# Digital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document?

Save this PDF as:

Size: px
Start display at page:

Download "Digital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document?"

## Transcription

1 Cryptography Digital Signatures Professor: Marius Zimand Digital signatures are meant to realize authentication of the sender nonrepudiation (Note that authentication of sender is also achieved by MACs.) How about this idea for a digital signature? Scan your handwritten signature and append it to the document? Does not work: anyone can take it and append it to their document. We see that the signature must depend on the document. A solution that works: use a PKC We have seen that a PKC which satisfies E k (D k (m)) = m can be used for digital signatures. Example: RSA. Alice has n A, p A, q A, e A, d A. She signs message m with m d A ( mod n A ). So she sends to Bob (m, m d A ( mod n A )) Bob receives this and he needs to check the validity of the signature. The verification algorithm consists in computing (m d A ) e A ( mod n A ). If this equal to m, the signature is valid, otherwise it is not. This is the idea for all digital signature algorithms: 1

2 producing the signature requires the knowledge of the secret key owned by the signer. the signature must have a property linking it to the message and anyone (having the public key) can test that property. In practice, people are using specialized digital signatures algorithms. The generic schema for a digital signature is as follows: digital signature schema: (M, A, K, S, V ), where M is the set of messages A is the set of signatures K is the set of keys S is the signing algorithm. Formally, S = {sig k one signature function for each key k K}, where each sig k maps messages to signatures. The signing algorithm should be efficient secret (only the owner of k can do it). V is the verification algorithm, Formally, V = {ver k one verification test for each key k K}, Requirement: true, ver k (x, y) = false, if y = sig k (x), if y sig k (x). 2

3 The verification test should be efficient public. Security requirement (informallly): It should be computationally infeasible for someone who does not have k to produce a y that passes the test ver(x, y). El Gamal Signature Algorithm parameters for the key p - large prime number α - primitive root of p a - randomly chosen in the range {2,..., p 2}. β = α a ( mod p) ( β is a in a box. ) (p, α, β) - public a - secret key. signature algorithm We want to sign message m. 1. select random k such that gcd(k, p 1) = compute r = α k ( mod p). 3. compute s = k 1 (m ar)( mod (p 1)). 4. The signature is the pair (r, s) and the signed message is (m, (r, s)). 3

4 verification algorithm The user who makes the verification knows (p, α, β) - the public key (m, (r, s)) - the signed message The verification is done by computing v 1 = β r r s ( mod p) and v 2 = α m ( mod p). If v 1 = v 2, then the signature is valid, otherwise it is not. Let s see that a good signature passes the verification test. s = k 1 (m ar)( mod (p 1)) sk = m ar( mod (p 1)) m = sk + ar( mod (p 1)) v 2 = α m = α sk+ar ( mod p) = (α a ) r (α k ) s = β r r s ( mod p). Informal analysis of security Suppose that Eve wants to sign as Alice (who is the owner of the secret key a). Eve needs an s that passes the verification test, i.e., she needs an s such that β r r s = α m ( mod p). So r s = β r α m ( mod p). This is similar to the Discrete Log problem, so Eve cannot find such an s. 4

5 Digital Signature Algorithm (DSA) proposed by NIST in 1991, and adopted as a federal standard in parameters for the key q - prime number having 160 bits p - prime such that q divides p 1. g - primitive root of p. α = g (p 1)/q ( mod p). Note that α q = 1( mod p). a - secret number randomly chosen in the range {0,..., p 1}. β = α a ( mod p) (p, q, α, β) - public key. a - secret key signature algorithm 1. select random secret k, k {1,..., q 1} 2. compute r = (α k ( mod p))( mod q). 3. compute s = k 1 (m + ar)( mod q). 4. the signature is (r, s); the signed message is (m, (r, s)). 5

6 verification algorithm 1. compute u 1 = s 1 m( mod q). 2. compute u 2 = s 1 r( mod q). 3. compute v = (α u 1 β u 2 ( mod p))( mod q). 4. if v = r, then accept the signature; otherwise reject the signature. Let s check that a good signature passes the verification test. m = (sk ar)( mod (q)) s 1 m = (k a r s 1 )( mod q) k = s 1 m + a r s 1 ( mod q) = u 1 + a u 2 ( mod q). So α k = α u 1+a u 2 ( mod p) = α u1 β u 2 ( mod p). The first equality holds because α q = 1( mod p). And finally: (α k ( mod p))( mod q) = (α u1 β u 2 ( mod p))( mod q). The LHS is r and the RHS is v, so r = v. DSA is very similar to the El Gamal signature scheme; the major difference is that the r and s, the components of the signature, are guaranteed to be smaller than q, which has the fixed size of 160 bits. On the other hand, the security is based on 6

7 working in the larger field Z p ; thus we can pick an appropriate large size for p for which the DLP is hard and the signature will still be short. (Toy) Example of DSA q = 13, p = 4q + 1 = 53, g = 2; g is a primitive root of p. α = g (p 1)/q ( mod p) = 2 4 ( mod 53) = 16. a = 3 (the secret key) β = α a ( mod p) = 16 3 ( mod 53) = 15. Suppose we wish to sign the message m = 5. For this we generate the ephemeral secret key k = 2 compute r = (α k ( mod p))( mod q) = (16 2 ( mod 53))( mod 13) = 5. compute s = k 1 (m + ar)( mod q) = 10. the signature is the pair (5, 10). To verify the signature the receiver computes u 1 = s 1 m( mod q) = 7 u 2 = s 1 r( mod q) = 7 v = (α u1 β u 2 ( mod p))( mod q) = 5. Note that v = r and thus the signature is accepted as valid. 7

8 In DSA, it is recommended that p has at least 512 bits, and some experts say it is more prudent to have p with 1024 bits. Thus the order of magnitude of DSA parameters is about the same as for RSA. However, DSA is slower than RSA signature algorithm, because the DSA operations are more complicated. The verification algorithm in RSA requires only one exponentiation modulo a number that is, say 1024 bits long. In DSA, verification requires two exponentiations. The main problem is that the DSA only needs working modulo a number with 160 bits, but because this size is too small for making the DLP hard, an extra layer has been introduced that works mod p, where p should have 512, or, better, 1024 bits. DSA works in the group generated by α (this group has q elements, because, α q = 1, and thus, the first q powers form a pattern that repeats itself). We have defined the DLP in the multiplicative group of nonzero residues mod p (this group is denoted Zp). However, DLP can be formulated in any group, not only in Zp. We can use in cryptography, any such group in which DLP is hard. Such a group exists in the world of elliptic curves (EC). Thus there exists an EC-DSA which is very similar to the DSA that we have seen, only that the arithmetic is done in the world of EC. The advantage of EC is that it needs smaller parameters, and operations can be implemented faster. The use of Hash Functions in Signature Schemes It is very inefficient to apply a signature algorithm to a long message. Consequently, in general, we do not sign the message m but its digest h(m) (where h is some hash function). For example this is how we typically sign a message m with RSA. 8

9 compute the digest h(m). compute s = h(m) d ( mod n). the message + signature is (m, s). The verification of (m, s) is done as follows: compute the digest h(m). compute s e ( mod n). check that the above two numbers are equal. Recall that we want a hash function to have the following properties: preimage resistant: given a digest, it should be hard to find a message having that digest. weak collision resistant: given a message m 1 it should be unfeasible to find another message m 2 that collides with m 1. strong collision resistant: it should be hard to find two messages that collide. Let s see why these properties are necessary: (1) suppose h is not preimage resistant. Then Eve can do the following attack (we assume that RSA signature is used). Eve computes h = r e ( mod n), for some random integer r. Eve also computes the pre-image of h under h, i.e., Eve computes m such that h(m) = h. Eve can do this under assumption (1). 9

10 Eve now has the signed message (m, r). In other words, Eve has first produced the digest, and then a message having that digest. Such a forgery is called an existential forgery in that the attacker may not have any control over the content of the message for which she has obtained a valid signature. A protocol allowing existential forgery is still considered to have a weakness. (2) suppose h is not weak collision resistant. Eve can do the following: Eve obtains your signature (m, s) on message m. Eve finds another message m with h(m) = h(m ). Eve now has a valid (m, s) on the message m. (3) suppose h is not strong collision resistant. The legitimate signer can do the following: the signer chooses two messages m and m with h(m) = h(m ). the signer signs m (using the digest h(m)) and sends (m, s). later the signer repudiates this signature claiming it was a signature on m. 10

11 The Birthday Paradox and attacks based on it 30 people are in a room. What is the probability that some pair of them have the same birthday? Answer: /2? For what (smallest) value of n people in a room is the above probability at least Answer: 23 Why: let s estimate the probability that no pair of people have the same birthday = So, Prob (there are 2 people with same birthday) = = Let s look at the general case. Suppose that there are n types of objects labelled {1, 2,..., n} and we randomly pick k objects (there is an unlimited supply of each type of object). Let P (n, k) be the probability that there is at least one duplicate. We will use the inequality (1 x) e x, for all x 0, which can easily be proved using some Calculus I stuff. 11

12 Then P (n, k) = 1 n 1 n 2 n n... n k + 1 n = 1 (1 1 n )(1 2 n )... (1 k 1 n ) > 1 e 1/n e 2/n e (k 1)/n = 1 e [1/n+2/n+...+(k 1)/n] = 1 e (k(k 1)/(2n) 1 e k2 /(2n). Let s estimate for what k is P (n, k) = 1/2. We need to have e k2 /(2n) = 1/2 k 2 /(2n) = ln 2 k = 2(ln 2)n = 1.18 n n. The rule of a thumb is that for k n, chances of a collision are approx. 1/2. If a hash functions produces digest of length m bits, then there are n = 2 m possible digests. If we randomly pick n = 2 m/2 random messages, there is a 1/2 chance that two of them collide. So if we want that the effort of the attacker is at least 2 80, we need to make the length of the digest m = 160. Also note that if k = 2λn for some λ, then the probability that there is a duplicate is 1 e λ. 12

13 For the birthday attack (to be presented next), the relevant situation is a follows: there are n types of objects, and an unlimited supply of objects of each type. there is a group 1 of k random choices of objects there is a group 2 of k random choices of objects let P (n, k) be the probability that a choice in group 1 = a choice in group 2. if k = λn, then P (n, k) 1 e λ. 13

14 Birthday paradox attack on signature schemes Recall the basic scheme m h(m) sign(h(m)). Suppose that digest h(m) is 50 bits long. Suppose m is a sale contract and Bob (the real-estate agent) asks Alice (the buyer) to sign it. m is a good contract and Alice wants to sign it. She thinks that Bob may change m to a bad contract m so that h(m) = h(m ), in which case sign(h(m)) = sign(h(m )). If h is a good hash function (looking like a random function) the chances that this happens are 1/2 50, which is negligible, so Alice is not too worried. However Bob does the following: He finds 30 places in the contract where he can make a small change. In this way he prepares: group 1: 2 30 contracts (essentially the same) that are good for Alice. group 2: 2 30 contracts (essentially the same) that are bad for Alice. So k = 2 30 and n = 2 50 and k = λn for λ = So with prob 1 e , there is m in group1 and m in group 2 that collide, i.e, h(m) = h(m ). Bob asks Alice to sign m, which she does, but then changes m with m. 14

15 Dear Anthony, This letter is I am writing to introduce you to to you Mr. -- Alfred P. -- Barton, the new newly appointed chief senior jewellery buyer for our the Northern European Europe area division. He will take has taken over the -- responsibility for all the whole of our interests in watches and jewellery jewellery and watches in the area region. Please afford give him every all the help he may need needs to seek out find the most modern up to date lines for the top high end of the market. He is empowered authorized to receive on our behalf samples specimens of the latest newest watch and jewellery jewellery and watch products, up subject to a limit maximum of ten thousand dollars. He will carry hold a signed copy of this letter document as proof of identity. An order with his signature, which is appended attached authorizes allows you to charge the cost to this company at the above head office address. We fully -- expect that our level volume of orders will increase in the following next year and trust hope that the new appointment will be prove advantageous an advantage to both our companies. Figure 11.8 A Letter in 2 37 Variations [DAVI89] 15

### 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

### Outline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures

Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike

### 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

### Digital Signatures. Good properties of hand-written signatures:

Digital Signatures Good properties of hand-written signatures: 1. Signature is authentic. 2. Signature is unforgeable. 3. Signature is not reusable (it is a part of the document) 4. Signed document is

### Signature Schemes. CSG 252 Fall 2006. Riccardo Pucella

Signature Schemes CSG 252 Fall 2006 Riccardo Pucella Signatures Signatures in real life have a number of properties They specify the person responsible for a document E.g. that it has been produced by

### Digital Signature. Raj Jain. Washington University in St. Louis

Digital Signature Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-11/

### Digital signatures. Informal properties

Digital signatures Informal properties Definition. A digital signature is a number dependent on some secret known only to the signer and, additionally, on the content of the message being signed Property.

### 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

### Cryptography Lecture 8. Digital signatures, hash functions

Cryptography Lecture 8 Digital signatures, hash functions A Message Authentication Code is what you get from symmetric cryptography A MAC is used to prevent Eve from creating a new message and inserting

### 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

### Crittografia e sicurezza delle reti. Digital signatures- DSA

Crittografia e sicurezza delle reti Digital signatures- DSA Signatures vs. MACs Suppose parties A and B share the secret key K. Then M, MAC K (M) convinces A that indeed M originated with B. But in case

### Computer Science 308-547A Cryptography and Data Security. Claude Crépeau

Computer Science 308-547A Cryptography and Data Security Claude Crépeau These notes are, largely, transcriptions by Anton Stiglic of class notes from the former course Cryptography and Data Security (308-647A)

### Digital Signatures. Prof. Zeph Grunschlag

Digital Signatures Prof. Zeph Grunschlag (Public Key) Digital Signatures PROBLEM: Alice would like to prove to Bob, Carla, David,... that has really sent them a claimed message. E GOAL: Alice signs each

### 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

### Authentication requirement Authentication function MAC Hash function Security of

UNIT 3 AUTHENTICATION Authentication requirement Authentication function MAC Hash function Security of hash function and MAC SHA HMAC CMAC Digital signature and authentication protocols DSS Slides Courtesy

### 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:

### 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

### Introduction to Computer Security

Introduction to Computer Security Hash Functions and Digital Signatures Pavel Laskov Wilhelm Schickard Institute for Computer Science Integrity objective in a wide sense Reliability Transmission errors

### 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

### 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

### Digital Signatures. What are Signature Schemes?

Digital Signatures Debdeep Mukhopadhyay IIT Kharagpur What are Signature Schemes? Provides message integrity in the public key setting Counter-parts of the message authentication schemes in the public

### Communications security

University of Roma Sapienza DIET Communications security Lecturer: Andrea Baiocchi DIET - University of Roma La Sapienza E-mail: andrea.baiocchi@uniroma1.it URL: http://net.infocom.uniroma1.it/corsi/index.htm

### Final Exam. IT 4823 Information Security Administration. Rescheduling Final Exams. Kerberos. Idea. Ticket

IT 4823 Information Security Administration Public Key Encryption Revisited April 5 Notice: This session is being recorded. Lecture slides prepared by Dr Lawrie Brown for Computer Security: Principles

### 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

### Digital Signature CHAPTER 13. Review Questions. (Solution to Odd-Numbered Problems)

CHAPTER 13 Digital Signature (Solution to Odd-Numbered Problems) Review Questions 1. We mentioned four areas in which there is a differences between a conventional and a digital signature: inclusion, verification,

### Practice Questions. CS161 Computer Security, Fall 2008

Practice Questions CS161 Computer Security, Fall 2008 Name Email address Score % / 100 % Please do not forget to fill up your name, email in the box in the midterm exam you can skip this here. These practice

### Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl

Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl www.crypto-textbook.com Chapter 10 Digital Signatures ver. October 29, 2009 These slides were prepared

### 1 Signatures vs. MACs

CS 120/ E-177: Introduction to Cryptography Salil Vadhan and Alon Rosen Nov. 22, 2006 Lecture Notes 17: Digital Signatures Recommended Reading. Katz-Lindell 10 1 Signatures vs. MACs Digital signatures

### Asymmetric Encryption. With material from Jonathan Katz, David Brumley, and Dave Levin

Asymmetric Encryption With material from Jonathan Katz, David Brumley, and Dave Levin Warmup activity Overview of asymmetric-key crypto Intuition for El Gamal and RSA And intuition for attacks Digital

### Asymmetric Cryptography. Mahalingam Ramkumar Department of CSE Mississippi State University

Asymmetric Cryptography Mahalingam Ramkumar Department of CSE Mississippi State University Mathematical Preliminaries CRT Chinese Remainder Theorem Euler Phi Function Fermat's Theorem Euler Fermat's Theorem

### Public Key Cryptography. Basic Public Key Cryptography

Public Key Cryptography EJ Jung Basic Public Key Cryptography public key public key? private key Alice Bob Given: Everybody knows Bob s public key - How is this achieved in practice? Only Bob knows the

### A New Generic Digital Signature Algorithm

Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

### Lecture Note 7 AUTHENTICATION REQUIREMENTS. Sourav Mukhopadhyay

Lecture Note 7 AUTHENTICATION REQUIREMENTS Sourav Mukhopadhyay Cryptography and Network Security - MA61027 In the context of communications across a network, the following attacks can be identified: 1.

### Introduction. Digital Signature

Introduction Electronic transactions and activities taken place over Internet need to be protected against all kinds of interference, accidental or malicious. The general task of the information technology

### Cryptographic Hash Functions Message Authentication Digital Signatures

Cryptographic Hash Functions Message Authentication Digital Signatures Abstract We will discuss Cryptographic hash functions Message authentication codes HMAC and CBC-MAC Digital signatures 2 Encryption/Decryption

### 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

### 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

### 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

### Implementation and Comparison of Various Digital Signature Algorithms. -Nazia Sarang Boise State University

Implementation and Comparison of Various Digital Signature Algorithms -Nazia Sarang Boise State University What is a Digital Signature? A digital signature is used as a tool to authenticate the information

### Capture Resilient ElGamal Signature Protocols

Capture Resilient ElGamal Signature Protocols Hüseyin Acan 1, Kamer Kaya 2,, and Ali Aydın Selçuk 2 1 Bilkent University, Department of Mathematics acan@fen.bilkent.edu.tr 2 Bilkent University, Department

### 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,

### Randomized Hashing for Digital Signatures

NIST Special Publication 800-106 Randomized Hashing for Digital Signatures Quynh Dang Computer Security Division Information Technology Laboratory C O M P U T E R S E C U R I T Y February 2009 U.S. Department

### Part VII. Digital signatures

Part VII Digital signatures CHAPTER 7: Digital signatures Digital signatures are one of the most important inventions/applications of modern cryptography. The problem is how can a user sign a message such

### Cryptography and Network Security

Cryptography and Network Security Spring 2012 http://users.abo.fi/ipetre/crypto/ Lecture 9: Authentication protocols, digital signatures Ion Petre Department of IT, Åbo Akademi University 1 Overview of

### CRYPTOGRAPHY IN NETWORK SECURITY

ELE548 Research Essays CRYPTOGRAPHY IN NETWORK SECURITY AUTHOR: SHENGLI LI INSTRUCTOR: DR. JIEN-CHUNG LO Date: March 5, 1999 Computer network brings lots of great benefits and convenience to us. We can

### MTAT.07.003 Cryptology II. Digital Signatures. Sven Laur University of Tartu

MTAT.07.003 Cryptology II Digital Signatures Sven Laur University of Tartu Formal Syntax Digital signature scheme pk (sk, pk) Gen (m, s) (m,s) m M 0 s Sign sk (m) Ver pk (m, s)? = 1 To establish electronic

### Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis

Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis by Susana Sin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master

### 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,

### Title Goes Here An Introduction to Modern Cryptography. Mike Reiter

Title Goes Here An Introduction to Modern Cryptography Mike Reiter 1 Cryptography Study of techniques to communicate securely in the presence of an adversary Traditional scenario Goal: A dedicated, private

### Overview of Cryptographic Tools for Data Security. Murat Kantarcioglu

UT DALLAS Erik Jonsson School of Engineering & Computer Science Overview of Cryptographic Tools for Data Security Murat Kantarcioglu Pag. 1 Purdue University Cryptographic Primitives We will discuss the

### Public Key Cryptography in Practice. c Eli Biham - May 3, 2005 372 Public Key Cryptography in Practice (13)

Public Key Cryptography in Practice c Eli Biham - May 3, 2005 372 Public Key Cryptography in Practice (13) How Cryptography is Used in Applications The main drawback of public key cryptography is the inherent

### Digital signatures are one of the most important inventions/applications of modern cryptography.

CHAPTER 7: DIGITAL SIGNATURES Digital signatures are one of the most important inventions/applications of modern cryptography. Part VII Digital signatures The problem is how can a user sign (electronically)

### 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

### 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

### Introduction to Cryptography, Part II

Introduction to Cryptography, Part II Mariana Raykova 1 Alice and Bob Alice wants to communicate securely with Bob (Cryptographers frequently speak of Alice and Bob instead of A and B... What key should

### Public Key Cryptography. Performance Comparison and Benchmarking

Public Key Cryptography Performance Comparison and Benchmarking Tanja Lange Department of Mathematics Technical University of Denmark tanja@hyperelliptic.org 28.08.2006 Tanja Lange Benchmarking p. 1 What

### Textbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures

Textbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝 陽 科 技 大 學 資 工

### CSC474/574 - Information Systems Security: Homework1 Solutions Sketch

CSC474/574 - Information Systems Security: Homework1 Solutions Sketch February 20, 2005 1. Consider slide 12 in the handout for topic 2.2. Prove that the decryption process of a one-round Feistel cipher

### 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

### Public Key Cryptography Overview

Ch.20 Public-Key Cryptography and Message Authentication I will talk about it later in this class Final: Wen (5/13) 1630-1830 HOLM 248» give you a sample exam» Mostly similar to homeworks» no electronic

### A Factoring and Discrete Logarithm based Cryptosystem

Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 11, 511-517 HIKARI Ltd, www.m-hikari.com A Factoring and Discrete Logarithm based Cryptosystem Abdoul Aziz Ciss and Ahmed Youssef Ecole doctorale de Mathematiques

### APNIC elearning: Cryptography Basics. Contact: esec02_v1.0

APNIC elearning: Cryptography Basics Contact: training@apnic.net esec02_v1.0 Overview Cryptography Cryptographic Algorithms Encryption Symmetric-Key Algorithm Block and Stream Cipher Asymmetric Key Algorithm

### Cryptosystems. Bob wants to send a message M to Alice. Symmetric ciphers: Bob and Alice both share a secret key, K.

Cryptosystems Bob wants to send a message M to Alice. Symmetric ciphers: Bob and Alice both share a secret key, K. C= E(M, K), Bob sends C Alice receives C, M=D(C,K) Use the same key to decrypt. Public

### 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

### Fast Batch Verification for Modular Exponentiation and Digital Signatures

An extended abstract of this paper appears in Advances in Cryptology Eurocrypt 98 Proceedings, Lecture Notes in Computer Science Vol. 1403, K. Nyberg ed., Springer-Verlag, 1998. This is the full version.

### DIGITAL SIGNATURES 1/1

DIGITAL SIGNATURES 1/1 Signing by hand COSMO ALICE ALICE Pay Bob \$100 Cosmo Alice Alice Bank =? no Don t yes pay Bob 2/1 Signing electronically Bank Internet SIGFILE } {{ } 101 1 ALICE Pay Bob \$100 scan

### Improved Online/Offline Signature Schemes

Improved Online/Offline Signature Schemes Adi Shamir and Yael Tauman Applied Math. Dept. The Weizmann Institute of Science Rehovot 76100, Israel {shamir,tauman}@wisdom.weizmann.ac.il Abstract. The notion

### 2. Cryptography 2.4 Digital Signatures

DI-FCT-UNL Computer and Network Systems Security Segurança de Sistemas e Redes de Computadores 2010-2011 2. Cryptography 2.4 Digital Signatures 2010, Henrique J. Domingos, DI/FCT/UNL 2.4 Digital Signatures

### Chapter 8 Security. IC322 Fall 2014. Computer Networking: A Top Down Approach. 6 th edition Jim Kurose, Keith Ross Addison-Wesley March 2012

Chapter 8 Security IC322 Fall 2014 Computer Networking: A Top Down Approach 6 th edition Jim Kurose, Keith Ross Addison-Wesley March 2012 All material copyright 1996-2012 J.F Kurose and K.W. Ross, All

### Cryptographic Algorithms and Key Size Issues. Çetin Kaya Koç Oregon State University, Professor http://islab.oregonstate.edu/koc koc@ece.orst.

Cryptographic Algorithms and Key Size Issues Çetin Kaya Koç Oregon State University, Professor http://islab.oregonstate.edu/koc koc@ece.orst.edu Overview Cryptanalysis Challenge Encryption: DES AES Message

### An Introduction to Digital Signature Schemes

An Introduction to Digital Signature Schemes Mehran Alidoost Nia #1, Ali Sajedi #2, Aryo Jamshidpey #3 #1 Computer Engineering Department, University of Guilan-Rasht, Iran m.alidoost@hotmail.com #2 Software

### Midterm Exam Solutions CS161 Computer Security, Spring 2008

Midterm Exam Solutions CS161 Computer Security, Spring 2008 1. To encrypt a series of plaintext blocks p 1, p 2,... p n using a block cipher E operating in electronic code book (ECB) mode, each ciphertext

### Chapter 7: Network security

Chapter 7: Network security Foundations: what is security? cryptography authentication message integrity key distribution and certification Security in practice: application layer: secure e-mail transport

### A New Efficient Digital Signature Scheme Algorithm based on Block cipher

IOSR Journal of Computer Engineering (IOSRJCE) ISSN: 2278-0661, ISBN: 2278-8727Volume 7, Issue 1 (Nov. - Dec. 2012), PP 47-52 A New Efficient Digital Signature Scheme Algorithm based on Block cipher 1

### 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

1 Introduction to Cryptography and Data Security 1 1.1 Overview of Cryptology (and This Book) 2 1.2 Symmetric Cryptography 4 1.2.1 Basics 4 1.2.2 Simple Symmetric Encryption: The Substitution Cipher...

### Some Identity Based Strong Bi-Designated Verifier Signature Schemes

Some Identity Based Strong Bi-Designated Verifier Signature Schemes Sunder Lal and Vandani Verma Department of Mathematics, Dr. B.R.A. (Agra), University, Agra-282002 (UP), India. E-mail- sunder_lal2@rediffmail.com,

### Message Authentication

Message Authentication message authentication is concerned with: protecting the integrity of a message validating identity of originator non-repudiation of origin (dispute resolution) will consider the

### ACTA UNIVERSITATIS APULENSIS No 13/2007 MATHEMATICAL FOUNDATION OF DIGITAL SIGNATURES. Daniela Bojan and Sidonia Vultur

ACTA UNIVERSITATIS APULENSIS No 13/2007 MATHEMATICAL FOUNDATION OF DIGITAL SIGNATURES Daniela Bojan and Sidonia Vultur Abstract.The new services available on the Internet have born the necessity of a permanent

### An Introduction to Cryptography as Applied to the Smart Grid

An Introduction to Cryptography as Applied to the Smart Grid Jacques Benoit, Cooper Power Systems Western Power Delivery Automation Conference Spokane, Washington March 2011 Agenda > Introduction > Symmetric

### Implementation of Elliptic Curve Digital Signature Algorithm

Implementation of Elliptic Curve Digital Signature Algorithm Aqeel Khalique Kuldip Singh Sandeep Sood Department of Electronics & Computer Engineering, Indian Institute of Technology Roorkee Roorkee, India

### 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

### Cryptography: Authentication, Blind Signatures, and Digital Cash

Cryptography: Authentication, Blind Signatures, and Digital Cash Rebecca Bellovin 1 Introduction One of the most exciting ideas in cryptography in the past few decades, with the widest array of applications,

### 1720 - Forward Secrecy: How to Secure SSL from Attacks by Government Agencies

1720 - Forward Secrecy: How to Secure SSL from Attacks by Government Agencies Dave Corbett Technical Product Manager Implementing Forward Secrecy 1 Agenda Part 1: Introduction Why is Forward Secrecy important?

### Cryptography and Network Security Digital Signature

Cryptography and Network Security Digital Signature Xiang-Yang Li Message Authentication Digital Signature Authentication Authentication requirements Authentication functions Mechanisms MAC: message authentication

### Message Authentication Codes (MACs)

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Introduction to Cryptography ECE 597XX/697XX Part 12 Message Authentication Codes (MACs) Israel Koren ECE597/697 Koren Part.12.1 Content

### 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

### Client Server Registration Protocol

Client Server Registration Protocol The Client-Server protocol involves these following steps: 1. Login 2. Discovery phase User (Alice or Bob) has K s Server (S) has hash[pw A ].The passwords hashes are

### 9/17/2015. Cryptography Basics. Outline. Encryption/Decryption. Cryptanalysis. Caesar Cipher. Mono-Alphabetic Ciphers

Cryptography Basics IT443 Network Security Administration Instructor: Bo Sheng Outline Basic concepts in cryptography system Secret cryptography Public cryptography Hash functions 1 2 Encryption/Decryption

### Authentication, digital signatures, PRNG

Multimedia Security Authentication, digital signatures, PRNG Mauro Barni University of Siena Beyond confidentiality Up to now, we have been concerned with protecting message content (i.e. confidentiality)

### 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

### In this paper a new signature scheme and a public key cryptotsystem are proposed. They can be seen as a compromise between the RSA and ElGamal-type sc

Digital Signature and Public Key Cryptosystem in a Prime Order Subgroup of Z n Colin Boyd Information Security Research Centre, School of Data Communications Queensland University of Technology, Brisbane

### CS 393 Network Security. Nasir Memon Polytechnic University Module 11 Secure Email

CS 393 Network Security Nasir Memon Polytechnic University Module 11 Secure Email Course Logistics HW 5 due Thursday Graded exams returned and discussed. Read Chapter 5 of text 4/2/02 Module 11 - Secure

### Cryptography and Network Security Chapter 11

Cryptography and Network Security Chapter 11 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 11 Cryptographic Hash Functions Each of the messages, like each

### 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

### 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

### ELECTRONIC POWER OF ATTORNEY PROTOCOL BY USING DIGITAL SIGNATURE ALGORITHM

ELECTRONIC POWER OF ATTORNEY PROTOCOL BY USING DIGITAL SIGNATURE ALGORITHM 1,3 WALIDATUSH SHOLIHAH, 2 SUGI GURITMAN, 3 HERU SUKOCO 1 Diploma Programme, Bogor Agricultural University 2 Mathematics Department,