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Slide 1 - Elliptic Curve Cryptography Part 1  Cryptography 1
Slide 2 - Elliptic Curve Crypto (ECC) “Elliptic curve” is not a cryptosystem Elliptic curves are a different way to do the math in public key system Elliptic curve versions of DH, RSA, etc. Elliptic curves may be more efficient Fewer bits needed for same security But the operations are more complex Part 1  Cryptography 2
Slide 3 - What is an Elliptic Curve? An elliptic curve E is the graph of an equation of the form y2 = x3 + ax + b Also includes a “point at infinity” What do elliptic curves look like? See the next slide! Part 1  Cryptography 3
Slide 4 - Elliptic Curve Picture Consider elliptic curve E: y2 = x3 - x + 1 If P1 and P2 are on E, we can define P3 = P1 + P2 as shown in picture Addition is all we need Part 1  Cryptography 4 P1 P2 P3 x y
Slide 5 - Points on Elliptic Curve Consider y2 = x3 + 2x + 3 (mod 5) x = 0  y2 = 3  no solution (mod 5) x = 1  y2 = 6 = 1  y = 1,4 (mod 5) x = 2  y2 = 15 = 0  y = 0 (mod 5) x = 3  y2 = 36 = 1  y = 1,4 (mod 5) x = 4  y2 = 75 = 0  y = 0 (mod 5) Then points on the elliptic curve are (1,1) (1,4) (2,0) (3,1) (3,4) (4,0) and the point at infinity:  Part 1  Cryptography 5
Slide 6 - Elliptic Curve Math Addition on: y2 = x3 + ax + b (mod p) P1=(x1,y1), P2=(x2,y2) P1 + P2 = P3 = (x3,y3) where x3 = m2 - x1 - x2 (mod p) y3 = m(x1 - x3) - y1 (mod p) And m = (y2-y1)(x2-x1)-1 mod p, if P1P2 m = (3x12+a)(2y1)-1 mod p, if P1 = P2 Special cases: If m is infinite, P3 = , and  + P = P for all P Part 1  Cryptography 6
Slide 7 - Elliptic Curve Addition Consider y2 = x3 + 2x + 3 (mod 5). Points on the curve are (1,1) (1,4) (2,0) (3,1) (3,4) (4,0) and  What is (1,4) + (3,1) = P3 = (x3,y3)? m = (1-4)(3-1)-1 = -32-1 = -3(3) = 1 (mod 5) x3 = 1 - 1 - 3 = 2 (mod 5) y3 = 1(1-2) - 4 = 0 (mod 5) On this curve, (1,4) + (3,1) = (2,0) Part 1  Cryptography 7
Slide 8 - ECC Diffie-Hellman Public: Elliptic curve and point (x,y) on curve Secret: Alice’s A and Bob’s B Part 1  Cryptography 8 Alice, A Bob, B A(x,y) B(x,y) Alice computes A(B(x,y)) Bob computes B(A(x,y)) These are the same since AB = BA
Slide 9 - ECC Diffie-Hellman Public: Curve y2 = x3 + 7x + b (mod 37) and point (2,5)  b = 3 Alice’s secret: A = 4 Bob’s secret: B = 7 Alice sends Bob: 4(2,7) = (7,32) Bob sends Alice: 7(2,7) = (18,35) Alice computes: 7(7,32) = (22,1) Bob computes: 4(18,35) = (22,1) Part 1  Cryptography 9
Slide 10 - Uses for Public Key Crypto Part 1  Cryptography 10
Slide 11 - Uses for Public Key Crypto Confidentiality Transmitting data over insecure channel Secure storage on insecure media Authentication (later) Digital signature provides integrity and non-repudiation No non-repudiation with symmetric keys Part 1  Cryptography 11
Slide 12 - Non-non-repudiation Alice orders 100 shares of stock from Bob Alice computes MAC using symmetric key Stock drops, Alice claims she did not order Can Bob prove that Alice placed the order? No! Since Bob also knows symmetric key, he could have forged message Problem: Bob knows Alice placed the order, but he can’t prove it Part 1  Cryptography 12
Slide 13 - Non-repudiation Alice orders 100 shares of stock from Bob Alice signs order with her private key Stock drops, Alice claims she did not order Can Bob prove that Alice placed the order? Yes! Only someone with Alice’s private key could have signed the order This assumes Alice’s private key is not stolen (revocation problem) Part 1  Cryptography 13
 

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