Elliptic Curve Cryptography (ECC)

1
Elliptic Curve Cryptography (ECC)

• Mustafa Demirhan
• Bhaskar Anepu
• Ajit Kunjal

2
Contents

• Introduction
• Addition Law
• Elliptic Curves Mod n
• Encryption Example
• Decryption Example
• General Diffie-Hellman Key Exchange Scheme
• Diffie Hellman Method with Elliptic Curves
• Conclusions

3
Introduction

• What is Elliptic Curve Cryptography (ECC)?
• ECC is an encryption technique based on elliptic
  curve theory that can be used as faster, smaller,
  and more efficient cryptosystems
• Who introduced it and when?
• Miller and Koblitz in mid 1980s and Lenstra
  showed how to use elliptic curves to factor
  integers
• What is the basic principle?
• Obtain same level of security as conventional
  cryptosystems but with much smaller key sizes

4
General Form of Elliptic Curve

• An elliptic curve
• E y2 x3 ax b
• (a, b) belong to any of the appropriate sets
  namely rational numbers, complex numbers,
  integers etc.
• More general form y2a1xya3y x3a2x2a4xa5

5
Addition Law

• Given two points P1 and P2 on E, we can find P3
  as follows
• Let P1 (2, 9) and P2 (3, 10) and E y2 x3
  73
• Find the equation of the line passing through P1
  and P2
• Find a point Q such that it lies on the line
  through P1 and P2 and the curve E

Q
P2
P1
P3
6
Addition Law Example

• Equation of the line y x 7
• For Q, substitute this eqn. in E.(x7)2 x3
  73
• Roots of this cubic P1,P2 and Q.
• Rule For a cubic polynomial of the form
  x3a2x2a1xa0 the roots r1,r2 and r3 are related
  by r1r2r3-a2
• Applying this to our cubicx3-x2-14x24 0, we
  obtain 23xQ1 ? xQ -4
• yQxQ7 ? yQ 3 ? Q (-4,3)
• P3 is the mirror image of Q. Thus P3 (-4, -3)

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Addition Law-Definition

• Define a law of addition on E byP1 P2 P3
• Addition Law Let E y2 x3 ax b and let
• P1 (x1, y1) P2 (x2, y2)
• Then P 1 P2 P3 (x3, y3) where
• x3 m2 - x1 - x2
• y3 m (x1 - x3) - y1
• and m (y2 - y1) / (x2 x1) if P1 ? P2
• m (3x12 a) / (2y1) if P1 P2

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Definition of P P

• Draw a tangent line through P, the point of
  intersection with the curve is defined as R,
  then PP 2P R

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Definition of P (-P)

• P (-P) O

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Multiplication

• k.P P P P . P (k times) where k is an
  integer

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Elliptic Curves Mod n

• Let E y2 x3 2x 3 (mod 5)
• The points on E are pairs (x, y) mod 5 that
  satisfy the equation
• The possible values are
• x 0 ? y2 3 (mod 5) ? no solutionsx 1 ? y2
  6 (mod 5) ? y 1, 4x 2 ? y2 15 (mod 5) ?
  y 0x 3 ? y2 36 (mod 5) ? y 1, 4x 4 ?
  y2 75 (mod 5) ? y 0
• Therefore the points on E are (1,1), (1,4),
  (2,0), (3,1), (3,4), (4,0)

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Encryption Example

• Let E y2 x3 2x 3 (mod 5)
• P (1, 4), K (3, 1)
• The cipher text is obtained as followsm
  (14)/(3-1) 1 (mod 5)x3 -1-1-3 -3 (mod 5)
  2y3 1(1-2)-4 0 (mod 5)
• Cipher Text C (2,0)

Q
K
P
C
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Decryption Example

• Let E y2 x3 2x 3 (mod 5)
• C (2, 0) K (3, 1)
• The decryption is same as encrypting with K
• -K (3,-1) (mod 5) (3,4)
• m 4/1 4x3 16-2-3 1 (mod 5)y3 4(2-1)-0
  4
• Hence, P (1,4)

Q
K
P
C
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An Example Usage of Elliptic Curves

• The crucial property of an elliptic curve is that
  we can define a rule for "adding" two points
  which are on the curve, to obtain a third point
  which is also on the curve
• Cryptography can be done as follows
• Alice, Bob, Cathy and David agree on a
  (non-secret) elliptic curve and a (non-secret)
  fixed curve point F. Alice chooses a secret
  random integer KA which is her secret key, and
  publishes the curve point PA KAF as her public
  key. Bob, Cathy and David do the same
• Now suppose Alice wishes to send a message to
  Bob. One method is for Alice to simply compute
  KAPB and use the result as the secret key for a
  conventional symmetric block cipher (say DES)
• Bob can compute the same number by calculating KB
  PA, since KBPA KB(KAF) (KBKA)F
  KA(KBF) KAPB
• The security of the scheme is based on the
  assumption that it is difficult to compute k
  given F and kF.

15
General Diffie-Hellman Key Exchange
(a, p)

• Alice and Bob chooses a large prime number p and
  a primitive root a (mod p). Both p and a can be
  made public.
• Alice chooses a secret random x and Bob chooses a
  secret random y.
• Alice sends ax (mod p) to Bob, and Bob sends ay
  (mod p) to Alice.
• Alice calculates key as K (ay)x (mod p) and Bob
  calculates K (ax)y (mod p)

Alice
Bob
x
ax (mod p)
y
ay (mod p)
(ay)x(mod p)
(ax)y(mod p)
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DH Key Exchange with Elliptic Curves
(E, P)

• Diffie-Hellman key exchange- another example
• Given elliptic curve E and a point P (public)
• Alice selects an a, computes AaP, send A to Bob
• Bob selects a b, computes BbP, sends B to Alice
• Then Alice can compute the key KaBabP,
  similarly, Bob computes the key KbAabP

Alice
Bob
a, A
A aP
b, B
B bP
abP
abP
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Using The Shared Keys

• The key that we obtained using Diffie-Hellman
  with elliptic curves can be used either directly
  in another elliptic curve cryptosystem, or in a
  conventional cryptosystem such as DES, RSA etc.
• However, for the latter, we need to convert the
  point in the elliptic curve system to a number.
  This can be done, but it is beyond the scope of
  this presentation

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Conclusions

• Elliptic Curves are just another way to map the
  data into another form. The power of the scheme
  comes from the fact that it is very hard to do
  the un-mapping without knowledge of the key
• Elliptic Curve Cryptosystems provide same level
  of security as other conventional cryptosystems
  but with a much smaller key size
• Smaller the key size, lesser the hardware required