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

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Course Information (Many Changes from Last Week) 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 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 Schedule 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 [1 st floor of this building] 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 [1 st floor of this building] Grading

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From Last Lecture… Scalar Multiplication on Elliptic Curve S = P + P + … + P = rP when r 1 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 P3P3P7P7P14P 6P6P2P2P 3 – 1 = 2 Point Additions 4 – 1 = 3 Point Doubles r times O Given P, aP - Compute a. Discrete Logarithm Problem

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Pollards Method [Pollard 1978] [Teske, 1998] (Semi-)Objective (Semi-) Algorithm (Real-)Objective Function f for Discrete Log (Real-)Algorithm

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Examples Example Algorithm

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Exercise Exercise 4

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The Pohlig-Hellman Method [Pohlig, Hellman 1978]

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The Pohlig-Hellman Method [cont.] Given P, Q = aP - Compute a. (Real-)Problem Given P, Q = aP - Compute a mod p k e k (Semi-)Problem Properties Algorithm

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The Pohlig-Hellman Method [cont.] Algorithm Given P, Q = aP - Compute a mod p k e k

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Chinese Remainder Theorem Given P, Q = aP - Compute a mod p k e k (Semi-)Problem Chinese Remainder Theorem

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Three-Pass Protocol [Shamir 1980] Private Key Cryptography Key Agreement Protocol kk M Encryption Algorithm E k (M) Decryption Algorithm D k (E k (M)) = M Three-pass Protocol k1k1 k2k2 M E k 1 (M) Encryption Algorithm E k 1 (M) Super-Encryption Algorithm E k 2 ( E k 1 (M)) Decryption Algorithm E k 2 (M)=D k 1 ( E k 2 ( E k 1 (M))) E k 2 (M) Super-Decryption Algorithm M

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Massey-Omura Protocol [Massey, Omura 1986] Three-pass Protocol k1k1 k2k2 M E k 1 (M) Encryption Algorithm E k 1 (M) Super-Encryption Algorithm E k 2 ( E k 1 (M)) Decryption Algorithm E k 2 (M) Super-Decryption Algorithm M Massey-Omura Protocol Encryption Algorithm Super-Encryption Algorithm Decryption Algorithm E k 2 (M) Super-Decryption Algorithm

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Massey-Omura Protocol [cont.] Massey-Omura Protocol Encryption Algorithm Super-Encryption Algorithm Decryption Algorithm E k 2 (M) Super-Decryption Algorithm Example Encryption Algorithm Super-Encryption Algorithm Decryption Algorithm Super-Decryption Algorithm

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Massey-Omura Protocol [cont.] Integer Point on Elliptic Curve Point on Elliptic Curve Integer Exercise 4Exercise 5

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Public Key Cryptography Private Key Cryptography Key Agreement Protocol kk M Encryption Algorithm E k (M) Decryption Algorithm D k (E k (M)) = M Public Key Cryptography k pub,k pri Certificate Authority (CA) k pub M Encryption Algorithm E k pub (M) Decryption Algorithm D k pri (E k pub (M)) = M

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ElGamal Public Key Encryption [ElGamal 1985] Public Key Cryptography k pub,k pri Certificate Authority (CA) k pub M Encryption Algorithm E k pub (M) Decryption Algorithm D k pri (E k pub (M)) = M Certificate Authority (CA) Encryption Algorithm E k pub (M) = M 1,M 2 M 1 = kP, M 2 = M + kB E k pub (M) = M 1,M 2 Decryption Algorithm D k pri (E k pub (M)) = M 2 -sM 1 = M ElGamal PKE

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ElGamal Public Key Encryption (cont.) Certificate Authority (CA) Encryption Algorithm E k pub (M) = M 1,M 2 M 1 = kP, M 2 = M + kB E k pub (M) = M 1,M 2 Decryption Algorithm D k pri (E k pub (M)) = M 2 -sM 1 = M ElGamal PKE Example Encryption Algorithm E k pub (M) = M 1,M 2 M 1 = kP = 7(0,1) = (4,3), M 2 = M + kB = (4,2)+7(3,1) = (0,1) E k pub (M) = M 1,M 2 M 1 = (4,3) M 2 = (0,1) Decryption Algorithm D k pri (E k pub (M)) = M 2 -sM 1 = (0,1)-5(4,3) = (4,2)

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ElGamal Public Key Encryption (cont.) Certificate Authority (CA) Encryption Algorithm E k pub (M) = M 1,M 2 M 1 = kP, M 2 = M + kB E k pub (M) = M 1,M 2 Decryption Algorithm D k pri (E k pub (M)) = M 2 -sM 1 = M ElGamal PKE Given P, sP (public key), kP, M + skP, Find M. ElGamal Problem Ver. I Given P, sP Find s. Discrete Log.

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Digital Signature [Diffie, Hellman 1976] Alice is sending a message M to Bob 1.Bob can be sure that the sender is really Alice. 2.Alice cannot refuse that she did send the message 3.No one can send a message claiming that they are Alice. Objective Digital Signature k pri,k pub Certificate Authority (CA) k pub M Signing Algorithm M,S k pri (M) Verification Algorithm V k pub (S k pri (M)) = M ? Public Key Cryptography k pub,k pri Certificate Authority (CA) k pub M Encryption Algorithm E k pub (M) Decryption Algorithm D k pri (E k pub (M)) = M

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ElGamal Digital Signatures [ElGamal 1985] Digital Signature k pri,k pub Certificate Authority (CA) k pub M Signing Algorithm M,S k pri (M) Verification Algorithm S k pri (M)) is signed by Alice??? ElGamals Protocol Certificate Authority (CA) k pub =(A,B) Signing Algorithm Verification Algorithm

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ElGamal Digital Signatures (cont.) ElGamals Protocol Certificate Authority (CA) k pub =(A,B) Signing Algorithm Verification Algorithm Example Signing Algorithm Verification Algorithm

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ElGamal Digital Signatures (cont.) ElGamals Protocol Certificate Authority (CA) k pub =(A,B) Signing Algorithm Verification Algorithm Given A, B=aA (public key), m (message), m (forged message) Find R,s such that ElGamal Problem Ver. II Given P, sP Find s. Discrete Log.

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

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Overview Discrete Logarithm Problem Massey- Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures Digital Signature Algorithm (DSA)

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Digital Signature Algorithm [Vanstone 1992] ElGamals Protocol Certificate Authority (CA) k pub =(A,B) Signing Algorithm Verification Algorithm DSAs Protocol Certificate Authority (CA) k pub =(A,B) Signing Algorithm Verification Algorithm 3 Scalar Multiplications 2 Scalar Multiplications

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Exercise Exercise 4 Exercise 5

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Exercise Exercise 6

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Pairing-Based Cryptography Bilinear Function Diffie-Hellman Exchange Protocol 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 P aP bP ALICEALICE ALICEALICE BOBBOB BOBBOB Three-Parties DHE ALICE BOBBOB BOBBOB CHALIECHALIE CHALIECHALIE a, aP b, bPc, cP bP aP cP ALICE BOBBOB BOBBOB CHALIECHALIE CHALIECHALIE a, aP, bP b, bP cP c, cP aP bcP abP acP Three-Parties DHE with Pairing ALICE BOBBOB BOBBOB CHALIECHALIE CHALIECHALIE a, aP b, bPc, cP bP aP cP bP cP aP

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Thank you for your attention Please feel free to ask questions or comment.

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