash University of Chicago Plan 1. A few points about hash functions 2. Introducing Public-Key Encryption 3. Math for
Public Key (asymmetric) Cryptography
Public-Key Cryptography Public Key (asymmetric) Cryptography Luca Veltri (mail.to: luca.veltri@.veltri@unipr.it) Course of Network Security, Spring 2013 http:// ://www.tlc.unipr.it it/veltri Also referred
ICS 180 May 4th, Guest Lecturer: Einar Mykletun
ICS 180 May 4th, 2004 Guest Lecturer: Einar Mykletun 1 Symmetric Key Crypto 2 Symmetric Key Two users who wish to communicate share a secret key Properties High encryption speed Limited applications: encryption
3/22/17. Admin. Assignment 6 ENCRYPTION. David Kauchak CS52 Spring Survey: How is the class going? Survey: respondents. 24 total respondents
Admin Assignment 6 ENCRYPTION David Kauchak CS52 Spring 2016 Survey: respondents Survey: How is the class going? 24 total respondents 1 Survey: How is the difficulty of the class? Survey: time spent per
Cryptography Lecture 4. Attacks against Block Ciphers Introduction to Public Key Cryptography. November 14, / 39
Cryptography 2017 Lecture 4 Attacks against Block Ciphers Introduction to Public Key Cryptography November 14, 2017 1 / 39 What have seen? What are we discussing today? What is coming later? Lecture 3
T Cryptography and Data Security
T-79.159 Cryptography and Data Security Lecture 10: 10.1 Random number generation 10.2 Key management - Distribution of symmetric keys - Management of public keys Kaufman et al: Ch 11.6; 9.7-9; Stallings:
1 Extended Euclidean Algorithm
CS 124 Section #8 RSA, Random Walks, Linear Programming 3/27/17 1 Extended Euclidean Algorithm Given a, b, find x, y such that ax + by = d where d is the GCD of a, b. This will be necessary in implementing
Algorithms (III) Yijia Chen Shanghai Jiaotong University
Algorithms (III) Yijia Chen Shanghai Jiaotong University Review of the Previous Lecture Factoring: Given a number N, express it as a product of its prime factors. Many security protocols are based on the
Encryption. INST 346, Section 0201 April 3, 2018
Encryption INST 346, Section 0201 April 3, 2018 Goals for Today Symmetric Key Encryption Public Key Encryption Certificate Authorities Secure Sockets Layer Simple encryption scheme substitution cipher:
ח'/סיון/תשע "א. RSA: getting ready. Public Key Cryptography. Public key cryptography. Public key encryption algorithms
Public Key Cryptography Kurose & Ross, Chapters 8.28.3 (5 th ed.) Slides adapted from: J. Kurose & K. Ross \ Computer Networking: A Top Down Approach (5 th ed.) AddisonWesley, April 2009. Copyright 19962010,
Algorithms (III) Yu Yu. Shanghai Jiaotong University
Algorithms (III) Yu Yu Shanghai Jiaotong University Review of the Previous Lecture Factoring: Given a number N, express it as a product of its prime factors. Many security protocols are based on the assumed
Cryptography MIS
Cryptography MIS-5903 http://community.mis.temple.edu/mis5903sec011s17/ Cryptography History Substitution Monoalphabetic Polyalphabetic (uses multiple alphabets) uses Vigenere Table Scytale cipher (message
What did we talk about last time? Public key cryptography A little number theory
Week 4 - Friday What did we talk about last time? Public key cryptography A little number theory If p is prime and a is a positive integer not divisible by p, then: a p 1 1 (mod p) Assume a is positive
Public-Key Cryptography
Multimedia Security Mauro Barni University of Siena Private-Key Cryptography Traditional secret key cryptography uses one key shared by both sender and receiver if this key is disclosed communication secrecy
A nice outline of the RSA algorithm and implementation can be found at:
Cryptography Lab: RSA Encryption and Decryption Lab Objectives: After this lab, the students should be able to Explain the simple concepts of encryption and decryption to protect information in transmission.
- 0 - CryptoLib: Cryptography in Software John B. Lacy 1 Donald P. Mitchell 2 William M. Schell 3 AT&T Bell Laboratories ABSTRACT
- 0 - CryptoLib: Cryptography in Software John B. Lacy 1 Donald P. Mitchell 2 William M. Schell 3 AT&T Bell Laboratories ABSTRACT With the capacity of communications channels increasing at the current
Elementary number theory
Elementary number theory The notion of primes, greatest common divisors, congruences and Euler s phi function. the number theoretic concepts and Sage commands Sage Implementation of the RSA algorithm.
CPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 7 January 30, 2012 CPSC 467b, Lecture 7 1/44 Public-key cryptography RSA Factoring Assumption Computing with Big Numbers Fast Exponentiation
Cryptography III. Public-Key Cryptography Digital Signatures. 2/1/18 Cryptography III
Cryptography III Public-Key Cryptography Digital Signatures 2/1/18 Cryptography III 1 Public Key Cryptography 2/1/18 Cryptography III 2 Key pair Public key: shared with everyone Secret key: kept secret,
Computer Security 3/23/18
s s encrypt a block of plaintext at a time and produce ciphertext Computer Security 08. Cryptography Part II Paul Krzyzanowski DES & AES are two popular block ciphers DES: 64 bit blocks AES: 128 bit blocks