Last Updated: Oct 14, 2017
|
|
- Dwain Sparks
- 6 years ago
- Views:
Transcription
1 RSA Last Updated: Oct 14, 2017
2 Recap Number thery What is a prime number? What is prime factrizatin? What is a GCD? What des relatively prime mean? What des c-prime mean? What des cngruence mean? What is the additive inverse f 13 % 17? What is the multiplicative inverse f 7 % 8?
3 Recap: Diffie-Hellman Yu re trapped in yur spaceship Yu have enugh energy t send a single message t yur HQ Yu have: HQ s public DH values g=5, p = g a = Yur AES implementatin frm Labs #1 & 2 An arbitrary precisin calculatr Hw can yu cnstruct yur message s that it will be safe frm eavesdrppers?
4 Asymmetric Encryptin
5 Public Key Terminlgy Public Key Private Key Digital Signature Yu encrypt with a public key, and yu decrypt with a private key Yu sign with a private key, and yu verify with a public key
6 Mdel fr Encryptin with Public Key Cryptgraphy Alice Bb s Public Key Bb Bb s Private Key Plaintext Ciphertext Plaintext Encryptin Algrithm Decryptin Algrithm
7 Mdel fr Digital Signature with Public Key Cryptgraphy Alice Alice s Private Key Bb Alice s Public Key Plaintext Signed Dcument Plaintext Signing Algrithm Signature Verificatin Algrithm
8 Histry f RSA Invented in 1977 by Rn Rivest Adi Shamir Lenard Adleman Patented until 2000 It s withstd years f extensive cryptanalysis Suggests a level f cnfidence in the algrithm Example f successful attacks against implementatins Side channel attacks Pr randm number generatrs
9 Textbk RSA m = message c = ciphertext e = public expnent d = private expnent n = mdulus RSA Encryptin c = m e % n RSA Decryptin m = c d % n
10 The Math Behind RSA RSA encrypt/decrypt peratins are simple The math t get t the pint where these peratins wrk is nt s simple (at first) Fermat s little therem Euler s generalizatin f Fermat s little therem
11 If Fermat s Little Therem p is prime a is relatively prime t p (c-prime) Then Fermat s therem states a p-1 º 1 (md p) fr all 0 < a < p This serves as the basis fr Fermat s primality test Which values f a aren t c-prime t p? Euler s generalizatin Pierre de Fermat ( )
12 Euler s Generalizatin f Fermat s Little Therem Euler said a phi(n) º 1 (md n) phi(n) Euler s ttient functin j(n) The number f values less than n which are relatively prime t n Multiplicative grup f integers (Z n *) n desn t need t be prime a must still be c-prime t n RSA is interested in values f n that are the prduct f tw prime numbers p and q Lenhard Euler ( )
13 Cmputing phi(n) in RSA phi(n) is the number f integers between 0 and n that are c-prime t n When p * q = n, and p and q are prime, what is the phi(n)? (p-1)(q-1) Prf (When p * q = n) Observatins 1) there are p-1 multiples f q between 1 and n 2) there are q-1 multiples f p between 1 and n These multiples are nt c-prime t n Definitin: phi(n) = # f values between 0 and n that are c-prime t n phi(n) = # f values between 0 and n minus # f values between 0 and n nt c-prime t n Why nt? phi(n) = [ n 1] [(p-1) + (q-1)] = [pq 1] (p-1) (q-1) = pq p q + 1 = (p-1)(q-1)
14 RSA Euler said: a phi(n) º 1 (md n) m (p-1)(q-1) º 1 (md n) Ntice: m (p-1)(q-1) * m º m (p-1)(q-1)+1 º m (md n) m phi(n)+1 º m (md n) Let e*d = k*phi(n) + 1 Then e*d º 1 (md phi(n)) Therefre m ed º m k*phi(n)+1 º m phi(n) *m phi(n) * * m º m (md n) RSA Encryptin m e = c (md n) RSA Decryptin c d = m (md n)
15 Steps fr RSA Encryptin Select p, q (large prime numbers) n=p*q phi(n) = (p-1)(q-1) Select integer e where e is relatively prime t phi(n) Cmmn values fr e are 3 and Why? Calculate d, where d*e = 1 (md phi(n)) Public key is KU = {e, n} Private key is KR = {d, n} RSA encryptin m e = c (md n) RSA decryptin c d = m (md n) Why is RSA Secure? Hard t factr large numbers Hard t cmpute d withut phi(n) Discrete lgs are hard (m d % n) Given signature, hard t find d
16 RSA Usage Given m e = c (md n) and c d = m (md n) What restrictins shuld be placed n m? Fr bulk encryptin (files, s, web pages, etc) Sme try using RSA as blck cipher Never, never, never encrypt data directly using RSA Inefficient Insecure Always use symmetric encryptin fr data, and use RSA t encrypt the symmetric key (after adding the apprpriate padding) Digital signatures D nt sign the entire dcument - t slw Sign ( encrypt ) a hash f the dcument using the private key The length f the hash is < n
17 Hw d we get p, q, e, & d? What is p? Hw d we get it? What is q? Hw d we get it? What is e? Hw d we get it? What is the relatinship f e and (p-1)(q-1)? What is d? Hw d we get it?
18 Multiplicative Inverses Use the extended Euclidean algrithm Based n the fact that GCD can be defined recursively If x > y, then GCD(x,y) = (recursively) GCD(y, x-y) Als if x > y, then GCD(x,y) = (recursively) GCD(y, x%y) GCD can als be used as fllws: Suppse ax + by = gcd(x,y) If x is the mdulus, and gcd (x,y) = 1 Then ax + by = 1 and b is y -1
19 Steps fr RSA Encryptin Select p, q (large prime numbers) n=p*q phi(n) = (p-1)(q-1) Select integer e where e is relatively prime t phi(n) Cmmn values fr e are 3 and Why? Calculate d, where d*e = 1 (md phi(n)) Public key is KU = {e, n} Private key is KR = {d, n} RSA encryptin m e = c (md n) RSA decryptin c d = m (md n)
20 Extended Euclidean algrithm We will use this algrithm t cmpute d Based n the fact that GCD can be defined recursively If x > y, then GCD(x,y) = (recursively) GCD(y, x-y) Als if x > y, then GCD(x,y) = (recursively) GCD(y, x%y)
21 GCD(x,y) = GCD(y, x%y) Apply this principle recursively GCD (120,23) 120 / 23 = 5 r 5 GCD ( 23, 5) 23 / 5 = 4 r 3 GCD ( 5, 3) 5 / 3 = 1 r 2 GCD ( 3, 2) 3 / 2 = 1 r 1 GCD ( 2, 1) 2 / 1 = 2 r 0 GCD is the last nn-zer remainder GCD is 1, 120 and 23 are c-prime
22 Sum f Prducts Each remainder r is equal t a sum f prducts f the riginal x and y GCD (x, y) = q remainder r, where r = x(1) + y(-q) GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(1) + 5(-4) 5 / 3 = 1 r 2 => 2 = 5(1) + 3(-1) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1) Ntice the first line is a sum f prducts invlving 120 and 23. We can derive a frmula fr each remainder t be a sum f prducts f 120,23.
23 Substitutin ** Substitute the equatin fr 5 the prir line and simplify t prduce a sum f prducts using the riginal numbers 120 and 23 t replace the sum f prducts f 23 and 5. GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(1) + 5(-4) ** 5 / 3 = 1 r 2 => 2 = 5(1) + 3(-1) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1) GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23 + [120(1) + 23(-5)] (-4) = 23 + (120(-4) + 23(20)) = 23(21) + 120(-4) 5 / 3 = 1 r 2 => 2 = 5(1) + 3(-1) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1)
24 Sum f Prducts ** Cntinue this pattern. Simplify the next line t be a sum f prducts f 120 and 23 using equatins already in the frm fr the prir remainders 5 and 3. Simplify. Be careful t get the signs crrect. Keep like terms (multiples) tgether. GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(21) + 120(-4) 5 / 3 = 1 r 2 => 2 = 5(1) + 3(-1) ** 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1) GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(21) + 120(-4) 5 / 3 = 1 r 2 => 2 = [120(1)+23(-5)](1) + [23(21)+120(-4)](-1) = 120(5) + 23(-26) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1)
25 Sum f Prducts GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(21) + 120(-4) 5 / 3 = 1 r 2 => 2 = 120(5) + 23(-26) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1) GCD (120, 23) 120 / 23 = 5 r 5 => 5 = 120(1) + 23(-5) 23 / 5 = 4 r 3 => 3 = 23(21) + 120(-4) 5 / 3 = 1 r 2 => 2 = 120(5) + 23(-26) 3 / 2 = 1 r 1 => 1 = [23(21) + 120(-4)](1) + [120(5) + 23(-26)](-1) = 23(47) + 120(-9) Ntice that 1 = 23* (-9) means that 47 is the multiplicative inverse f 23 (md 120)
26 Cmputing d Fr RSA, calculate GCD(phi(n), e) t find d using extended Euclidean algrithm (see handut n Lab #4 page) Manual iterative methd fr the exam Use the table methd in yur lab Fr RSA, the GCD(phi(n),e) will result in an equatin f the frm 1 = e*d + phi(n)*k Where d r k is negative If d is negative cnvert it t an equivalent psitive number (md n) using phi(n) + d
27 Steps fr determining d Calculate phi(n) and select an e (3, 5, 7 usually fr ur ty prblems) Find the GCD f (phi(n), e) which shuld always be 1 if e is valid. If the GCD is nt 1, repeat fr a different value f e. Create a sum f prducts expressin fr each remainder frm x/y f the frm [ remainder = X(1) + y(-qutient) ] Starting frm the first expressin, use substitutin t create an equatin f the frm: remainder = phi(n)*k + e(d) If d is negative, cnvert t a psitive number d = phi(n) + d
28 p=5, q=11, e=3 Practice
29 Practice p=5, q=11, e=3 n = p*q = 5*11 = 55 phi(n) = (p-1)(q-1) = 4*10 = 40 Calculate d GCD (phi(n), e) = GCD (40, 3) 40/ 3 = 13 r 1 => 1 = 40(1) + 3(-13) Since d is negative, d = phi(n) + d = 40 + (-13) = 27 Public Key = {3, 55} Private Key = {27, 55}
30 p=11, q=13, e= number <= 8 Practice
31 Practice p=11, q=13, e= number <= 8 n = p*q = 11*13 = 143 phi(n) = (p-1)(q-1) = 10*12 = 120 Calculate d GCD (phi(n),e) = GCD (120,3/5/7) ** nte that 3,5 nt c-prime t / 7 = 17 r 1 => 1 = 120(1) + 7(-17) Since d is negative, d = phi(n) + d = (-17) = 103 Public Key = {7, 143} Private Key = {103, 143}
32 p=5, q=13, e=5 Practice
33 Practice p=5, q=13, e=5 n = p*q = 5*13 = 65 phi(n) = (p-1)(q-1) = 4*12 = 48 Calculate d GCD (phi(n),e) = GCD (48,5) 48/ 5 = 9 r 3 => 3 = 48(1) + 5(-9) 5 / 3 = 1 r 2 => 2 = 5(1) + 3(-1) = 5(1) + [48(1) + 5(-9)](-1) 5(10) + 48(-1) 3 / 2 = 1 r 1 => 1 = 3(1) + 2(-1) = [48(1) + 5(-9)](1) + [5(10) + 48(-1)](-1) 48(2) + 5(-19) Since d is negative, d = phi(n) + d = 48 + (-19) = 29 Public Key = {5, 65} Private Key = {29, 65}
34 p=17, q=11, e=7 Practice
35 Practice p=17, q=11, e=7 n = p*q = 17*11 = 187 phi(n) = (p-1)(q-1) = 16*10 = 160 Calculate d GCD (phi(n), e) = GCD (160, 7) 160/ 7 = 22 r 6 => 6 = 160(1) + 7(-22) 7 / 6 = 1 r 1 => 1 = 7(1) + 6(-1) 7(1) + [160(1) + 7(-22)](-1) 160(-1) + 7(23) Public Key = {7, 187} Private Key = {23, 187}
Slides by Kent Seamons and Tim van der Horst Last Updated: Oct 1, 2013
RSA Slides by Kent Seamons and Tim van der Horst Last Updated: Oct 1, 2013 Recap Recap Number theory o What is a prime number? o What is prime factorization? o What is a GCD? o What does relatively prime
More informationPublic Key Cryptography. Tim van der Horst & Kent Seamons
Public Key Cryptgraphy Tim van der Hrst & Kent Seamns Last Updated: Oct 5, 2017 Asymmetric Encryptin Why Public Key Crypt is Cl Has a linear slutin t the key distributin prblem Symmetric crypt has an expnential
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationDefinition: For a positive integer n, if 0<a<n and gcd(a,n)=1, a is relatively prime to n. Ahmet Burak Can Hacettepe University
Number Theory, Public Key Cryptography, RSA Ahmet Burak Can Hacettepe University abc@hacettepe.edu.tr The Euler Phi Function For a positive integer n, if 0
More informationChapter 8 Public-key Cryptography and Digital Signatures
Chapter 8 Public-key Cryptography and Digital Signatures v 1. Introduction to Public-key Cryptography 2. Example of Public-key Algorithm: Diffie- Hellman Key Exchange Scheme 3. RSA Encryption and Digital
More informationPublic Key Algorithms
Public Key Algorithms 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/cse571-09/
More information5 th grade Common Core Standards
5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin
More informationRSA ENCRYPTION USING THREE MERSENNE PRIMES
Int. J. Chem. Sci.: 14(4), 2016, 2273-2278 ISSN 0972-768X www.sadgurupublications.com RSA ENCRYPTION USING THREE MERSENNE PRIMES Ch. J. L. PADMAJA a*, V. S. BHAGAVAN a and B. SRINIVAS b a Department of
More informationApplication Of Mealy Machine And Recurrence Relations In Cryptography
Applicatin Of Mealy Machine And Recurrence Relatins In Cryptgraphy P. A. Jytirmie 1, A. Chandra Sekhar 2, S. Uma Devi 3 1 Department f Engineering Mathematics, Andhra University, Visakhapatnam, IDIA 2
More informationThe RSA cryptosystem and primality tests
Mathematics, KTH Bengt Ek November 2015 Supplementary material for SF2736, Discrete mathematics: The RSA cryptosystem and primality tests Secret codes (i.e. codes used to make messages unreadable to outsiders
More informationPublic Key Cryptography
Public Key Cryptography Spotlight on Science J. Robert Buchanan Department of Mathematics 2011 What is Cryptography? cryptography: study of methods for sending messages in a form that only be understood
More informationCRYPTOGRAPHY AND NUMBER THEORY
CRYPTOGRAPHY AND NUMBER THEORY XINYU SHI Abstract. In this paper, we will discuss a few examples of cryptographic systems, categorized into two different types: symmetric and asymmetric cryptography. We
More informationRSA Algorithm. Factoring, EulerPhi, Breaking RSA. Çetin Kaya Koç Spring / 14
RSA Algorithm http://koclab.org Çetin Kaya Koç Spring 2018 1 / 14 Well-Known One-Way Functions Discrete Logarithm: Given p, g, and x, computing y in y = g x (mod p) is EASY Given p, g, y, computing x in
More informationCryptography. Course 1: Remainder: RSA. Jean-Sébastien Coron. September 21, Université du Luxembourg
Course 1: Remainder: RSA Université du Luxembourg September 21, 2010 Public-key encryption Public-key encryption: two keys. One key is made public and used to encrypt. The other key is kept private and
More informationMath 105: Review for Exam I - Solutions
1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all
More informationB. Definition of an exponential
Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.
More informationNumber Theory & Modern Cryptography
Number Theory & Modern Cryptography Week 12 Stallings: Ch 4, 8, 9, 10 CNT-4403: 2.April.2015 1 Introduction Increasing importance in cryptography Public Key Crypto and Signatures Concern operations on
More informationRSA. Ramki Thurimella
RSA Ramki Thurimella Public-Key Cryptography Symmetric cryptography: same key is used for encryption and decryption. Asymmetric cryptography: different keys used for encryption and decryption. Public-Key
More informationCryptography IV: Asymmetric Ciphers
Cryptography IV: Asymmetric Ciphers Computer Security Lecture 7 David Aspinall School of Informatics University of Edinburgh 31st January 2011 Outline Background RSA Diffie-Hellman ElGamal Summary Outline
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationdit-upm RSA Cybersecurity Cryptography
-upm Cybersecurity Cryptography José A. Mañas < http://www.dit.upm.es/~pepe/> Information Technology Department Universidad Politécnica de Madrid 4 october 2018 public key (asymmetric) public key secret
More informationAlgorithmic Number Theory and Public-key Cryptography
Algorithmic Number Theory and Public-key Cryptography Course 3 University of Luxembourg March 22, 2018 The RSA algorithm The RSA algorithm is the most widely-used public-key encryption algorithm Invented
More informationCIS 551 / TCOM 401 Computer and Network Security
CIS 551 / TCOM 401 Computer and Network Security Spring 2008 Lecture 15 3/20/08 CIS/TCOM 551 1 Announcements Project 3 available on the web. Get the handout in class today. Project 3 is due April 4th It
More informationFunction notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property
Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY
More informationLecture V : Public Key Cryptography
Lecture V : Public Key Cryptography Internet Security: Principles & Practices John K. Zao, PhD (Harvard) SMIEEE Amir Rezapoor Computer Science Department, National Chiao Tung University 2 Outline Functional
More informationPublic Key Cryptography
T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Public Key Cryptography EECE 412 1 What is it? Two keys Sender uses recipient s public key to encrypt Receiver uses his private key to decrypt
More informationLecture 1: Introduction to Public key cryptography
Lecture 1: Introduction to Public key cryptography Thomas Johansson T. Johansson (Lund University) 1 / 44 Key distribution Symmetric key cryptography: Alice and Bob share a common secret key. Some means
More informationAsymmetric Encryption
-3 s s Encryption Comp Sci 3600 Outline -3 s s 1-3 2 3 4 5 s s Outline -3 s s 1-3 2 3 4 5 s s Function Using Bitwise XOR -3 s s Key Properties for -3 s s The most important property of a hash function
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationCorrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec
U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.
More informationPublic Key 9/17/2018. Symmetric Cryptography Review. Symmetric Cryptography: Shortcomings (1) Symmetric Cryptography: Analogy
Symmetric Cryptography Review Alice Bob Public Key x e K (x) y d K (y) x K K Instructor: Dr. Wei (Lisa) Li Department of Computer Science, GSU Two properties of symmetric (secret-key) crypto-systems: The
More information5199/IOC5063 Theory of Cryptology, 2014 Fall
5199/IOC5063 Theory of Cryptology, 2014 Fall Homework 2 Reference Solution 1. This is about the RSA common modulus problem. Consider that two users A and B use the same modulus n = 146171 for the RSA encryption.
More informationNetwork Security Technology Spring, 2018 Tutorial 3, Week 4 (March 23) Due Date: March 30
Network Security Technology Spring, 2018 Tutorial 3, Week 4 (March 23) LIU Zhen Due Date: March 30 Questions: 1. RSA (20 Points) Assume that we use RSA with the prime numbers p = 17 and q = 23. (a) Calculate
More informationAddition. Ch1 - Algorithms with numbers. Multiplication. al-khwārizmī. al-khwārizmī. Division 53+35=88. Cost? (n number of bits) 13x11=143. Cost?
Ch - Algorithms with numbers Addition Basic arithmetic Addition ultiplication Division odular arithmetic factoring is hard Primality testing 53+35=88 Cost? (n number of bits) O(n) ultiplication al-khwārizmī
More informationMathematical Foundations of Public-Key Cryptography
Mathematical Foundations of Public-Key Cryptography Adam C. Champion and Dong Xuan CSE 4471: Information Security Material based on (Stallings, 2006) and (Paar and Pelzl, 2010) Outline Review: Basic Mathematical
More informationDepartment: MATHEMATICS
Cde: MATH 022 Title: ALGEBRA SKILLS Institute: STEM Department: MATHEMATICS Curse Descriptin: This curse prvides students wh have cmpleted MATH 021 with the necessary skills and cncepts t cntinue the study
More information8 th Grade Math: Pre-Algebra
Hardin Cunty Middle Schl (2013-2014) 1 8 th Grade Math: Pre-Algebra Curse Descriptin The purpse f this curse is t enhance student understanding, participatin, and real-life applicatin f middle-schl mathematics
More informationCryptography. P. Danziger. Transmit...Bob...
10.4 Cryptography P. Danziger 1 Cipher Schemes A cryptographic scheme is an example of a code. The special requirement is that the encoded message be difficult to retrieve without some special piece of
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationCOMP 551 Applied Machine Learning Lecture 11: Support Vector Machines
COMP 551 Applied Machine Learning Lecture 11: Supprt Vectr Machines Instructr: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted fr this curse
More informationMathematics of Cryptography
UNIT - III Mathematics of Cryptography Part III: Primes and Related Congruence Equations 1 Objectives To introduce prime numbers and their applications in cryptography. To discuss some primality test algorithms
More informationSecurity Issues in Cloud Computing Modern Cryptography II Asymmetric Cryptography
Security Issues in Cloud Computing Modern Cryptography II Asymmetric Cryptography Peter Schwabe October 21 and 28, 2011 So far we assumed that Alice and Bob both have some key, which nobody else has. How
More informationAnalysis of the RSA Encryption Algorithm
Analysis of the RSA Encryption Algorithm Betty Huang June 16, 2010 Abstract The RSA encryption algorithm is commonly used in public security due to the asymmetric nature of the cipher. The procedure is
More informationCryptography. pieces from work by Gordon Royle
Cryptography pieces from work by Gordon Royle The set-up Cryptography is the mathematics of devising secure communication systems, whereas cryptanalysis is the mathematics of breaking such systems. We
More informationRSA RSA public key cryptosystem
RSA 1 RSA As we have seen, the security of most cipher systems rests on the users keeping secret a special key, for anyone possessing the key can encrypt and/or decrypt the messages sent between them.
More informationChapter 4 Asymmetric Cryptography
Chapter 4 Asymmetric Cryptography Introduction Encryption: RSA Key Exchange: Diffie-Hellman [NetSec/SysSec], WS 2008/2009 4.1 Asymmetric Cryptography General idea: Use two different keys -K and +K for
More informationAsymmetric Cryptography
Asymmetric Cryptography Chapter 4 Asymmetric Cryptography Introduction Encryption: RSA Key Exchange: Diffie-Hellman General idea: Use two different keys -K and +K for encryption and decryption Given a
More informationMa/CS 6a Class 3: The RSA Algorithm
Ma/CS 6a Class 3: The RSA Algorithm By Adam Sheffer Reminder: Putnam Competition Signup ends Wednesday 10/08. Signup sheets available in all Sloan classrooms, Math office, or contact Kathy Carreon, kcarreon@caltech.edu.
More informationConcept Category 2. Trigonometry & The Unit Circle
Cncept Categry 2 Trignmetry & The Unit Circle Skill Checklist Use special right triangles t express values f fr the six trig functins Evaluate sine csine and tangent using the unit circle Slve tw-step
More informationRevisiting the Socrates Example
Sectin 1.6 Sectin Summary Valid Arguments Inference Rules fr Prpsitinal Lgic Using Rules f Inference t Build Arguments Rules f Inference fr Quantified Statements Building Arguments fr Quantified Statements
More informationMath.3336: Discrete Mathematics. Mathematical Induction
Math.3336: Discrete Mathematics Mathematical Induction Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationMath Foundations 10 Work Plan
Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame
More informationLemma 1.2. (1) If p is prime, then ϕ(p) = p 1. (2) If p q are two primes, then ϕ(pq) = (p 1)(q 1).
1 Background 1.1 The group of units MAT 3343, APPLIED ALGEBRA, FALL 2003 Handout 3: The RSA Cryptosystem Peter Selinger Let (R, +, ) be a ring. Then R forms an abelian group under addition. R does not
More informationPublic-key Cryptography and elliptic curves
Public-key Cryptography and elliptic curves Dan Nichols nichols@math.umass.edu University of Massachusetts Oct. 14, 2015 Cryptography basics Cryptography is the study of secure communications. Here are
More informationOverview. Background / Context. CSC 580 Cryptography and Computer Security. March 21, 2017
CSC 580 Cryptography and Computer Security Math for Public Key Crypto, RSA, and Diffie-Hellman (Sections 2.4-2.6, 2.8, 9.2, 10.1-10.2) March 21, 2017 Overview Today: Math needed for basic public-key crypto
More informationIntro to Public Key Cryptography Diffie & Hellman Key Exchange
Introduction to Modern Cryptography Lecture 5 Number Theory: 1. Quadratic residues. 2. The discrete log problem. Intro to Public Key Cryptography Diffie & Hellman Key Exchange Course Summary - Math Part
More informationPublic-Key Cryptosystems CHAPTER 4
Public-Key Cryptosystems CHAPTER 4 Introduction How to distribute the cryptographic keys? Naïve Solution Naïve Solution Give every user P i a separate random key K ij to communicate with every P j. Disadvantage:
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationSection 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law
Sectin 5.8 Ntes Page 1 5.8 Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential
More informationECEN 4872/5827 Lecture Notes
ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals
More informationPublic Key Cryptography. All secret key algorithms & hash algorithms do the same thing but public key algorithms look very different from each other.
Public Key Cryptography All secret key algorithms & hash algorithms do the same thing but public key algorithms look very different from each other. The thing that is common among all of them is that each
More informationTi Secured communications
Ti5318800 Secured communications Pekka Jäppinen September 20, 2007 Pekka Jäppinen, Lappeenranta University of Technology: September 20, 2007 Relies on use of two keys: Public and private Sometimes called
More informationIntroduction to Cryptography. Susan Hohenberger
Introduction to Cryptography Susan Hohenberger 1 Cryptography -- from art to science -- more than just encryption -- essential today for non-military applications 2 Symmetric Crypto Shared secret K =>
More informationCryptography and RSA. Group (1854, Cayley) Upcoming Interview? Outline. Commutative or Abelian Groups
Great Theoretical Ideas in CS V. Adamchik CS 15-251 Upcoming Interview? Lecture 24 Carnegie Mellon University Cryptography and RSA How the World's Smartest Company Selects the Most Creative Thinkers Groups
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationPublic Key Cryptography
Public Key Cryptography Ali El Kaafarani 1 Mathematical Institute 2 PQShield Ltd. 1 of 44 Outline 1 Public Key Encryption: security notions 2 RSA Encryption Scheme 2 of 44 Course main reference 3 of 44
More informationIndividual Events 5 I3 a 6 I4 a 8 I5 A Group Events
Answers: (993-94 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 SI a I a 6 I A Individual Events I3 a 6 I4 a 8 I A 6 b 4 b B 60 b *9 see the remarks b 0 B 36 c c 0 C c 6 c 3
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus
More informationReview. CS311H: Discrete Mathematics. Number Theory. Computing GCDs. Insight Behind Euclid s Algorithm. Using this Theorem. Euclidian Algorithm
Review CS311H: Discrete Mathematics Number Theory Instructor: Işıl Dillig What does it mean for two ints a, b to be congruent mod m? What is the Division theorem? If a b and a c, does it mean b c? What
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More information10 Public Key Cryptography : RSA
10 Public Key Cryptography : RSA 10.1 Introduction The idea behind a public-key system is that it might be possible to find a cryptosystem where it is computationally infeasible to determine d K even if
More informationEncryption: The RSA Public Key Cipher
Encryption: The RSA Public Key Cipher Michael Brockway March 5, 2018 Overview Transport-layer security employs an asymmetric public cryptosystem to allow two parties (usually a client application and a
More informationAlgebra2/Trig: Trig Unit 2 Packet
Algebra2/Trig: Trig Unit 2 Packet In this unit, students will be able t: Learn and apply c-functin relatinships between trig functins Learn and apply the sum and difference identities Learn and apply the
More informationCHM112 Lab Graphing with Excel Grading Rubric
Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline
More informationExperiment #3. Graphing with Excel
Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-
More informationTheme : Cryptography. Instructor : Prof. C Pandu Rangan. Speaker : Arun Moorthy CS
1 C Theme : Cryptography Instructor : Prof. C Pandu Rangan Speaker : Arun Moorthy 93115 CS 2 RSA Cryptosystem Outline of the Talk! Introduction to RSA! Working of the RSA system and associated terminology!
More informationPublic Key Cryptography
Public Key Cryptography Ali El Kaafarani Mathematical Institute Oxford University 1 of 74 Outline 1 Complexity measures 2 Algebra and Number Theory Background 3 Public Key Encryption: security notions
More informationIntroduction to Public-Key Cryptosystems:
Introduction to Public-Key Cryptosystems: Technical Underpinnings: RSA and Primality Testing Modes of Encryption for RSA Digital Signatures for RSA 1 RSA Block Encryption / Decryption and Signing Each
More informationMath 412: Number Theory Lecture 13 Applications of
Math 412: Number Theory Lecture 13 Applications of Gexin Yu gyu@wm.edu College of William and Mary Partition of integers A partition λ of the positive integer n is a non increasing sequence of positive
More informationNUMBER THEORY AND CODES. Álvaro Pelayo WUSTL
NUMBER THEORY AND CODES Álvaro Pelayo WUSTL Talk Goal To develop codes of the sort can tell the world how to put messages in code (public key cryptography) only you can decode them Structure of Talk Part
More informationLecture 5: Arithmetic Modulo m, Primes and Greatest Common Divisors Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 5: Arithmetic Modulo m, Primes and Greatest Common Divisors Lecturer: Lale Özkahya Resources: Kenneth Rosen,
More informationLecture Notes, Week 6
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467b: Cryptography and Computer Security Week 6 (rev. 3) Professor M. J. Fischer February 15 & 17, 2005 1 RSA Security Lecture Notes, Week 6 Several
More informationMa/CS 6a Class 4: Primality Testing
Ma/CS 6a Class 4: Primality Testing By Adam Sheffer Send anonymous suggestions and complaints from here. Email: adamcandobetter@gmail.com Password: anonymous2 There aren t enough crocodiles in the presentations
More information2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS
2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 6. An electrchemical cell is cnstructed with an pen switch, as shwn in the diagram abve. A strip f Sn and a strip f an unknwn metal, X, are used as electrdes.
More informationDiscrete Mathematics GCD, LCM, RSA Algorithm
Discrete Mathematics GCD, LCM, RSA Algorithm Abdul Hameed http://informationtechnology.pk/pucit abdul.hameed@pucit.edu.pk Lecture 16 Greatest Common Divisor 2 Greatest common divisor The greatest common
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationIntroduction to Modern Cryptography. Lecture RSA Public Key CryptoSystem 2. One way Trapdoor Functions
Introduction to Modern Cryptography Lecture 7 1. RSA Public Key CryptoSystem 2. One way Trapdoor Functions Diffie and Hellman (76) New Directions in Cryptography Split the Bob s secret key K to two parts:
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More information1 PreCalculus AP Unit G Rotational Trig (MCR) Name:
1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will
More informationCredits: 4 Lecture Hours: 4 Lab/Studio Hours: 0
Cde: MATH 025 Title: ELEMENTARY ALGEBRA Divisin: MATHEMATICS Department: MATHEMATICS Curse Descriptin: This curse is a review f elementary algebra and requires previus experience in algebra. The curse
More informationMa/CS 6a Class 4: Primality Testing
Ma/CS 6a Class 4: Primality Testing By Adam Sheffer Reminder: Euler s Totient Function Euler s totient φ(n) is defined as follows: Given n N, then φ n = x 1 x < n and GCD x, n = 1. In more words: φ n is
More informationLECTURE 5: APPLICATIONS TO CRYPTOGRAPHY AND COMPUTATIONS
LECTURE 5: APPLICATIONS TO CRYPTOGRAPHY AND COMPUTATIONS Modular arithmetics that we have discussed in the previous lectures is very useful in Cryptography and Computer Science. Here we discuss several
More informationCHAPTER 2 Algebraic Expressions and Fundamental Operations
CHAPTER Algebraic Expressins and Fundamental Operatins OBJECTIVES: 1. Algebraic Expressins. Terms. Degree. Gruping 5. Additin 6. Subtractin 7. Multiplicatin 8. Divisin Algebraic Expressin An algebraic
More informationIntroduction to Cybersecurity Cryptography (Part 4)
Introduction to Cybersecurity Cryptography (Part 4) Review of Last Lecture Blockciphers Review of DES Attacks on Blockciphers Advanced Encryption Standard (AES) Modes of Operation MACs and Hashes Message
More informationECE596C: Handout #11
ECE596C: Handout #11 Public Key Cryptosystems Electrical and Computer Engineering, University of Arizona, Loukas Lazos Abstract In this lecture we introduce necessary mathematical background for studying
More informationIntroduction to Cybersecurity Cryptography (Part 5)
Introduction to Cybersecurity Cryptography (Part 5) Prof. Dr. Michael Backes 13.01.2017 February 17 th Special Lecture! 45 Minutes Your Choice 1. Automotive Security 2. Smartphone Security 3. Side Channel
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More information