Lecture 9 Elliptic Curves

1
Lecture 9 Elliptic Curves
2

• In 1984, Hendrik Lenstra described an
  ingenious algorithm for factoring integers that
  relies on properties of elliptic curves. This
  discovery prompted researchers to investigate
  other applications of elliptic curves in
  cryptography and computational number theory.

3

• Elliptic curve cryptography (ECC) was
  discovered in 1985 by Neal Koblitz and Victor
  Miller. Elliptic curve cryptographic schemes are
  public-key mechanisms that provide the same
  functionality as RSA schemes. However, their
  security is based on the hardness of a different
  problem, namely the elliptic curve discrete
  logarithm problem (ECDLP).

4

• Currently the best algorithms known to solve
  the ECDLP have fully exponential running time, in
  contrast to the subexponential-time algorithms
  known for the integer factorization problem. This
  means that a desired security level can be
  attained with significantly smaller keys in
  elliptic curve systems than is possible with
  their RSA counterparts.

5

• For example, it is generally accepted that a
  160-bit elliptic curve key provides the same
  level of security as a 1024-bit RSA key. The
  advantages that can be gained from smaller key
  sizes include speed and efficient use of power,
  bandwidth, and storage.

6
Outline

• Weierstrass Equation
• Elliptic Curves over R
• Elliptic Curves over Finite Field
• Elliptic Curve Cryptosystems
• Factoring with Elliptic Curves

7
1 Weierstrass Equation
8
(No Transcript)
9
2 Elliptic Curves Over R2.1 Simplified
Weierstrass Equations
10

• 2.2 Elliptic Curves over R

11

• 2.3 Addition Law

12

• Chord-and-Tangent Rule

13

• Chord-and-Tangent Rule (Continued)

14
(No Transcript)
15
(No Transcript)
16

• Algebraic Formulas

17
(No Transcript)
18
3 Elliptic Curves over Finite Field

• 3.1 Elliptic Curves Mod p, p?2,3
• 3.1.2 Addition Law

19
3.1.2 Example
20
(No Transcript)
21
3.2 Elliptic Curves over GF(2n)
22
3.2.1Simplified Weierstrass Equations
23
3.2.2 Group law
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
3.2.3 Example
28

• 3.3 Number of Points

29
(No Transcript)
30

• 3.4 Discrete Logarithms on Elliptic Curves

31
(No Transcript)
32
4 Elliptic Curve Cryptosystems

• 4.1 Representing Plaintext

33
(No Transcript)
34

• 4.2 An Elliptic Curve ElGamal Cryptosystem

35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38

• 4.3 An Elliptic Curve Digital Signature Algorithm
  (ECDSA)

39
(No Transcript)
40
5 Factoring with Elliptic Curves5.1 The Elliptic
Curve Factoring Algorithm
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46

• 5.2 Degenerate Curves

47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
Thank You!