Download presentation

Presentation is loading. Please wait.

Published byBrittney Dimit Modified over 3 years ago

1
Discrete Methods in Mathematical Informatics Lecture 3: Other Applications of Elliptic Curve 23h October 2012 Vorapong Suppakitpaisarn Eng. 6 Room 363 Download: Lecture 1: Lecture 2: Lecture 3:

2
**Course Information (Many Changes from Last Week)**

Schedule Grading 10/9 – Elliptic Curve I (2 Exercises) (What is Elliptic Curve?) 10/16 – Elliptic Curve II (1 Exercises) (Elliptic Curve Cryptography[1]) 10/23 – Elliptic Curve III (3 Exercises) (Elliptic Curve Cryptography[2]) 10/30 – Cancelled 11/7 – Online Algorithm I (Prof. Han) 11/14 – Online Algorithm II (Prof. Han) 11/21 – Elliptic Curve IV (2 Exercises) (ECC Implementation I) 11/28 – Elliptic Curve V (2 Exercises) (ECC Implementation II) 12/4 – Cancelled From 12/11 – To be Announced For my part, you need to submit 2 Reports. Report 1: Select 3 from 6 exercises in Elliptic Curve I – III Submission Deadline: 14 November Report 2: Select 2 from 4 exercises in Elliptic Curve IV – V Submission Deadline: TBD Submit your report at Department of Mathematical Informatics’ office [1st floor of this building]

3
**Discrete Logarithm Problem**

From Last Lecture… Scalar Multiplication on Elliptic Curve S = P + P + … + P = rP when r1 is positive integer, S,P is a member of the curve Double-and-add method Let r = 14 = (01110)2 Compute rP = 14P r = 14 = ( )2 r times P 3P 7P 14P O 2P 6P 14P 3 – 1 = 2 Point Additions 4 – 1 = 3 Point Doubles Discrete Logarithm Problem Given P, aP - Compute a.

4
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

5
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

6
**Pollard’s Method [Pollard 1978]**

(Semi-)Objective [Teske, 1998] (Real-)Algorithm (Semi-) Algorithm (Real-)Objective Function f for Discrete Log

7
Examples Algorithm Example

8
Exercise Exercise 4

9
**The Pohlig-Hellman Method [Pohlig, Hellman 1978]**

10
**The Pohlig-Hellman Method [cont.]**

Algorithm (Real-)Problem Given P, Q = aP - Compute a. (Semi-)Problem Given P, Q = aP - Compute a mod pkek Properties

11
**The Pohlig-Hellman Method [cont.]**

Given P, Q = aP - Compute a mod pkek Algorithm

12
**Chinese Remainder Theorem**

(Semi-)Problem Given P, Q = aP - Compute a mod pkek

13
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

14
**Three-Pass Protocol [Shamir 1980]**

Private Key Cryptography Three-pass Protocol k1 k2 M Key Agreement Protocol Encryption Algorithm k k Ek1(M) Ek1 (M) Super-Encryption Algorithm M Dk(Ek(M)) = M Ek2 ( Ek1 (M)) Encryption Algorithm Ek2 ( Ek1 (M)) Decryption Algorithm Decryption Algorithm Ek(M) Ek(M) Ek2 (M)=Dk1 ( Ek2 ( Ek1 (M))) Ek2(M) Super-Decryption Algorithm M

15
**Massey-Omura Protocol [Massey, Omura 1986]**

Three-pass Protocol Massey-Omura Protocol k1 k2 M Encryption Algorithm Encryption Algorithm Ek1(M) Ek1 (M) Super-Encryption Algorithm Super-Encryption Algorithm Ek2 ( Ek1 (M)) Ek2 ( Ek1 (M)) Decryption Algorithm Decryption Algorithm Ek2(M) Ek2(M) Super-Decryption Algorithm Super-Decryption Algorithm M

16
**Massey-Omura Protocol [cont.]**

Example Encryption Algorithm Encryption Algorithm Super-Encryption Algorithm Super-Encryption Algorithm Decryption Algorithm Ek2(M) Decryption Algorithm Super-Decryption Algorithm Super-Decryption Algorithm

17
**Massey-Omura Protocol [cont.]**

Integer Point on Elliptic Curve Point on Elliptic Curve Integer Exercise 4 Exercise 5

18
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

19
**Public Key Cryptography**

Private Key Cryptography Public Key Cryptography Certificate Authority (CA) Key Agreement Protocol kpub,kpri kpub k k Dkpri (Ekpub (M)) = M M M Dk(Ek(M)) = M Encryption Algorithm Encryption Algorithm Decryption Algorithm Decryption Algorithm Ekpub(M) Ekpub (M) Ek(M) Ek(M)

20
**ElGamal Public Key Encryption [ElGamal 1985]**

Public Key Cryptography ElGamal PKE Certificate Authority (CA) Certificate Authority (CA) kpub,kpri kpub Dkpri (Ekpub (M)) = M2-sM = M Dkpri (Ekpub (M)) = M M Encryption Algorithm Encryption Algorithm Decryption Algorithm Decryption Algorithm Ekpub(M) = M1,M2 Ekpub(M) = M1,M2 Ekpub(M) Ekpub (M) M1 = kP, M2 = M + kB

21
**ElGamal Public Key Encryption (cont.)**

Example ElGamal PKE Certificate Authority (CA) Dkpri (Ekpub (M)) = M2-sM = M Dkpri (Ekpub (M)) = M2-sM1 = (0,1)-5(4,3) = (4,2) Encryption Algorithm Encryption Algorithm Decryption Algorithm Decryption Algorithm Ekpub(M) = M1,M2 Ekpub(M) = M1,M2 Ekpub(M) = M1,M2 Ekpub(M) = M1,M2 M1 = (4,3) M2 = (0,1) M1 = kP, M2 = M + kB M1 = kP = 7(0,1) = (4,3), M2 = M + kB = (4,2)+7(3,1) = (0,1)

22
**ElGamal Public Key Encryption (cont.)**

ElGamal PKE ElGamal Problem Ver. I Given P, sP (public key), kP, M + skP, Find M. Certificate Authority (CA) Dkpri (Ekpub (M)) = M2-sM = M Discrete Log. Given P, sP Find s. Encryption Algorithm Decryption Algorithm Ekpub(M) = M1,M2 Ekpub(M) = M1,M2 M1 = kP, M2 = M + kB

23
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

24
**Digital Signature [Diffie, Hellman 1976]**

Public Key Cryptography Digital Signature Certificate Authority (CA) Certificate Authority (CA) kpub,kpri kpub kpri,kpub kpub Dkpri (Ekpub (M)) = M M Encryption Algorithm Decryption Algorithm Vkpub (Skpri(M)) = M ? M Ekpub(M) Ekpub (M) Signing Algorithm Objective Verification Algorithm Alice is sending a message M to Bob Bob can be sure that the sender is really Alice. Alice cannot refuse that she did send the message No one can send a message claiming that they are Alice. M,Skpri(M) M, Skpri(M)

25
**ElGamal Digital Signatures [ElGamal 1985]**

ElGamal’s Protocol Certificate Authority (CA) Certificate Authority (CA) kpub=(A,B) kpri,kpub kpub Signing Algorithm Skpri(M)) is signed by Alice??? M Signing Algorithm Verification Algorithm Verification Algorithm M,Skpri(M) M, Skpri(M)

26
**ElGamal Digital Signatures (cont.)**

Example ElGamal’s Protocol Certificate Authority (CA) kpub=(A,B) Signing Algorithm Signing Algorithm Verification Algorithm Verification Algorithm

27
**ElGamal Digital Signatures (cont.)**

ElGamal’s Protocol ElGamal Problem Ver. II Given A, B=aA (public key), m (message), m‘ (forged message) Find R,s such that Certificate Authority (CA) kpub=(A,B) Signing Algorithm Discrete Log. Given P, sP Find s. Verification Algorithm

28
**Exercise Given A, B=aA (public key), m (message), m‘ (forged message)**

ElGamal Problem Ver. II Given A, B=aA (public key), m (message), m‘ (forged message) Find R,s such that Discrete Log. Given P, sP Find s. Exercise 6

29
**Overview Discrete Logarithm Problem Massey-Omura Encryption**

ElGamal Public Key Encryption Digital Signature Algorithm (DSA) ElGamal Digital Signatures

30
**Digital Signature Algorithm [Vanstone 1992]**

ElGamal’s Protocol DSA’s Protocol Certificate Authority (CA) Certificate Authority (CA) kpub=(A,B) kpub=(A,B) 2 Scalar Multiplications 3 Scalar Multiplications Signing Algorithm Signing Algorithm Verification Algorithm Verification Algorithm

31
Exercise Exercise 4 Exercise 4 Exercise 5

32
Exercise Exercise 6

33
**Pairing-Based Cryptography**

Diffie-Hellman Exchange Protocol Three-Parties DHE P 1. Generate P 2 E(F) 2. Generate positive integers a 3. Receive Q = bP 4. Compute aQ = abP 1. Receive P 2. Receive S = aP 3. Generate positive integer b 4. Compute bS = abP B O ALICE A L I C E aP a, aP bP aP C H A L I E bP B O b, bP cP c, cP Bilinear Function ALICE Three-Parties DHE with Pairing a, aP, bP ALICE abP C H A L I E bcP a, aP B O C H A L I E b, bP cP acP c, cP aP aP aP bP cP B O b, bP cP c, cP bP

34
**Thank you for your attention**

Please feel free to ask questions or comment.

Similar presentations

Presentation is loading. Please wait....

OK

Public Key Model 8. Cryptography part 2.

Public Key Model 8. Cryptography part 2.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian defence forces Ppt on addition of integers Ppt on review writing for kids Ppt on brand marketing manager Ppt on producers consumers and decomposers in the desert Ppt on statistics in maths pie Ppt on aggregate production planning Ppt on earthquake in japan Well made play ppt on dvd Ppt on revolution of the earth and seasons solar