Elliptical Curve Cryptography

1
Elliptical Curve Cryptography

• Manish Kumar
• Roll No - 43
• CS-A, S-7
• SOE, CUSAT

2
Outline

• Introduction
• Cryptography
• Mathematical Background
• Elliptic Curves
• Elliptic Curves Arithmetic
• Elliptical Curve Cryptography(ECC)
• Applications
• Conclusion
• References

3
Introduction

• Cryptography
• Cryptography is science of using mathematics
  to
• encrypt and decrypt data.
• Cryptography provide us mechanism to send,
  sensitive
• data through insecure network (like
  internet).

4
Introduction

• Secret key cryptography
• The encryption key and decryption key are
  the
• same.
• Key Distribution Problem.

5
Introduction

• Public key cryptography
• Different key for encryption and decryption
• Public-key and private-key
• Key distribution problem is solved.

6
Introduction

• A comparison of public key Cryptosystems

7
Introduction

• Elliptical Curve Cryptography
• ECC was introduced by Victor Miller and
  Neal Koblitz in
• 1985.
• Its new approach to Public key
  cryptography.
• ECC requires significantly smaller key size
  with same
• level of security.
• Benefits of having smaller key sizes
  faster
• computations, need less storage space.
• ECC ideal for Pagers PDAs Cellular
  Phones
• Smart Cards.

8
Mathematical Background

• A group is an algebric system consisting of a
  set G together with a binary operation defined
  on G satisfying the following axioms
• Closure for all x, y in G we have x y ?
  G
• Associativity for all x, y and z in G we
  have
• (x y) z x (y z)
• Identity element There is an element e in
  G such
• that a e e a a for all a in G.
• Inverse element For each a in G there is
  an
• element a' in G such that a a' a' a
  e.

9
Mathematical Background

• In addition if for x, y in G we have x y y
  x then we say that group G is abelian.
• A finite field is an algebraic system consisting
  of a set F together with a binary operations
  and defined on F satisfying the following
  axioms
• F is an abelian group with respect to .
• F \ 0 is an abelian group with respect
  to .

10
Mathematical Background

• For all x, y and z in F we have
  
  x ( y z) (x y) (x z)
  (x y) z (x z) (y z)
• The order of the finite field is the number of
  elements in the field.

11
Elliptic Curves

• Elliptic curves are not ellipses (the name comes
  from elliptic integrals)
• Standard Form Equation
• y2 x3 a.x b
• where x, y, a and b are
• real numbers.
• Each choice of the numbers a and b yields a
  different elliptic curve.

12
Elliptic Curves

• If 4a3 27b2 is not 0 (i.e. x3 a x b
  contains no repeated factors), then the elliptic
  curve can be used to form a group
• An elliptic curve group consists of the points on
  the curve and a special point O, meeting point of
  curve with a straight line at infinity.

13
Elliptic curve Arithmetic

• Point Addition
• Draw a line that intersects distinct points P and
  Q
• The line will intersect a third point -R
• Draw a vertical line through point -R
• The line will intersect a fourth point R
• Point R is defined as the summation of points P
  and Q
• R P Q

14
Elliptic curve Arithmetic

• Draw a line that intersects points P and
• -P
• The line will not intersect a third point
• For this reason, elliptic curves include O, a
  point at infinity
• P (-P) O
• O is the additive identity

15
Elliptic curve Arithmetic

• Point Doubling
• Draw a line tangent to point P
• The line will intersect a second point -R
• Draw a vertical line through point -R
• The line will intersect a third point R
• Point R is defined as the summation of point P
  with itself
• R 2P

16
Elliptical Curve Cryptography

• Point Multiplication
• The main cryptographic operation in ECC is
  point multiplication.
• Point multiplication is performed through a
  combination of point additions and point
  doublings,
• e.g.11P 2((2(2P)) P) P.
• Point multiplication is simply calculating Qk
  . P, where k is an integer and P is a point on
  the curve called as base point.

17
Elliptical Curve Cryptography

• Point Multiplication
• Each curve has a specially designated point
  P called
• the base point chosen such that a large
  fraction of the
• elliptic curve points are multiples of it.
• To generate a key pair, one selects a random
  integer k
• which serves as the private key, and
  computes k P
• which serves as the corresponding public
  key.

18
Elliptical Curve Cryptography

• The Elliptic curve discrete logarithm problem
• The discrete logarithm problem for ECC is
  the
• inverse of point multiplication.
• Given points P and Q, find a number k such
  that
• k P Q
• where P and Q are points on the
  elliptic curve
• Q is the public key
• k is the private key (very large prime
  number)

19
Elliptic Curve Discrete Logarithm

• We can find the value of k by adding P,
  k-times.
• This is called Brute-force Method (not work
  when
• k is large)
• Pollards rho is best method to solve DLP.
• Running time of Pollards rho is
  exponential.

20
Elliptical Curve Cryptography

• What makes ECC hard to crack?
• The security of ECC relies on the
  difficulty of
• solving the Elliptic Curve Discrete
  Logarithm
• Problem (ECDLP)
• i.e. finding k, given P and Q k P. The
  problem is
• computationally intractable for large
  values of k.

21
Performance Comparison

• ADVANTAGES OF ECC OVER RSA
• Smaller key size for equivalent
  security.
• Faster and Less computations.
• Less memory.

22
Applications

• Significant performance benefits from using ECC
  in secure web transaction.
• Elliptic Curve Digital Signature Algorithm(ECDSA)
  
• ECC can be used in constrained Environments
  Pagers PDAs Cellular Phones Smart Cards
  where traditional public-key mechanisms are
  simply impractical.

23
Conclusion

• ECC uses groups and a logarithm problem.
• ECC is a stronger option than the RSA and
  discrete logarithm systems for the future.
• Due to small key size, implementation is easy.
• ECC is excellent choice for portable,
  communicati-on devices.
• ECCs main advantage as key length increases, so
  does the difficulty of the inversion process.

24
References

• Cryptography and Network Security Principles and
  Practices, Fourth Edition,PHI, By William
  Stallings.
• Guide to Elliptic Curve Cryptography By Darrel
  Hankerson, Alfred Menezes, Scott Vanstone.
• Elliptic Curve Cryptography How it Works
  Sheueling Chang, Hans Eberle, Vipul Gupta, Nils
  Gura, Sun Microsystems Laboratories.
• The Elliptic Curve Cryptosystem For Smart Cards,
  A Certicom White Paper, Published May 1998 .
• Elliptic Curve Cryptography An Implementation
  Guide By Anoop MS anoopms_at_tataelxsi.co.in .