498Elliptic Curves and Elliptic Curve Cryptography

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498-Elliptic Curves and Elliptic Curve
Cryptography

• Michael Karls
• Spring 2005

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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 SchemeDiffie-Hellm
  an Key Exchange

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Group Definition

• 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 ab in G
• Associativity (ab)c a(bc)
• Identity There exists an element e in G such
  that ae a ea. We call e the identity
  element of G.
• Inverse For each a in G, there exists an
  element d in G such that ad e da. We call
  d the inverse of a.

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Group Definition (cont.)

• If a group G also satisfies the following axiom
  for all a, b in G
• Commutativity ab ba,
• we say G is an abelian group.
• The order of a group G, denoted G is the number
  of elements in G. If G lt 1, we say G has
  finite order.

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Group Examples

• One example of a group is the set of real numbers
  with addition.
• The set of 2 x 2 matrices with real number
  entries and non-zero determinant forms a group
  under matrix multiplication.
• Another group can be made from the set of
  permutations on the set T 1, 2, , n. This
  group is denoted by Sn.
• 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).

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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?

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Field Definition

• 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
• ab is in F
• (ab)c a(bc)
• ab ba
• There exists 0F in F such that a0F a 0Fa.
  We call 0F the additive identity.
• For each a in F, there exists an element x in F
  such that ax 0F xa. We call x the additive
  inverse of a and write x -a.

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Field Definition (cont.)

• Field axioms (cont.) For all a, b, c in F,
• ab in F
• (ab)c a(bc)
• ab ba
• There exists 1F in F, 1F ? 0F, such that for each
  a in F, a1F a 1Fa. We call 1F the
  multiplicative identity.
• For each a ? 0F in F, there exists an element y
  in F such that ay 1F ya. We call y the
  multiplicative inverse of a and write y a-1.
• a(bc) ab ac and (bc)a ba ca.
  (Distributive Law)

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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.

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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 1, called the
  point at infinity.
• There are other types of elliptic curves, but
  well only consider elliptic curves of this form.
• If the cubic polynomial x3axb has no repeated
  roots, we say the elliptic curve is non-singular.
• A necessary and sufficient condition for the
  cubic polynomial x3axb to have distinct roots
  is 4a3 27 b2 ? 0.
• In what follows, well always assume the elliptic
  curves are non-singular.

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Examples of Elliptic Curves

• y2 x3-7x6

• y2 x3-2x4

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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.

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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 1 P 1 P.
• Next, define the negative of the point at
  infinity to be -1 1.

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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.

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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
• 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.

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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.
• PQ R

• y2 x3-7x6

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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 ?.
• PQ P(-P) ?
• It follows that the additive inverse of P is -P.

• y2 x3-2x4

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Geometric Case 3 xPxQ 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
  PP 1.
• 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.
• PP R
• Notation 2P PP

• y2 x3-7x6

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Geometric Elliptic Curve Model

• For an interactive illustration of how geometric
  elliptic addition works, a great resource is
  Certicoms Geometric Elliptic Curve Model.
• For the elliptic curves y2 x3-7x6 and y2
  x3-2x4, try adding points P and Q or doubling P
  (i.e. 2 P PP), graphically.

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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, -1 1, and for any
  point P in E, -P is the reflection of P about the
  x-axis.

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Algebraic Elliptic Curve Addition (cont.)

• 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.

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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

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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

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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

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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 PQ
  P(-P) 1, which makes P and -P additive
  inverses.

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Algebraic Case 3 xPxQ and yP 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
  PP 1.
• 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).

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Algebraic Case 3 xPxQ and yP 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
• PP 2P R, with R (xR,yR), where

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Elliptic Curve Groups

• From these definitions of addition on an elliptic
  curve, it follows that
• Addition is closed on the set E.
• Addition is commutative.
• 1 is the identity with respect to addition.
• Every point P in E has an inverse with respect to
  addition, namely -P.
• The associative axiom also holds, but is hard
  to prove.

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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, pgt3.
  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 1, called the
  point at infinity.
• As in the real case, to get a non-singular
  elliptic curve, well require 4a3 27 b2 (mod p)
  ? 0 (mod p).
• Elliptic curves over Zp will consist of a finite
  set of points!

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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 pgt3.
• This is done in the essentially the same way as
  the real case, with appropriate modifications.

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Addition on Elliptic Curves over Zp (cont.)

• Suppose P and Q are points in E.
• Define P 1 1 P P for all P in E.
• If Q -P (mod p), then PQ 1.
• Otherwise, PQ R (xR,yR), where

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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 x316x6 and y2
  x321x4 over the field Z23, try adding points P
  and Q or doubling P (i.e. 2P
  PP).

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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.

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Diffie-Hellman Key Exchange via Colors of Paint

• 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.

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Diffie-Hellman Key Exchange via Colors of Paint
(cont.)

• 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 colorthis 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 Alices or Bobs secret color!

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Diffie-Hellman Key Exchange via an Elliptic Curve

• 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 1lt?ltE, computes
  P ? B, and sends P to Bob. Alice keeps her
  choice of ? secret.
• Bob chooses a random integer 1lt?ltE, computes Q
  ? B, and sends Q to Alice. Bob keeps his
  choice of ? secret.

• Alice and Bob choose E to be the curve y2
  x3x6 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.

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Diffie-Hellman Key Exchange via an Elliptic Curve
(cont.)

• 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!

• 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).

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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.