Lecture 9 Elliptic Curves. In 1984, Hendrik Lenstra described an ingenious algorithm for factoring integers that relies on properties of elliptic curves.

Slides:

Advertisements

Similar presentations

Public Key Cryptosystem

Advertisements

Are standards compliant Elliptic Curve Cryptosystems feasible on RFID?

Elliptic curve arithmetic and applications to cryptography By Uros Abaz Supervised by Dr. Shaun Cooper and Dr. Andre Barczak.

Advanced Information Security 4 Field Arithmetic

Efficient generation of cryptographically strong elliptic curves Shahar Papini Michael Krel Instructor : Barukh Ziv 1.

Elliptic Curve Cryptography (ECC) Mustafa Demirhan Bhaskar Anepu Ajit Kunjal.

CS470, A.SelcukElGamal Cryptosystem1 ElGamal Cryptosystem and variants CS 470 Introduction to Applied Cryptography Instructor: Ali Aydin Selcuk.

Dr. Lo’ai Tawalbeh Fall 2005 Chapter 10 – Key Management; Other Public Key Cryptosystems Dr. Lo’ai Tawalbeh Computer Engineering Department Jordan University.

ELECTRONIC PAYMENT SYSTEMSFALL 2001COPYRIGHT © 2001 MICHAEL I. SHAMOS Electronic Payment Systems Lecture 6 Epayment Security II.

ASYMMETRIC CIPHERS.

Introduction to Public Key Cryptography

-Anusha Uppaluri.  ECC- A set of algorithms for key generation, encryption and decryption (public key encryption technique)  ECC was introduced by Victor.

Lecture 8 Digital Signatures. This lecture considers techniques designed to provide the digital counterpart to a handwritten signature. A digital signature.

By Abhijith Chandrashekar and Dushyant Maheshwary.

Ipsita Sahoo 10IT61B05 School of Information Technology IIT Kharagpur October 29, 2011 E LLIPTIC C URVES IN C RYPTOGRAPHY.

Elgamal Public Key Encryption CSCI 5857: Encoding and Encryption.

Lecture slides prepared for “Computer Security: Principles and Practice”, 2/e, by William Stallings and Lawrie Brown, Chapter 21 “Public-Key Cryptography.

Elliptic Curve Cryptography

Lecture 10: Elliptic Curve Cryptography Wayne Patterson SYCS 653 Fall 2009.

Problems with symmetric (private-key) encryption 1) secure distribution of keys 2) large number of keys Solution to both problems: Public-key (asymmetric)

Lecture 7 Discrete Logarithms

Cryptography and Network Security Third Edition by William Stallings Lecture slides by Lawrie Brown.

Application of Elliptic Curves to Cryptography

Research on the Discrete Logarithm Problem Wang Ping Meng Xuemei

CS 627 Elliptic Curves and Cryptography Paper by: Aleksandar Jurisic, Alfred J. Menezes Published: January 1998 Presented by: Sagar Chivate.

Asymmetric Key Signatures David Evans and Samee Zahur CS4501, Fall 2015.

Applied Cryptography Spring 2015 Asymmetric ciphers.

Chapter 21 Public-Key Cryptography and Message Authentication.

Introduction to Modern Cryptography Sharif University Spring 2015 Data and Network Security Lab Sharif University of Technology Department of Computer.

Scott CH Huang COM5336 Cryptography Lecture 10 Elliptic Curve Cryptography Scott CH Huang COM 5336 Cryptography Lecture 10.

Cryptography and Network Security (CS435) Part Eight (Key Management)

Cryptography and Network Security Chapter 10 Fifth Edition by William Stallings Lecture slides by Lawrie Brown.

Elliptical Curve Cryptography Manish Kumar Roll No - 43 CS-A, S-7 SOE, CUSAT.

Public Key Cryptography. symmetric key crypto requires sender, receiver know shared secret key Q: how to agree on key in first place (particularly if.

1 Public-Key Cryptography and Message Authentication.

Cryptography and Network Security Chapter 13 Fifth Edition by William Stallings Lecture slides by Lawrie Brown.

PUBLIC-KEY CRYPTOGRAPH IT 352 : Lecture 2- part3 Najwa AlGhamdi, MSc – 2012 /1433.

Chapter 3 (B) – Key Management; Other Public Key Cryptosystems.

Cryptography and Network Security Key Management and Other Public Key Cryptosystems.

Scott CH Huang COM 5336 Cryptography Lecture 6 Public Key Cryptography & RSA Scott CH Huang COM 5336 Cryptography Lecture 6.

Understanding Cryptography by Christof Paar and Jan Pelzl These slides were prepared by Tim Güneysu, Christof Paar and Jan Pelzl.

Elliptic Curve Cryptography

Cryptography and Network Security

Public Key Cryptosystem Introduced in 1976 by Diffie and Hellman [2] In PKC different keys are used for encryption and decryption 1978: First Two Implementations.

Elliptic Curve Cryptography Celia Li Computer Science and Engineering November 10, 2005.

Cryptography issues – elliptic curves Presented by Tom Nykiel.

Lecture 11: Elliptic Curve Cryptography Wayne Patterson SYCS 653 Fall 2008.

1 Network Security Dr. Syed Ismail Shah

Introduction to Pubic Key Encryption CSCI 5857: Encoding and Encryption.

1 Cryptanalysis Lab Elliptic Curves. Cryptanalysis Lab Elliptic Curves 2 Outline [1] Elliptic Curves over R [2] Elliptic Curves over GF(p) [3] Properties.

Introduction to Elliptic Curve Cryptography CSCI 5857: Encoding and Encryption.

Elgamal Public Key Encryption CSCI 5857: Encoding and Encryption.

Cryptography Hyunsung Kim, PhD University of Malawi, Chancellor College Kyungil University February, 2016.

Efficient Montgomery Modular Multiplication Algorithm Using Complement and Partition Techniques Speaker: Te-Jen Chang.

Motivation Basis of modern cryptosystems

Key Management public-key encryption helps address key distribution problems have two aspects of this: – distribution of public keys – use of public-key.

Public Key Cryptography. Asymmetric encryption is a form of cryptosystem in which Encryption and decryption are performed using the different keys—one.

Information Security Lab. Dept. of Computer Engineering 251/ 278 PART II Asymmetric Ciphers Key Management; Other CHAPTER 10 Key Management; Other Public.

Giuseppe Bianchi Lecture 8: Elliptic Curve Crypto A (minimal) introduction.

Elliptic Curve Public Key Cryptography Why ? ● ECC offers greater security for a given key size. ● The smaller key size also makes possible much more compact.

Asymmetric-Key Cryptography

Public Key Cryptosystem

Network Security Design Fundamentals Lecture-13

D. Cheung – IQC/UWaterloo, Canada D. K. Pradhan – UBristol, UK

RSA and El Gamal Cryptosystems

The Application of Elliptic Curves Cryptography in Embedded Systems

Lattices. Svp & cvp. lll algorithm. application in cryptography

Cryptology Design Fundamentals

Network Security Design Fundamentals Lecture-13

Presentation transcript:

Lecture 9 Elliptic Curves

In 1984, Hendrik Lenstra described an ingenious algorithm for factoring integers that relies on properties of elliptic curves. This discovery prompted researchers to investigate other applications of elliptic curves in cryptography and computational number theory.

Elliptic curve cryptography (ECC) was discovered in 1985 by Neal Koblitz and Victor Miller. Elliptic curve cryptographic schemes are public-key mechanisms that provide the same functionality as RSA schemes. However, their security is based on the hardness of a different problem, namely the elliptic curve discrete logarithm problem (ECDLP).

Currently the best algorithms known to solve the ECDLP have fully exponential running time, in contrast to the subexponential-time algorithms known for the integer factorization problem. This means that a desired security level can be attained with significantly smaller keys in elliptic curve systems than is possible with their RSA counterparts.

For example, it is generally accepted that a 160-bit elliptic curve key provides the same level of security as a 1024-bit RSA key. The advantages that can be gained from smaller key sizes include speed and efficient use of power, bandwidth, and storage.

Outline  Weierstrass Equation  Elliptic Curves over R  Elliptic Curves over Finite Field  Elliptic Curve Cryptosystems  Factoring with Elliptic Curves

1 Weierstrass Equation

2 Elliptic Curves Over R 2.1 Simplified Weierstrass Equations

2.2 Elliptic Curves over R

2.3 Addition Law

Chord-and-Tangent Rule

Chord-and-Tangent Rule (Continued)

Algebraic Formulas

3 Elliptic Curves over Finite Field 3.1 Elliptic Curves Mod p, p≠2, Addition Law

3.1.2 Example

3.2 Elliptic Curves over GF(2 n )

3.2.1Simplified Weierstrass Equations

3.2.2 Group law

3.2.3 Example

3.3 Number of Points

3.4 Discrete Logarithms on Elliptic Curves

4 Elliptic Curve Cryptosystems 4.1 Representing Plaintext

4.2 An Elliptic Curve ElGamal Cryptosystem

4.3 An Elliptic Curve Digital Signature Algorithm (ECDSA)

5 Factoring with Elliptic Curves 5.1 The Elliptic Curve Factoring Algorithm

5.2 Degenerate Curves

Thank You!