elliptic curve cryptography: arithmetic behind
TRANSCRIPT
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Arithmetic of Elliptic Curves
Ayan Sengupta
May 5, 2015
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Overview
1 Group Structure of Elliptic Curves
2 Rational Points of Finite Order on Elliptic Curve
3 Group of Rational Points on Elliptic Curve
4 Application in Cryptography
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Motivation
Very important concept and major area of current researchin Number Theory.
Andrew Wiles used in his famour proof of Fermat’s lasttheorem.
They are vividly used in many algorithms:- Lenstra elliptic curve factorization.- Elliptic curve primality testing.
Elliptic curve cryptography (ECC) is based on the ellipticcurve discrete logarithm problem.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
What is Elliptic Curve ?
An algebraic curve of the form
Y 2 = X 3 + aX 2 + bX + c (1)
where a, b, c ∈ K , field (most popular are Q, Fp), such thatf (X ) = X 3 + aX 2 + bX + c has no repeated root in C.
We also assume a point at infinity O included in elliptic curve,that is the point where the vertical lines in XY -plane meet.
(a) One real root of f (X ) (b) Three real roots of f (X )
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
What is Elliptic Curve ?
An algebraic curve of the form
Y 2 = X 3 + aX 2 + bX + c (1)
where a, b, c ∈ K , field (most popular are Q, Fp), such thatf (X ) = X 3 + aX 2 + bX + c has no repeated root in C.
We also assume a point at infinity O included in elliptic curve,that is the point where the vertical lines in XY -plane meet.
(a) One real root of f (X ) (b) Three real roots of f (X )
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
What is Elliptic Curve ?
A smooth, projective algebraic curve of genus one with apre-assumed point O.
It is nothing related to ellipses!
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Figure : Addition operation on elliptic curve
Explicitely,x3 = λ2 − a− x1 − x2 (2)
y3 = λx3 + ν (3)
where, λ and ν are respectively the slope and intercept of theline joining P1,P2.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Figure : Doubling a point
x3 =x41−2bx21−8cx1+b2−4ac
4x31+4ax21+4bx1+4c(duplication formula)
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Figure : Inverse of a point
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Using Nine intersection theorem, associativity can be proved.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Points on an elliptic curve form an abelian group under theabove mentioned addition operation.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Group Structure of Elliptic Curves
Concentrate on elliptic curve C over Q and points (x1, y1) suchthat both x1, y1 ∈ Q.It can be shown that such points (rational points) on C form asubgroup under the same addition operation.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Order of a Point on Elliptic Curve
P is a point (x1, y1) on elliptic curve C with order m if
mP = P + P + · · ·+ P︸ ︷︷ ︸m
= O (4)
such that m′P 6= O for all integers 1 ≤ m
′< m.
If no such m exists then P is of infinite order.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Order of a Point on Elliptic Curve
P is a point (x1, y1) on elliptic curve C with order m if
mP = P + P + · · ·+ P︸ ︷︷ ︸m
= O (4)
such that m′P 6= O for all integers 1 ≤ m
′< m.
If no such m exists then P is of infinite order.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Points of Order 2
2P = O if and only if P = −P, i.e. y1 = −y1. So, y1 = 0.
Number of rational points of order 2 depends on the number ofsolutions of the equation f (x) = 0 in Q.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Points of Order 2
2P = O if and only if P = −P, i.e. y1 = −y1. So, y1 = 0.Number of rational points of order 2 depends on the number ofsolutions of the equation f (x) = 0 in Q.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Points of Order 3
3P = O if and only if 2P = P.From duplication formula,
x41 − 2bx2
1 − 8cx1 + b2 − 4ac
4x31 + 4ax2
1 + 4bx1 + 4c= x1 (5)
So, x1 is a root of the equation3X 4 + 4aX 3 + 6bX 2 + 12cX + (4ac − b2) which is same as
2f (X )f′′
(X )− f′(X )
2.
For each x1 we can get two distinct y1s. So, total there are 9points in complex field of order 3 (including O).
These points are precisely all the inflection points i.e., thepoints on the curve C , such that the tangent at that point hasmultiplicity 3.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Points of Order 3
3P = O if and only if 2P = P.From duplication formula,
x41 − 2bx2
1 − 8cx1 + b2 − 4ac
4x31 + 4ax2
1 + 4bx1 + 4c= x1 (5)
So, x1 is a root of the equation3X 4 + 4aX 3 + 6bX 2 + 12cX + (4ac − b2) which is same as
2f (X )f′′
(X )− f′(X )
2.
For each x1 we can get two distinct y1s. So, total there are 9points in complex field of order 3 (including O).These points are precisely all the inflection points i.e., thepoints on the curve C , such that the tangent at that point hasmultiplicity 3.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Nagell-Lutz Theorem
This theorem gives the overview of all the rational points thatcan have finite order.
Theorem
(Nagell-Lutz) Let
Y 2 = f (X ) = X 3 + aX 2 + bX + c (6)
be a non-singular cubic curve with integer coefficients a, b, c;and let D be the discriminant of the cubic polynomial f (x),
D = −4a3c + a2b2 + 18abc − 4b3 − 27c2. (7)
Let P = (x , y) be a rational point of finite order. Then x and yare integers; and either y = 0, or else y |D.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Nagell-Lutz Theorem
Nagell-Lutz theorem is not an if and only ifcondition!
To find whether a particular point on C has finite order or not,we need to check all of its multiples to find the order. Mazur’stheorem is a very strong result which makes our life easier.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Nagell-Lutz Theorem
Nagell-Lutz theorem is not an if and only ifcondition!
To find whether a particular point on C has finite order or not,we need to check all of its multiples to find the order. Mazur’stheorem is a very strong result which makes our life easier.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Mazur’s Theorem
Theorem
Let C be a non-singular rational cubic curve, and suppose thatC (Q) contans a point of finite order m. Then either
1 ≤ m ≤ 10 or m = 12.
More precisely, the set of all points of finite order in C (Q)forms a subgroup, which has one of the following two forms:a) A cyclic group of order N with 1 ≤ N ≤ 10 or N = 12.b) The product of a cyclic group of order two and a cyclicgroup of order 2N with 1 ≤ N ≤ 4.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Example
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Mordell’s Theorem
Theorem
Let C be a non-singular cubic curve with rational coefficientsand has a rational point. Then the group of rational pointsC (Q) is finitely generated.
This theorem tells us that starting from a single rational pointon an elliptic curve and using only the group laws (addition,duplication, inversion) we can generate the whole set ofrational points.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Mordell’s Theorem
We define a map H : C −→ [0,∞) such that
H(x , y) = max{|m|, |n|}
where, x = mn in its irreducible form.
If x = 0, we define H(x , y) = 1. Also H(O) = 1.We call this map “height”of a point.
Define “small height”h(x , y) = logH(x , y).
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Mordell’s Theorem
We define a map H : C −→ [0,∞) such that
H(x , y) = max{|m|, |n|}
where, x = mn in its irreducible form.
If x = 0, we define H(x , y) = 1. Also H(O) = 1.We call this map “height”of a point.Define “small height”h(x , y) = logH(x , y).
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Proof of Mordell’s Theorem
Theorem
(Descent’s Theorem) If Γ is a abelian group with a functionh : Γ −→ [0,∞) such thata) For every real number n, the set {P ∈ Γ : h(P) ≤ n} is finite.b) For every P0 ∈ Γ, there is a constant k0 such that
h(P + P0) ≤ 2h(P) + k0 (8)
for every P ∈ Γ.c) There is a constant k such that
h(2P) ≥ 4h(P)− k (9)
for all P ∈ Γ.d) The subgroup 2Γ has finite index in Γ.Then Γ is finitely generated.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Proof of Mordell’s Theorem
It can be proved explicitely that C (Q) and the map “littleheight”h satisfy the above conditions.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Mordell’s Theorem
We have
C (Q) ∼= Z⊕ Z⊕ · · · ⊕ Z︸ ︷︷ ︸r
⊕Zp1d1⊕ Zp2d2
⊕ · · · ⊕ Zps ds . (10)
r is called rank of Γ and the subgroupZp1d1
⊕ Zp2d2⊕ · · · ⊕ Zps ds correspondes to the elements of
finite order in C (Q).
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Example
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Basics of Cryptography
Cryptography is the study of message hiding. The basic modelof cryptography is
Figure : Adversarial model of cryptography
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Secure Systems
For most secure and robust system, we assume that theadversary has considerable capabilites. He is able to read allthe data transmitted over the channel, has significantcomputational resources and has complete descriptions of thecommunications protocols and any cryptographic mechanismsdeployed (except for secret keying informations). The challengeis to design a robust mechanism to secure the communicationfrom such powerful adversaries.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Public-Key Cryptography
It is a part of cryptography where each entity selects a pair ofkeys, consisting of a public key, which is used for encryptionand a private key which is used for decryption. The keys havethe property that the actual plain text can not be computedeffeciently from the knowledge of only cipher text and thepublic keys. Public-key cryptosystems rely on the hardness ofsome very popular number theoretic problems. e.g.-
RSA scheme is based on the intractibility of integerfactorization problem for semiprimes.
ECC schemes depends totally on the hardness of ellipticcurve discrete logarithm problem (ECDLP).
Merkle-Hellman knapsack cryptosystem is based on integerknapsack problem (also called subset sum problem).
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Public-Key Cryptography
It is a part of cryptography where each entity selects a pair ofkeys, consisting of a public key, which is used for encryptionand a private key which is used for decryption. The keys havethe property that the actual plain text can not be computedeffeciently from the knowledge of only cipher text and thepublic keys. Public-key cryptosystems rely on the hardness ofsome very popular number theoretic problems. e.g.-
RSA scheme is based on the intractibility of integerfactorization problem for semiprimes.
ECC schemes depends totally on the hardness of ellipticcurve discrete logarithm problem (ECDLP).
Merkle-Hellman knapsack cryptosystem is based on integerknapsack problem (also called subset sum problem).
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
ECDLP
Definition
For a point P of order n and a pointQ ∈ {O,P, 2P, · · · , (n − 1)P} find the integer d ∈ [0, n − 1]such that Q = dP.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
ElGamal Elliptic Curve Cryptographic System
Suppose we have an elliptic curve C defined over a finite fieldFq, where q is a large prime. C , q and a point P ∈ C withlarge order n are publicly known. We first represent ourmessage m as a point M in C (Fq). When A wants tocommunicate secretly with B, they proceed thus:
B choose a random integer b ∈ [0, n − 1] and publishesthe point bP as public key and keeps b to himself as theprivate key.
A chooses a random integer a ∈ [0, n − 1] and publishesthe point aP. He then sends the pair (aP,M + a(bP)) toB, where M + a(bP) is the ciphertext. A keeps his secretkey, a to himself.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
ElGamal Elliptic Curve Cryptographic System
To decrypt the message, B first calculates b(aP) using A’spublic key and B’s own private key. As C is an abeliangroup, a(bP) = b(aP).
Now, B gets back the message fromM + a(bP)− b(aP) = M. From M, B gets back theoriginal message m by reversing the imbedding.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
ECDLP
Many protocols like - Elliptic Curve Integrated EncryptionScheme, Elliptic Curve Digital Signature Algorithm are basedon the intractibility of ECDLP.
There are several algorithms such as Number field sieve,Pohlig-Hellman algorithm, Pollard’s rho algorithm, Shor’salgorithm solve this problem. But the best known algorithm sofar is of complexity O(
√p), where p is the largest prime divisor
of n. But yet no one has been able to prove mathematically theintractibility of ECDLP.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
ECDLP
Many protocols like - Elliptic Curve Integrated EncryptionScheme, Elliptic Curve Digital Signature Algorithm are basedon the intractibility of ECDLP.There are several algorithms such as Number field sieve,Pohlig-Hellman algorithm, Pollard’s rho algorithm, Shor’salgorithm solve this problem. But the best known algorithm sofar is of complexity O(
√p), where p is the largest prime divisor
of n. But yet no one has been able to prove mathematically theintractibility of ECDLP.
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Arithmetic ofElliptic Curves
AyanSengupta
GroupStructure ofElliptic Curves
RationalPoints ofFinite Orderon EllipticCurve
Group ofRationalPoints onElliptic Curve
Application inCryptography
Acknoweledgement
1. http://en.wikipedia.org/wiki2. https://www.nsa.gov/ia/programs/suitebcryptography/index.shtml