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Basic s of Elliptic Curve Cryptography

Basic s of Elliptic Curve Cryptography. İbrahim EBELER BİL617 / Spring ‘ 10. Content Overview. Elliptic Curve (Mathematical) Overview Key Development Encryption Scheme Why Elliptic Curve?. What is an elliptic curve?. A type of cubic curve General elliptic curve Over a field K

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Basic s of Elliptic Curve Cryptography

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  1. Basics of Elliptic Curve Cryptography İbrahim EBELER BİL617 / Spring ‘10

  2. Content Overview • Elliptic Curve (Mathematical) Overview • Key Development • Encryption Scheme • Why Elliptic Curve?

  3. What is an elliptic curve? • A type of cubic curve • General elliptic curve • Over a field K • Field Characteristic ¹ 2,3 • Can be expressed in the form: y2 = x3 + ax + b • Usually denoted as E(a,b) y2 = x3 - 4x + .67

  4. Elliptic curves defined by y2=x3+ax+b

  5. Law of Addition • P1 + P2 = P3

  6. Law of Addition • P1 + P2 = P3 • Establish P1 and P2

  7. Law of Addition • P1 + P2 = P3 • Establish P1 and P2 • Draw a line between the two

  8. Law of Addition • P1 + P2 = P3 • Establish P1 and P2 • Draw a line between the two • Let the intersect point be Q

  9. Law of Addition • P1 + P2 = P3 • Establish P1 and P2 • Draw a line between the two • Let the intersect point be Q • Drop down from Q vertically to find P3

  10. Law of Addition (Numerical)

  11. Law of Addition – Special Cases • Identity Element – ¥ • Adding a point to itself • Take a TANGENT line to the curve at that point • Now consider all of this Modulo a prime!

  12. Multiplication on Elliptic Curves • Multiplication is intuitive • Take a point P • 3P = (P + P) + P

  13. Key Generation • Alice chooses two large primes • Such that p º q º 2 (mod 3) • Alice calculates n = p * q • Alice calculates Nn = lcm( p+1, q+1 ) • Alice chooses e such that gcd( e, Nn ) = 1

  14. More Key Generation • Alice computes d such that… • e*d º 1 (mod Nn) • Alice’s Private Key : d, p, q, and Nn • Alice’s Public Key : n, e

  15. Encryption Scheme • Plaintext M = ( mx, my ) where mx, myÎ Zn • M must be on the Elliptic Curve En(0,b) • b is determined by M • Bob encrypts M to Alice • C = E(M) = e * M over En(0,b) • Bob sends the ciphertext C = ( cx, cy ) to Alice

  16. Decryption Scheme • Alice decrypts C from Bob • M = D(C) = d * C over En(0,b)

  17. Encryption Sample

  18. Encryption Sample

  19. Diffie-Hellman Key Exchange (E,P) • Public: Elliptic curve E and point P • Private • Alice: a • Bob: b • Agreed upon key is K=abP Alice Bob a b A=aP B=bP K=a(B)=abP=b(A)

  20. Why Elliptic Curve? • It seems so complex… • Why go to all the trouble…

  21. Comparison… • Lets look at RSA! • Widely accepted • Still used • Growing size of keys to accommodate increased computing power

  22. Key Size: Equivalent Strength Comparison

  23. Why Elliptic Curve? • It is strong for its size! • Easily implemented in embedded systems • NSA Suite B uses this for half of its algorithms

  24. Suite B Algorithms

  25. Basics of Elliptic Curve Cryptography İbrahim EBELER BİL617 / Spring ‘10

  26. Resources • “New Public-Key Schemes Based on Elliptic Curves over the Ring Zn” by Koyama et ali. • “The State of Elliptic Curve Cryptography” by Koblitz et ali. • MathWorld Online • ICSA Guide to Cryptography (Tables) • IEEE Standard 1364

  27. More Resources • Introduction to Cryptography with Coding Theory by Wade Trappe and Lawrence Washington This document is a reassessment over ECC2006 by Peter Hefley.

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