A Note on Point Multiplication on Supersingular Elliptic Curve over Ternary Fields

1. A Note on Point Multiplication on Supersingular Elliptic Curve over Ternary Fields Kwang Ho Kim The State Academy of Sciences, D.P.R. of Korea Cryptology ePrintArchive, 2007 Ha Donghun, Kim Hangyu

2. Contents • Overview of Elliptic Curve • Elliptic Curve Cryptosystem • Curve Arithmetic • New Algorithm

3. Overview of Elliptic Curve(1/2) • Elliptic Curve An elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form 1

4. Overview of Elliptic Curve(2/2) • Elliptic Curve Addition in Elliptic Curve Q -(P+Q) P P+Q 2

5. Elliptic Curve Cryptosystem Alice(sender) Message Bob(receiver) ※ private key: M public key: G, kG 3

6. Curve Arithmetic • Supersingular elliptic curve: • Arithmetic operations on the curve: 4

7. Curve Arithmetic • Projective 1.Ordinary projective 2.Lopez-Dahab projective 3.Jacobian projective 4.ML-projective ※ 1I(inversion) ≥ 15M(Multiplication), 1M ≥ 300C(Cubing) 5

8. New Algorithm(1/2) • Tripling ※Cost is 8C • Doubling ※Cost is 6M+4C 6

9. New Algorithm(2/2) • Addition ※Cost is 7M+3C 7

10. Thank You