Download Elliptic Curves and Elliptic Curve Cryptography

390-Elliptic Curves
and Elliptic Curve
Cryptography
Michael Karls
1
Outline
Groups, Abelian Groups, and Fields
 Elliptic Curves Over the Real Numbers
 Elliptic Curve Groups
 Elliptic Curves Over a Finite Field
 An Elliptic Curve Cryptography Scheme—
Diffie-Hellman Key Exchange

2
Group Definition

1.
2.
3.
4.
A group is a non-empty set G equipped with a binary
operation * that satisfies the following axioms for all a,
b, c in G:
Closure: a*b in G
Associativity: (a*b)*c = a*(b*c)
Identity: There exists an element e in G such that
a*e = a = e*a. We call e the identity element of G.
Inverse: For each a in G, there exists an element d in
G such that a*d = e = d*a. We call d the inverse of a.
3
Group Definition (cont.)

5.

If a group G also satisfies the following
axiom for all a, b in G:
Commutativity: a*b = b*a,
we say G is an abelian group.
The order of a group G, denoted |G| is
the number of elements in G. If |G| < ,
we say G has finite order.
4
Group Examples







One example of a group is the set of real numbers with addition.
Another group can be made from the set of permutations on the set
T = {1, 2, … , n}.
Recall that a permutation is a 1-1 onto function from T ! T.
When n = 3, the set of permutations on T is S3 = {(1) , (12), (13), (23), (123),
(132)}.
Recall that in cycle notation, for  = (12), (1) = 2, (2) = 1, and (3) = 3.
For permutations  and , define the product   to be the permutation
obtained by applying  first, then .
For example, with  = (13) and  = (12),
  = (13)(12) = (132) and   = (12)(13) = (123).
5
Group Examples






Here is the “multiplication”
table for S3:
From the table, we see that S3
is closed under this product,
the identity element is (1),
each element has an inverse,
and the product is associative.
Therefore, S3 is a group!
We call Sn the Symmetric
Group on n elements.
Which of these examples are
finite?
Which are abelian?
(1)
(12)
(13)
(23)
(123)
(132)
(1)
(1)
(12)
(13)
(23)
(123)
(132)
(12)
(12)
(1)
(123)
(132)
(13)
(23)
(13)
(13)
(132)
(1)
(123)
(23)
(12)
(23)
(23)
(123)
(132)
(1)
(12)
(13)
(123)
(123)
(23)
(12)
(13)
(132)
(1)
(132)
(132)
(13)
(23)
(12)
(1)
(123)
6
Field Definition

1.
2.
3.
4.
5.
A field F is a non-empty set with two binary operations,
usually denoted + and *, which satisfy the following
axioms for all a, b, c in F:
a+b is in F
(a+b)+c = a+(b+c)
a+b = b+a
There exists 0F in F such that a+0F = a = 0F+a. We call
0F the additive identity.
For each a in F, there exists an element x in F such
that a+x = 0F = x+a. We call x the additive inverse of a
and write x = -a.
7
Field Definition (cont.)

6.
7.
8.
9.
10.
11.
Field axioms (cont.): For all a, b, c in F,
a*b in F
(a*b)*c = a*(b*c)
a*b = b*a
There exists 1F in F, 1F  0F, such that for each a in F,
a*1F = a = 1F*a. We call 1F the multiplicative identity.
For each a  0F in F, there exists an element y in F
such that a*y = 1F = y*a. We call y the multiplicative
inverse of a and write y = a-1.
a*(b+c) = a*b + a*c and (b+c)*a = b*a + c*a.
(Distributive Law)
8
Field Examples





Note that any field is an abelian group under +
and the non-zero elements of a field form an
abelian group under *.
Some examples of fields:
Real numbers
Zp, the set of integers modulo p, where p is a prime
number is a finite field.
For example,
Z7 = {0, 1, 2, 3, 4, 5, 6} and Z23 = {0, 1, 2, 3, … , 22}.
9
Elliptic Curves Over the Real
Numbers





Let a and b be real numbers. An elliptic curve E over the field of real
numbers R is the set of points (x,y) with x and y in R that satisfy the
equation
together with a single element , called the point at infinity.
There are other types of elliptic curves, but we’ll only consider elliptic curves
of this form.
If the cubic polynomial x3+ax+b has no repeated roots, we say the elliptic
curve is non-singular.
A necessary and sufficient condition for the cubic polynomial x3+ax+b to
have distinct roots is 4a3 + 27 b2  0.
In what follows, we’ll always assume the elliptic curves are non-singular.
10
Examples of Elliptic Curves

y2 = x3-7x+6

y2 = x3-2x+4
4
4
3
2
2
1
-3
-2
-1
1
2
3
4
-2
-1
1
2
3
4
-1
-2
-2
-4
-3
-4
11
An Elliptic Curve Lemma

The next result provides a way to turn the set of
points on a non-singular elliptic curve into an
abelian group!
Elliptic Curve Lemma: Any line containing two
points of a non-singular elliptic curve contains a
unique third point of the curve, where
Any vertical line contains , the point at infinity.

Any tangent line contains the point of tangency twice.


12
Geometric Elliptic Curve
Addition



Using the Elliptic Curve Lemma, we can define
a way to geometrically “add” points P and Q on
a non-singular elliptic curve E!
First, define the point at infinity to be the
additive identity, i.e. for all P in E,
P +  = P =  + P.
Next, define the negative of the point at infinity
to be -  = .
13
Geometric Elliptic Curve
Addition (cont.)


For P = (xP,yP), define the negative of P
to be -P = (xP,-yP), the reflection of P
about the x-axis.
From the elliptic curve equation,
we see that whenever P is in E, -P is also
in E.
14
Geometric Elliptic Curve
Addition (cont.)
In what follows, assume that neither P nor
Q is the point at infinity.
 For P = (xP,yP) and Q = (xQ,yQ) in E, there
are three cases to consider:
1. P and Q are distinct points with xP  xQ.
2. Q = -P, so xP = xQ and yP = - yQ.
3. Q = P, so xP = xQ and yP = yQ.

15
Geometric Case 1: xP  xQ




By the Elliptic Curve
Lemma, the line L
through P and Q will
intersect the curve at
one other point.
Call this third point -R.
Reflect the point -R
about the x-axis to point
R.
P+Q = R

y2 = x3-7x+6
4
R
2
Q
P
-3
-2
-1
1
2
3
4
-2
R
-4
16
Geometric Case 2: xP = xQ and
yP = - yQ




In this case, the line L
through P and Q = -P is
vertical.
By the Elliptic Curve
Lemma, L will also
intersect the curve at .
P+Q = P+(-P) = 
It follows that the
additive inverse of P is
-P.

y2 = x3-2x+4
4
3
P
2
1
-2
-1
1
2
3
4
-1
Q
-2
-3
-4
17
Geometric Case 3: xP=xQ and yP
= yQ






Since P = Q, the line L
through P and Q is tangent to
the curve at P.
If yP = 0, then P = -P, so we
are in Case 2, and P+P =
.
For yP  0, the Elliptic Curve
Lemma says that L will
intersect the curve at another
point, -R.
As in Case 1, reflect -R about
the x-axis to point R.
P+P = R
Notation: 2P = P+P

y2 = x3-7x+6
4
R
2
-3
-2
-1
1
2
3
4
-2
R
P
-4
18
Geometric Elliptic Curve Model
For an interactive illustration of how
geometric elliptic addition works, a great
resource is Certicom’s Geometric Elliptic
Curve Model.
 For the elliptic curves y2 = x3-7x+6 and y2
= x3-2x+4, try adding points P and Q or
doubling P (i.e. 2 P = P+P), graphically.

19
Algebraic Elliptic Curve Addition



Geometric elliptic curve addition is useful for
illustrating the idea of how to add points on an
elliptic curve.
Using algebra, we can make this definition more
rigorous!
As in the geometric definition, the point at infinity
is the identity, - = , and for any point P in E,
-P is the reflection of P about the x-axis.
20
Algebraic Elliptic Curve Addition
(cont.)


1.
2.
3.
In what follows, assume that neither P nor Q is
the point at infinity.
As in the geometric case, for P = (xP,yP) and Q =
(xQ,yQ) in E, there are three cases to consider:
P and Q are distinct points with xP  xQ.
Q = -P, so xP = xQ and yP = - yQ.
Q = P, so xP = xQ and yP = yQ.
21
Algebraic Case 1: xP  xQ


First we consider the case where P = (xP,yP) and Q = (xQ,yQ) with xP
 xQ.
The equation of the line L though P and Q is y =  x+, where

In order to find the points of intersection of L and E, substitute
 x +  for y in the equation for E to obtain the following:

The roots of (2) are the x-coordinates of the three points of
intersection.
Expanding (2), we find:

22
Algebraic Case 1: xP  xQ
(cont.)


Since a cubic equation over the real numbers has either one or
three real roots, and we know that xP and xQ are real roots, it follows
that (3) must have a third real root, xR.
Writing the cubic on the left-hand side of (3) in factored form
we can expand and equate coefficients of like terms to find
23
Algebraic Case 1: xP  xQ
(cont.)



We still need to find the y-coordinate of the third point, -R = (xR,-yR)
on the curve E and line L.
To do this, we can use the fact that the slope of line L is determined
by the points P and -R, both of which are on L:
Thus, the sum of P and Q will be the point R = (xR, yR) with
where
24
Algebraic Case 2: xP = xQ and
yP = - yQ


In this case, the line L through P and
Q = -P is vertical, so L contains the point
at infinity.
As in the geometric case, we define
P+Q = P+(-P) = , which makes P and
-P additive inverses.
25
Algebraic Case 3: xP=xQ and
y P = yQ
Finally, we need to look at the case when
Q = P.
 If yP = 0, then P = -P, so we are in Case 2,
and P+P = .
 Therefore, we can assume that yP  0.
 Since P = Q, the line L through P and Q is
the line tangent to the curve at (xP,yP).

26
Algebraic Case 3: xP=xQ and
y P = yQ

The slope of L can be found by implicitly
differentiating the equation y2 = x3 + ax + b and
substituting in the coordinates of P:

Arguing as in Case 1, we find that
P+P = 2P = R, with R = (xR,yR), where
27
Elliptic Curve Groups

From these definitions of addition on an elliptic
curve, it follows that:
1.
Addition is closed on the set E.
Addition is commutative.
 is the identity with respect to addition.
Every point P in E has an inverse with respect to
addition, namely -P.
2.
3.
4.

The associative axiom also holds, but is “hard”
to prove.
28
Elliptic Curves Over Finite
Fields

Instead of choosing the field of real numbers, we can
create elliptic curves over other fields!
Let a and b be elements of Zp for p prime, p>3. An
elliptic curve E over Zp is the set of points (x,y) with x and
y in Zp that satisfy the equation

together with a single element , called the point at
infinity.
As in the real case, to get a non-singular elliptic curve,
we’ll require 4a3 + 27 b2 (mod p)  0 (mod p).
Elliptic curves over Zp will consist of a finite set of points!


29
Addition on Elliptic Curves over
Zp
Just as in the real case, we can define
addition of points on an elliptic curve E
over Zp, for prime p>3.
 This is done in the essentially the same
way as the real case, with appropriate
modifications.

30
Addition on Elliptic Curves over
Zp (cont.)




Suppose P and Q are points in E.
Define P +  =  + P = P for all P in E.
If Q = -P (mod p), then P+Q = .
Otherwise, P+Q = R = (xR,yR), where
31
Elliptic Curves Over Z23 Model
Again, Certicom provides a model for an
elliptic curve over a finite field: Finite
Geometric Elliptic Curve Model.
 For the elliptic curves y2 = x3+16x+6 and
y2 = x3+21x+4 over the field Z23, try
adding points P and Q or doubling P
(i.e. 2P =P+P).

32
Cryptography on an Elliptic
Curve
Using an elliptic curve over a finite field,
we can exchange information securely!
 For example, we can implement a scheme
invented by Whitfield Diffie and Martin
Hellman in 1976 for exchanging a secret
key.

33
Diffie-Hellman Key Exchange
via Colors of Paint
1.
2.
3.
4.
5.
6.
Alice and Bob each have a three-gallon bucket that
holds paint.
Alice and Bob choose a public color of paint, such as
yellow.
Alice chooses a secret color, red.
Alice mixes one gallon of her secret color, red, with one
gallon of yellow and sends the mixture to Bob.
Bob chooses a secret color, purple.
Bob mixes one gallon of his secret color, purple, with
one gallon of yellow and sends the mixture to Alice.
34
Diffie-Hellman Key Exchange
via Colors of Paint (cont.)
7.
8.
9.

Alice adds one gallon of her secret color, red to the
mixture from Bob. Alice ends up with a bucket of one
gallon each of yellow, purple, and red paint.
Bob adds one gallon of his secret color, purple, to the
mixture from Alice. Bob ends up with a bucket one
gallon each of yellow, red, and purple paint.
Both Alice and Bob will have a bucket of paint with the
same color—this common color is the key!
Note that even if eavesdropper Eve knows that the
common color is yellow, or intercepts the paint mixtures
from Alice or Bob, she will not be able to figure out
Alice’s or Bob’s secret color!
35
Diffie-Hellman Key Exchange
via an Elliptic Curve
1.
2.
3.
4.
Alice and Bob publicly agree
on an elliptic curve E over a
finite field Zp.
Next Alice and Bob choose a
public base point B on the
elliptic curve E.
Alice chooses a random
integer 1<<|E|, computes P
=  B, and sends P to Bob.
Alice keeps her choice of 
secret.
Bob chooses a random
integer 1<<|E|, computes Q
=  B, and sends Q to Alice.
Bob keeps his choice of 
secret.
1.
2.
3.
4.
Alice and Bob choose E to
be the curve y2 = x3+x+6
over Z7.
Alice and Bob choose the
public base point to be
B=(2,4).
Alice chooses  = 4,
computes P = B = 4(2,4) =
(6,2), and sends P to Bob.
Alice keeps  secret.
Bob chooses  = 5,
computes Q = B = 5(2,4) =
(1,6), and sends Q to Alice.
Bob keeps  secret.
36
Diffie-Hellman Key Exchange
via an Elliptic Curve (cont.)
5.
6.
7.

Alice computes
KA = Q = (B).
Bob computes
KB = P = (B).
The shared secret key
is K = KA = KB.
Even if Eve knows the
base point B, or P or Q,
she will not be able to
figure out  or , so K
remains secret!
5.
6.
7.
Alice computes
KA=Q = 4(1,6) =
(4,2).
Bob computes
KB = P = 5(6,2) =
(4,2).
The shared secret
key is K = (4,2).
37
References




Hungerford, Thomas W. Abstract Algebra: An
Introduction Second Edition. New York: Saunders
College Publishing, 1997.
Koblitz, Neal. Algebraic Aspects of Cryptography.
Berlin: Springer-Verlag, 1999.
“Online ECC Tutorial.” Certicom. www.certicom.com
Stinson, Douglas R. Cryptography Theory and
Practice Second Edition. New York: Chapman &
Hall/CRC, 2002.
38