1. Basic Elliptic Curve Cryptography 1Lt Peter Hefley 90 OSS Instructor Fall ‘06

2. Content Overview Elliptic Curve 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 y 2 = x 3 + ax + b  Usually denoted E(a,b) y 2 = x 3 - 4x +.67

4. Law of Addition P 1 + P 2 = P 3

5. Law of Addition P 1 + P 2 = P 3 Establish P 1 and P 2

6. Law of Addition P 1 + P 2 = P 3 Establish P 1 and P 2 Draw a line between the two

7. Law of Addition P 1 + P 2 = P 3 Establish P 1 and P 2 Draw a line between the two Let the intersect point be Q

8. Law of Addition P 1 + P 2 = P 3 Establish P 1 and P 2 Draw a line between the two Let the intersect point be Q Drop down from Q vertically to find P 3

9. 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!

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

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

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

13. Encryption Scheme Plaintext M = ( m x, m y ) where m x, m y Î Z n  M must be on the Elliptic Curve E n (0,b)  b is determined by M Bob encrypts M to Alice  C = E(M) = e * M over E n (0,b) Bob sends the ciphertext C = ( c x, c y ) to Alice

14. Decryption Scheme Alice decrypts C from Bob  M = D(C) = d * C over E n (0,b)

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

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

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

18. Key Size: Equivalent Strength Comparison Time to Break (MIPS/Yrs) RSA/DSA Key Size ECC Key Size RSA:ECC Key Size Ratio 10 4 5121065:1 10 8 7681326:1 10 11 10241607:1 10 20 204821010:1 10 78 2100060035:1

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

20. Suite B Algorithms Encryption:AES (FIPS-197) Digital Signature:Elliptic Curve Digital Signature Algorithm (FIPS 186-2) Key Exchange:Elliptic Curve Diffie-Hellman or MQV (Draft NIST SP 800-56) Hashing:Secure Hash Algorithm (FIPS 180-2)

21. Basic Elliptic Curve Cryptography 1Lt Peter Hefley 90 OSS Instructor Fall ‘06 Peter.Hefley@warren.af.mil www.cryptografix.net Cell: (412) 721-7631

22. Resources “New Public-Key Schemes Based on Elliptic Curves over the Ring Z n ” by Koyama et ali. “The State of Elliptic Curve Cryptography” by Koblitz et ali. MathWorld Online

23. More Resources Introduction to Cryptography with Coding Theory by Wade Trappe and Lawrence Washington ICSA Guide to Cryptography (Tables) IEEE Standard 1364