This document discusses elliptic curve cryptography and defines some key concepts. It provides an example of a private key, public key, and generator point G for an elliptic curve. It then gives an example of finding the first 20 points on the elliptic curve y2=x3+7 modulo a prime number 38047. Finally, it states that in general, an elliptic curve is defined as y2=x3+ax+b modulo a prime number p.

1. Elliptic Curve Points in
Finite Fields
Prof Bill Buchanan, The Cyber Academy
http://asecuritysite.com

2. Elliptic Curves
Private key:
0xc9f4f55bdeb5ba0bd337f2dbc952a5439e20ef9a
f6203d25d014e7102d86aaeeL
Public key:
0xc44370819cb3b7b57b2aa7edf550a9a5410c234
d27aff497458bbbfec8b6a327,
0x52a1a3e222cd89cbd2764b69bd9b0ea5c4fd6ca
28861e1f2140eeff9c2e76487
G:
(506626302227734366957871889516853432625
0603453777594175500187360389116729240L,
3267051002075881697808308513050704318447
1273380659243275938904335757337482424L)

3. Elliptic Curves
Private key:
0xc9f4f55bdeb5ba0bd337f2dbc952a5439e20ef9a
f6203d25d014e7102d86aaeeL
Public key:
0xc44370819cb3b7b57b2aa7edf550a9a5410c234
d27aff497458bbbfec8b6a327,
0x52a1a3e222cd89cbd2764b69bd9b0ea5c4fd6ca
28861e1f2140eeff9c2e76487
G:
(506626302227734366957871889516853432625
0603453777594175500187360389116729240L,
3267051002075881697808308513050704318447
1273380659243275938904335757337482424L)

4. Finite Fields with Prime
• A: 0
• B: 7
• Prime number: 38047
• Elliptic curve is: y2=x3+ 7
• Finding the first 20 points
• (83, 33) (180, 104) (296, 156) (491, 81) (896,
81) (905, 155) (1307, 86) (1563, 41) (1570,
136) (1857, 56) (1904, 25) (2004, 119)
(2011, 170) (2209, 75) (2219, 90) (2447, 51)
(2767, 40) (2843, 170) (3138, 29) (3485, 33)
(3713, 109)
• [Link]

5. Real Elliptic Curves
• Elliptic curve is:
• y2=x3+ax+ b (mod p)

6. Real Elliptic Curves
• Elliptic curve is:
• y2=x3+ax+ b (mod p)

7. Real Elliptic Curves
• Elliptic curve is:
• y2=x3+ax+ b (mod p)

8. Real Elliptic Curves
• Elliptic curve is:
• y2=x3+ax+ b (mod p)

9. Real Elliptic Curves
• Elliptic curve is:
• y2=x3+ax+ b (mod p)

10. Elliptic Curve Points in
Finite Fields
Prof Bill Buchanan, The Cyber Academy
http://asecuritysite.com