Understanding_Cryptography_Chptr_9---ECC

Understanding_Cryptography_Chptr_9---ECC - Understanding...

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Understanding Cryptography by Christof Paar and Jan Pelzl www.crypto-textbook.com These slides were prepared by Tim Güneysu, Christof Paar and Jan Pelzl Chapter 9 – Elliptic Curve Cryptography ver. November 3rd, 2009
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The slides can be used free of charge. All copyrights for the slides remain with Christof Paar and Jan Pelzl. The title of the accompanying book “Understanding Cryptography” by Springer and the author’s names must remain on each slide. If the slides are modified, appropriate credits to the book authors and the book title must remain within the slides. It is not permitted to reproduce parts or all of the slides in printed form whatsoever without written consent by the authors. Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl U Some legal stuff (sorry): Terms of Use 2/24
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s Introduction s Computations on Elliptic Curves s The Elliptic Curve Diffie-Hellman Protocol s Security Aspects s Implementation in Software and Hardware 3/24 Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl U Content of this Chapter
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s Introduction s Computations on Elliptic Curves s The Elliptic Curve Diffie-Hellman Protocol s Security Aspects s Implementation in Software and Hardware Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl U Content of this Chapter 4/24
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Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl s Problem: Asymmetric schemes like RSA and Elgamal require exponentiations in integer rings and fields with parameters of more than 1000 bits. s High computational effort on CPUs with 32-bit or 64-bit arithmetic s Large parameter sizes critical for storage on small and embedded s Motivation: Smaller field sizes providing equivalent security are desirable s Solution: Elliptic Curve Cryptography uses a group of points (instead of integers) for cryptographic schemes with coefficient sizes of 160-256 bits, reducing significantly the computational effort. U Motivation 5/24
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s Introduction s Computations on Elliptic Curves s The Elliptic Curve Diffie-Hellman Protocol s Security Aspects s Implementation in Software and Hardware Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl U Content of this Chapter 6/24
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U Computations on Elliptic Curves Elliptic curves are polynomials that define points based on the (simplified) Weierstraß equation: y 2 = x 3 + ax + b for parameters a,b that specify the exact shape of the curve On the real numbers and with parameters a, b R, an elliptic curve looks like this b Elliptic curves can not just be defined over the real numbers R but over many other types of finite fields. Example : y 2 = x 3 -3x+3 over R Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl 7/24
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Chapter 9 of Understanding Cryptography by Christof Paar and Jan Pelzl U Computations on Elliptic Curves (ctd.) s In cryptography, we are interested in elliptic curves module a prime p : s Note that Z p = {0,1,…, p -1}
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Understanding_Cryptography_Chptr_9---ECC - Understanding...

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