2.2 Group law on elliptic curves
8 min read•august 21, 2024
Elliptic curves are fascinating mathematical objects that blend algebra and geometry. They form abelian groups under , allowing for rich structure and applications in cryptography and number theory.
The group law on elliptic curves defines how to combine points, forming the basis for their group structure. Understanding this law is crucial for working with elliptic curves and unlocking their potential in various mathematical and practical contexts.
Definition of elliptic curves
- Fundamental objects in arithmetic geometry combine algebraic and geometric properties
- Serve as a bridge between number theory and algebraic geometry
- Provide rich structure for studying solutions to certain polynomial equations
Weierstrass form
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- Standard representation of elliptic curves
- Coefficients a and b determine the curve's shape and properties
- Discriminant must be non-zero for a non-singular curve
- Short simplifies calculations and analysis
Singular vs nonsingular curves
- Nonsingular curves have smooth, continuous graphs without self-intersections
- Singular curves contain at least one point where the curve is not smooth
- Nonsingularity ensures the curve forms an
- Singularities occur when partial derivatives vanish simultaneously (cusps or nodes)
Affine vs projective models
- Affine model uses two-dimensional coordinate system (x, y)
- Projective model introduces a third coordinate (X : Y : Z) to represent points at infinity
- Projective model provides a more complete geometric picture
- Homogeneous equation for projective model
Group structure
- Elliptic curves form abelian groups under point addition
- Group structure allows for algebraic operations and study of curve properties
- Crucial for applications in cryptography and number theory
Point at infinity
- Serves as the of the group
- Represented as (0 : 1 : 0) in projective coordinates
- Allows for a well-defined group operation on all points of the curve
- Conceptualized as the point where all vertical lines intersect
Abelian group properties
- ensures sum of any two points remains on the curve
- (P + Q) + R = P + (Q + R) holds for all points P, Q, R
- Commutativity P + Q = Q + P simplifies calculations
- Inverse element -P exists for every point P on the curve
Geometric interpretation
- Addition of points visualized using lines intersecting the curve
- used for doubling a point (adding a point to itself)
- Reflects the deep connection between algebraic and geometric properties
- Provides intuitive understanding of group operations on elliptic curves
Addition law
- Defines how to combine points on an elliptic curve
- Forms the basis for the group structure and cryptographic applications
- Allows for efficient computation of multiples of points
Chord-and-tangent method
- Geometric approach to adding two distinct points P and Q
- Draw a line through P and Q, find the third intersection point R
- Reflect R across the x-axis to obtain P + Q
- For doubling, use the tangent line at P to find 2P
Explicit formulas
- Algebraic expressions for computing coordinates of P + Q
- For P = (x1, y1) and Q = (x2, y2), slope m = (y2 - y1) / (x2 - x1)
- x3 = m^2 - x1 - x2, y3 = m(x1 - x3) - y1
- Doubling formula when P = Q uses m = (3x1^2 + a) / (2y1)
Special cases
- Adding P and -P results in the O
- Adding any point P to O returns P (O acts as identity element)
- Vertical line intersects curve at P, -P, and O
- Handling of these cases ensures well-defined group operation
Torsion points
- Points of finite order on an elliptic curve
- Play crucial role in understanding the structure of elliptic curves
- Connected to important theorems in arithmetic geometry
Definition and properties
- Torsion point P satisfies nP = O for some positive integer n
- Order of P defined as smallest positive integer n such that nP = O
- form a finite subgroup of the elliptic curve
- Rational torsion points have coordinates in the base field
Torsion subgroups
- Consist of all points of a given order n
- Denoted as E[n] for the subgroup of n-torsion points
- Structure theorem E[n] ≅ Z/nZ ⊕ Z/nZ over algebraically closed fields
- provide insight into the curve's algebraic structure
Nagell-Lutz theorem
- Characterizes rational torsion points on elliptic curves over Q
- States that rational torsion points have integer coordinates
- y-coordinate of a rational torsion point (x, y) ≠ O is either 0 or divides 2a^3 + 27b^2
- Provides a method for finding all rational torsion points on a curve
Mordell-Weil theorem
- Fundamental result in arithmetic geometry
- Describes the structure of rational points on elliptic curves
- Has far-reaching consequences in number theory and cryptography
Statement and significance
- Rational points on an elliptic curve over a number field form a finitely generated abelian group
- Group can be expressed as E(K) ≅ Z^r ⊕ T, where r is the rank and T is the torsion subgroup
- Provides a powerful tool for studying Diophantine equations
- Connects elliptic curves to important conjectures (Birch and Swinnerton-Dyer conjecture)
Weak Mordell-Weil theorem
- Intermediate step in proving the full
- States that the quotient group E(K)/mE(K) is finite for any m > 1
- Relies on the finiteness of the Selmer group
- Provides a foundation for the descent method used in the full proof
Height functions
- Assign a non-negative real number to each rational point on the curve
- Canonical height h(P) measures the arithmetic complexity of a point P
- Satisfies h(nP) = n^2h(P) for any integer n
- Used in the proof of the Mordell-Weil theorem and for finding generators of E(K)
Computational aspects
- Focus on algorithms and methods for working with elliptic curves
- Essential for practical applications in cryptography and number theory
- Enable efficient implementation of elliptic curve-based systems
Point counting algorithms
- Determine the number of points on an elliptic curve over a
- Naive approach checks each possible x-coordinate for corresponding y-values
- More efficient methods include and its improvements
- Point counting crucial for assessing cryptographic strength of curves
Schoof's algorithm
- Polynomial-time algorithm for counting points on elliptic curves over finite fields
- Uses the action of on torsion points
- Computes the trace of Frobenius modulo small primes and combines results using CRT
- Complexity O(log^8 q) for curves over F_q, significant improvement over naive methods
Applications in cryptography
- (ECC) offers smaller key sizes compared to RSA
- (ECDH) for key exchange
- (ECDSA) for digital signatures
- Point multiplication serves as the basis for many cryptographic protocols
Isogenies
- Morphisms between elliptic curves preserving the group structure
- Provide a way to study relationships between different elliptic curves
- Play crucial role in various aspects of elliptic curve theory
Definition and properties
- Rational map φ : E1 → E2 between elliptic curves that preserves the group structure
- Degree of an isogeny defined as the size of its kernel
- Separable have kernels of order equal to their degree
- Inseparable isogenies occur in characteristic p > 0 and involve p-th power maps
Dual isogenies
- For every isogeny φ : E1 → E2, there exists a dual isogeny φ^ : E2 → E1
- Composition φ^ ∘ φ = [deg φ] (multiplication by degree map on E1)
- Dual isogeny has the same degree as the original isogeny
- Allows for the study of relationships between isogenous curves
Isogeny classes
- Set of all elliptic curves isogenous to a given curve E
- Curves in the same isogeny class share many arithmetic properties
- Over finite fields, characterized by the trace of Frobenius
- Important in the study of L-functions and
Elliptic curves over finite fields
- Study of elliptic curves defined over fields with finite number of elements
- Crucial for applications in cryptography and coding theory
- Exhibit unique properties not present in curves over infinite fields
Hasse's theorem
- Bounds the number of points on an elliptic curve E over F_q
- States that |#E(F_q) - (q + 1)| ≤ 2√q
- Provides a tight estimate for the group order
- Fundamental result in the theory of elliptic curves over finite fields
Frobenius endomorphism
- Endomorphism φ : E → E defined by (x, y) ↦ (x^q, y^q) over F_q
- Satisfies the characteristic equation φ^2 - tφ + q = 0 on E
- Trace t determines the number of points #E(F_q) = q + 1 - t
- Eigenvalues of Frobenius provide information about the curve's structure
Supersingular vs ordinary curves
- have trace of Frobenius t ≡ 0 (mod p)
- have t ≢ 0 (mod p)
- Supersingular curves have maximum p-torsion and special endomorphism rings
- Distinction important for cryptographic applications and theoretical study
Complex multiplication
- Theory of elliptic curves with extra endomorphisms
- Connects elliptic curves to algebraic number theory
- Provides powerful tools for generating cryptographically strong curves
Endomorphism ring
- Ring of endomorphisms End(E) of an elliptic curve E
- For generic curves over C, End(E) ≅ Z
- CM curves have larger endomorphism rings isomorphic to orders in imaginary quadratic fields
- Structure of End(E) determines many arithmetic properties of the curve
CM fields
- Imaginary quadratic fields K = Q(√-d) where d > 0 is square-free
- Elliptic curves with CM by K have isomorphic to an order in K
- classified by their discriminant and class number
- Provide a rich source of elliptic curves with special properties
Class number formula
- Relates the class number of a CM field to special values of L-functions
- For imaginary quadratic field K, h(K) = w√|D| / (2π) L(1, χ)
- w is the number of roots of unity in K, D is the discriminant
- Connects complex analysis, algebraic number theory, and elliptic curves
Modular curves
- Parametrize elliptic curves with additional structure
- Fundamental objects in the theory of modular forms
- Provide geometric interpretation of modular equations
Moduli interpretation
- classify isomorphism classes of elliptic curves with level structure
- X(N) parametrizes elliptic curves with full level N structure
- X0(N) parametrizes elliptic curves with a cyclic subgroup of order N
- Points on modular curves correspond to elliptic curves with specific properties
Modular forms
- Holomorphic functions on the upper half-plane with transformation properties
- Closely related to modular curves through q-expansions
- Eisenstein series and cusp forms provide examples of modular forms
- Modular forms of weight 2 correspond to differential forms on modular curves
Connection to elliptic curves
- Modularity theorem states that every elliptic curve over Q is modular
- Implies that L-functions of elliptic curves are L-functions of modular forms
- Allows for the study of arithmetic properties of elliptic curves using modular forms
- Crucial in the proof of Fermat's Last Theorem and other important results
Key Terms to Review (43)
Abelian group: An abelian group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility, and additionally, the operation is commutative. This means that for any two elements in the group, the order in which they are combined does not affect the result. This structure is essential when discussing the group law on elliptic curves, where points on the curve can be added together to form a new point while adhering to the properties of an abelian group.
Abelian group properties: Abelian group properties refer to the characteristics of a mathematical structure known as an abelian group, where the group operation is both associative and commutative, and there exists an identity element along with inverses for every element. This structure is crucial when studying elliptic curves, as it allows for a well-defined addition operation on the points of the curve, making them behave like a group in a coherent and predictable way.
Associativity: Associativity is a fundamental property of binary operations that states the way in which operations are grouped does not affect the result. In the context of group law on elliptic curves, this means that when adding three points together, the order in which you perform the addition does not change the final outcome, allowing for a consistent and well-defined group structure.
Carl Friedrich Gauss: Carl Friedrich Gauss was a prominent German mathematician and scientist, known for his contributions across various fields, including number theory, statistics, and algebra. His work laid the groundwork for many modern mathematical concepts, influencing areas such as elliptic curves, units in algebraic number theory, and the principles behind Diophantine approximation.
Chord and Tangent Rule: The chord and tangent rule states that if two points on an elliptic curve are connected by a chord, and a tangent is drawn at one of those points, then the intersection of the tangent with the curve creates a third point. This concept is crucial in understanding how points on elliptic curves interact, forming the basis for the group law, which allows for the addition of points on the curve.
Class Number Formula: The class number formula relates the class number of a number field to its L-functions and regulators, serving as a critical bridge between algebraic number theory and analytic properties of L-functions. This formula provides insight into the distribution of ideals in a number field and connects the arithmetic of the field with its geometric properties, such as those found in elliptic curves. By analyzing the relationships between class groups and L-functions, this formula highlights the interplay between algebraic structures and their analytical counterparts.
Closure: In the context of elliptic curves, closure refers to the concept of forming a complete group by including all limits of sequences of points under the group operation. This notion is essential for understanding the group law on elliptic curves, as it ensures that operations on points remain within the curve, allowing us to treat points as elements of a complete algebraic structure. Closure is also closely tied to properties like associativity and identity, which are crucial for defining how points interact within this mathematical framework.
Cm fields: CM fields, or complex multiplication fields, are a special class of number fields that have a certain algebraic structure related to complex tori and elliptic curves. They are characterized by the presence of an endomorphism ring that contains the integers of an imaginary quadratic field, allowing for rich interactions with the theory of elliptic curves, particularly in terms of their group laws and rational points.
Dual Isogenies: Dual isogenies are a special type of morphism between elliptic curves that relate two curves via their group structures. They essentially provide a way to connect two elliptic curves through a map that preserves the group operation, offering insights into the underlying arithmetic properties of these curves.
Edwards Curve: An Edwards curve is a type of elliptic curve defined by a specific mathematical equation that facilitates efficient arithmetic operations, making it especially useful in cryptographic applications. These curves exhibit desirable properties such as fast addition and point doubling, which are crucial for implementing cryptographic protocols securely and efficiently. Their structure allows for a simplified group law, which is central to understanding elliptic curve operations.
Elliptic Curve Cryptography: Elliptic Curve Cryptography (ECC) is a method of public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows for secure communication by using smaller key sizes compared to other encryption methods, while still providing a high level of security. This efficiency makes it particularly useful in environments with limited resources, like mobile devices, and relies on the group law defined on elliptic curves for encryption and decryption processes.
Elliptic Curve Diffie-Hellman: Elliptic Curve Diffie-Hellman (ECDH) is a key exchange protocol that allows two parties to generate a shared secret over an insecure channel using elliptic curves. The protocol relies on the mathematical properties of elliptic curves and their associated group laws, enabling secure communication by allowing parties to exchange keys without directly sharing the keys themselves.
Elliptic Curve Digital Signature Algorithm: The Elliptic Curve Digital Signature Algorithm (ECDSA) is a cryptographic algorithm used to create digital signatures based on the mathematics of elliptic curves. It leverages the properties of elliptic curves over finite fields to provide a high level of security with shorter keys, making it efficient and suitable for various applications like secure communications and blockchain technologies. The security of ECDSA is based on the difficulty of the Elliptic Curve Discrete Logarithm Problem, connecting it to the underlying group law on elliptic curves.
Endomorphism Ring: The endomorphism ring is a mathematical structure that consists of all endomorphisms of an object, such as an elliptic curve, along with the operations of addition and composition. This ring captures the symmetries of the object and provides important insights into its structure, particularly in the context of group laws, isogenies, and complex multiplication, which can lead to a deeper understanding of the underlying algebraic geometry.
Finite Field: A finite field is a set equipped with two operations, addition and multiplication, that satisfy the field properties and contains a finite number of elements. This concept is crucial in number theory and algebra, particularly because every finite field can be constructed from a prime power, which allows for consistent arithmetic operations. Finite fields are especially significant in the study of elliptic curves as they provide a structured environment where group laws can be defined and analyzed.
Frobenius Endomorphism: The Frobenius endomorphism is a fundamental operation in algebraic geometry and number theory, particularly relating to the structure of varieties over finite fields. It maps a point in an algebraic variety to its 'p-th power,' where 'p' is the characteristic of the field, thereby providing insights into the properties of the variety and its points. This endomorphism plays a critical role in understanding group laws on elliptic curves, l-adic representations, and the relationships between different algebraic structures.
Hasse's Theorem: Hasse's Theorem refers to a fundamental result in the theory of elliptic curves that connects the properties of the group law on these curves with the number of rational points over finite fields. This theorem provides a way to estimate the number of solutions to the equation defining an elliptic curve, thereby linking algebraic geometry with number theory and giving insight into the structure of elliptic curves over finite fields.
Height Functions: Height functions are mathematical tools used to measure the complexity of algebraic numbers and points on varieties. They help quantify how 'large' or 'complicated' these numbers or points are, often in relation to their coordinates. This concept is especially useful in number theory and geometry, where understanding the properties of points on elliptic curves, complex tori, arithmetic surfaces, and dynamical systems is crucial for deeper insights into their structure and relationships.
Identity element: An identity element is a special type of element in a mathematical structure that, when combined with any other element in that structure, leaves the other element unchanged. In the context of elliptic curves, this concept is crucial to understanding the group law because it helps define how points on the curve interact with each other through addition. The identity element essentially serves as the 'zero' in the group, allowing for a consistent way to perform operations involving the curve's points.
Intersection Points: Intersection points are the specific coordinates where two or more curves meet or cross each other in a given plane. In the context of elliptic curves, these points play a crucial role in defining the group law, where the geometric properties of the curves and their intersections are used to establish addition operations between points on the curve.
Isogenies: Isogenies are morphisms between elliptic curves that preserve the group structure, meaning they provide a way to map one curve to another while maintaining the properties of the group operation. They are vital in understanding the relationships between elliptic curves, as they can help establish connections, classify them, and enable the transfer of information from one curve to another. Isogenies are particularly significant in arithmetic geometry for studying the rational points on elliptic curves and their applications in number theory.
Isogeny Classes: Isogeny classes are sets of elliptic curves that are related to each other through isogenies, which are non-constant morphisms that preserve the group structure of the curves. Each isogeny class contains curves that share the same number of points over a given finite field, highlighting deep connections in their algebraic structure. This concept plays a crucial role in understanding how elliptic curves can be transformed into one another while maintaining essential properties, and it connects to both the group laws on elliptic curves and the endomorphism algebras associated with these curves.
Modular Curves: Modular curves are algebraic curves that parametrize the isomorphism classes of elliptic curves with certain level structures, providing a bridge between number theory and geometry. They play a crucial role in understanding the properties of modular forms and their connection to elliptic curves, linking the theory of modular groups, group laws on elliptic curves, the Modularity Theorem, and comparison theorems.
Modular forms: Modular forms are complex analytic functions on the upper half-plane that are invariant under the action of a modular group and exhibit specific transformation properties. They play a central role in number theory, especially in connecting various areas such as elliptic curves, number fields, and the study of automorphic forms.
Moduli interpretation: Moduli interpretation refers to the way in which objects, such as curves or surfaces, can be classified and parametrized by their geometric or algebraic properties. In the context of elliptic curves, this means understanding how different curves can be represented within a certain family and how their structures relate to one another. This approach allows mathematicians to study the complex relationships and transformations between these curves, revealing deep insights into their behavior and classification.
Mordell-Weil Group: The Mordell-Weil Group is an important concept in arithmetic geometry, specifically concerning the study of rational points on elliptic curves. It describes the group of rational points on an elliptic curve defined over a number field, equipped with a group structure based on the curve's addition law. Understanding the Mordell-Weil Group provides insights into the number of rational solutions and their algebraic properties, tying together various aspects of elliptic curves and number theory.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over a number field is finitely generated. This fundamental result connects the theory of elliptic curves with algebraic number theory, revealing the structure of rational solutions and their relationship to torsion points and complex multiplication.
Mordell's Theorem: Mordell's Theorem states that any elliptic curve defined over a number field has a finite number of rational points. This powerful result connects the world of elliptic curves to the study of rational solutions of equations, showing that while elliptic curves can exhibit complex behavior, the rational points on them are surprisingly limited. Understanding this theorem provides key insights into the properties of elliptic curves and their relation to Diophantine equations, as well as influences on broader questions in number theory.
Nagell-Lutz Theorem: The Nagell-Lutz Theorem is a result in arithmetic geometry that provides conditions for when a point on an elliptic curve, defined over the integers, has an integer coordinate. Specifically, it states that if the order of a point on an elliptic curve is finite, then the coordinates of that point are integers if they lie on the Weierstrass form of the elliptic curve. This theorem plays a crucial role in understanding the structure and properties of elliptic curves and their rational points.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking work in the field of algebra and the theory of elliptic functions. His contributions laid essential groundwork for understanding elliptic curves, which are pivotal in the study of group law on these curves. Abel's work demonstrated that certain equations could not be solved by radicals, which influenced the development of modern algebra and number theory.
Ordinary Curves: Ordinary curves are a special class of algebraic curves characterized by having a single point of singularity or none at all. These curves are important in the study of elliptic curves because their properties help establish the group structure needed to define operations such as addition and scalar multiplication on the points of the curve, ultimately leading to the development of the group law on elliptic curves.
Point Addition: Point addition is a mathematical operation defined on elliptic curves that allows for the combination of two points to yield a third point on the curve. This operation is fundamental to the group law of elliptic curves, which endows the set of points on an elliptic curve with the structure of a group, enabling various applications in number theory and cryptography.
Point at Infinity: The point at infinity is a crucial concept in the study of elliptic curves, representing a unique point that acts as the identity element in the group law on these curves. It allows us to complete the set of points on an elliptic curve, providing a way to define addition of points that is consistent and well-behaved. This point essentially ensures that every line drawn on the elliptic curve intersects it at least once, thus playing a vital role in the structure of the curve's group.
Point Counting Algorithms: Point counting algorithms are computational methods used to determine the number of rational points on algebraic varieties, particularly on elliptic curves over finite fields. These algorithms are crucial for cryptographic applications and number theory, as they help in analyzing the structure of elliptic curves and understanding their group properties. They enable mathematicians to efficiently calculate how many solutions exist to an equation defining an elliptic curve modulo a prime number.
Schoof's Algorithm: Schoof's Algorithm is an efficient method for counting the number of points on an elliptic curve over a finite field. It uses properties of the elliptic curve and modular arithmetic to compute the number of points without directly enumerating them. This method is especially important in cryptography, as it provides a way to determine the group structure of elliptic curves, which is essential for understanding their applications in secure communications.
Secp256k1: secp256k1 is a widely used elliptic curve defined over a finite field that is particularly notable for its application in cryptographic systems, especially Bitcoin. The curve is part of the Standards for Efficient Cryptography (SEC) and provides a secure method for generating public-private key pairs through its group law properties, making it essential for digital signatures and key exchanges.
Supersingular Curves: Supersingular curves are a special class of elliptic curves defined over a finite field, characterized by their unique properties in the context of the group law on elliptic curves. These curves have a singular point that plays a crucial role in their arithmetic properties, particularly in how they interact with the Frobenius endomorphism. Supersingular curves have applications in cryptography and coding theory due to their distinct features, such as non-ordinary behavior in their point counting.
Symmetric point: A symmetric point refers to a point on an elliptic curve that is directly opposite another point with respect to the x-axis. In the context of elliptic curves, if you have a point P represented as (x, y), the symmetric point is typically denoted as -P, which corresponds to (x, -y). This concept is crucial in understanding how points interact under the group law on elliptic curves, as it plays a key role in defining the addition operation and the geometric interpretation of these curves.
Tangent Line: A tangent line is a straight line that touches a curve at a single point without crossing it at that point. In the context of elliptic curves, the tangent line plays a crucial role in defining the group law, allowing for the geometric interpretation of addition of points on the curve and the construction of new points from existing ones.
Torsion Points: Torsion points are points on an algebraic group, such as an elliptic curve, that have finite order, meaning they generate a subgroup of the group that repeats after a certain number of additions. They play a crucial role in understanding the structure of elliptic curves, their isogenies, and the behavior of rational points on these curves. Torsion points also relate to the study of complex tori and can influence the properties of abelian varieties and Jacobian varieties.
Torsion Subgroups: A torsion subgroup is a subset of an algebraic group consisting of all the elements that have finite order, meaning they yield the identity element when multiplied by some integer. This concept is crucial in the study of elliptic curves and their group structure, as it helps identify points on the curve that exhibit periodic behavior under the group operation. The torsion subgroup provides insight into the overall structure of the elliptic curve's group and plays a vital role in understanding the solutions to equations defined over various fields.
Weak Mordell-Weil Theorem: The Weak Mordell-Weil Theorem states that for an elliptic curve defined over a number field, the group of rational points on the curve is finitely generated. This theorem provides a significant result in understanding the structure of the group of rational points and connects closely to the broader implications of the Mordell-Weil Theorem, which deals with the group of rational points being isomorphic to a direct sum of a finite group and a free abelian group. The weak version ensures that despite potential complexities in the elliptic curve, one can still ascertain important information about its rational solutions.
Weierstrass Form: The Weierstrass form is a specific equation used to represent elliptic curves, typically given by the equation $$y^2 = x^3 + ax + b$$ where $a$ and $b$ are constants. This form is crucial in studying the properties of elliptic curves, including their group structure, isogenies, and rational points. It serves as a standard representation that simplifies the analysis of elliptic curves and their applications in number theory and algebraic geometry.