1.4 Elliptic curve point doubling

Elliptic curve point doubling is a key operation in elliptic curve arithmetic. It involves taking a point on the curve and computing its double, which is crucial for many cryptographic algorithms and mathematical applications.

Understanding point doubling is essential for grasping the group law of elliptic curves. It forms the basis for generating cyclic subgroups and plays a vital role in elliptic curve cryptography, factorization methods, and primality proving.

Definition of point doubling

• Point doubling is a fundamental operation in elliptic curve arithmetic that takes a point $P$ on an elliptic curve and computes $2P$, which is the result of adding $P$ to itself
• This operation is crucial for many elliptic curve based algorithms, including key generation, encryption, and digital signature schemes

Algebraic definition

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• Algebraically, point doubling is defined as follows: given a point $P = (x_1, y_1)$ on an elliptic curve $E$, the double of $P$, denoted as $2P = (x_2, y_2)$, is the point obtained by drawing the tangent line to $E$ at $P$ and finding the other point of intersection of this line with $E$
• The resulting point is then reflected across the $x$-axis to obtain $2P$
• The algebraic formula for point doubling depends on the specific form of the elliptic curve equation (short Weierstrass, Montgomery, twisted Edwards)

Geometric interpretation

• Geometrically, point doubling can be visualized as follows: consider an elliptic curve $E$ and a point $P$ on $E$
• To find $2P$, draw the tangent line to $E$ at $P$, which intersects $E$ at another point, say $Q$
• Reflect $Q$ across the $x$-axis to obtain $2P$
• This geometric interpretation provides an intuitive understanding of the point doubling operation and its relationship to the shape of the elliptic curve

Point doubling formula

• The point doubling formula is a set of equations used to compute the coordinates of $2P$ given the coordinates of $P$
• The specific formula depends on the form of the elliptic curve equation being used (short Weierstrass, Montgomery, twisted Edwards)
• These formulas are derived using the algebraic definition of point doubling and the properties of the elliptic curve

Derivation of formula

• The derivation of the point doubling formula involves finding the equation of the tangent line at $P$, substituting it into the elliptic curve equation, and solving for the coordinates of the other point of intersection
• This process involves algebraic manipulation and the use of the elliptic curve's group law properties
• The resulting equations are then simplified to obtain the final point doubling formula

Formula for short Weierstrass form

• For an elliptic curve in short Weierstrass form, $y^2 = x^3 + ax + b$, the point doubling formula is given by:
  • $x_2 = \lambda^2 - 2x_1$
  • $y_2 = \lambda(x_1 - x_2) - y_1$
  • where $\lambda = \frac{3x_1^2 + a}{2y_1}$

Formula for Montgomery form

• For an elliptic curve in Montgomery form, $By^2 = x^3 + Ax^2 + x$, the point doubling formula is given by:
  • $x_2 = \frac{(x_1^2 - 1)^2}{4x_1(x_1^2 + Ax_1 + 1)}$
  • $y_2 = \frac{(2x_1 + x_2 + A)(x_1 - x_2) - 2y_1}{B}$

Formula for twisted Edwards form

• For an elliptic curve in twisted Edwards form, $ax^2 + y^2 = 1 + dx^2y^2$, the point doubling formula is given by:
  • $x_2 = \frac{2xy}{ax^2 + y^2}$
  • $y_2 = \frac{y^2 - ax^2}{2 - ax^2 - y^2}$

Point doubling in group law

• Point doubling plays a crucial role in the group law of elliptic curves, which defines the rules for point addition and multiplication
• The group law states that for any two points $P$ and $Q$ on an elliptic curve, there exists a unique point $R$ such that $P + Q = R$
• Point doubling is a special case of point addition where $P = Q$

Role in generating cyclic subgroup

• Point doubling is essential for generating cyclic subgroups of elliptic curve groups
• Given a point $P$ on an elliptic curve, the cyclic subgroup generated by $P$ is the set of all points that can be obtained by repeatedly applying the point doubling operation to $P$
• The cyclic subgroup generated by $P$ is denoted as $\langle P \rangle = \{P, 2P, 4P, 8P, \ldots\}$

Relationship to point addition

• Point doubling is closely related to point addition, as both operations are part of the elliptic curve group law
• Point addition is the operation of adding two distinct points on an elliptic curve, while point doubling is the operation of adding a point to itself
• The formulas for point addition and point doubling share some similarities, but they differ in the specific equations used and the handling of special cases (point at infinity, singularities)

Efficient computation of point doubling

• Efficient computation of point doubling is crucial for the performance of elliptic curve based algorithms, as point doubling is a frequently used operation
• Several techniques can be employed to optimize the computation of point doubling, including the use of different coordinate systems and the application of explicit formulas

Affine coordinates vs projective coordinates

• Elliptic curve points can be represented using different coordinate systems, such as affine coordinates and projective coordinates
• Affine coordinates represent points using two values $(x, y)$, while projective coordinates use three values $(X, Y, Z)$ to represent points
• Projective coordinates can help avoid the need for expensive field inversions during point doubling, which can improve performance

Explicit formulas for point doubling

• Explicit formulas for point doubling are optimized equations that minimize the number of field operations (additions, multiplications, inversions) required to compute $2P$
• These formulas are derived by applying algebraic manipulations and substitutions to the standard point doubling formulas
• Explicit formulas can be tailored to specific elliptic curve forms (short Weierstrass, Montgomery, twisted Edwards) and coordinate systems (affine, projective)

Cost analysis of point doubling

• Cost analysis involves evaluating the computational cost of point doubling in terms of the number and type of field operations required
• The cost of point doubling depends on the elliptic curve form, coordinate system, and the specific explicit formula used
• By analyzing the cost of different point doubling methods, one can choose the most efficient approach for a given elliptic curve based application

Applications of point doubling

• Point doubling is a fundamental operation in elliptic curve arithmetic and has several important applications in cryptography and computational number theory
• These applications rely on the security and efficiency of elliptic curve based algorithms, which in turn depend on the performance of point doubling

Elliptic curve cryptography

• Elliptic curve cryptography (ECC) is a public-key cryptography approach that uses the algebraic structure of elliptic curves over finite fields
• Point doubling is a crucial operation in ECC, as it is used in key generation, encryption, decryption, and digital signature algorithms
• The security of ECC relies on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP), which involves finding the scalar multiple of a given point

Elliptic curve factorization method

• The elliptic curve factorization method (ECM) is an integer factorization algorithm that uses elliptic curves to find small factors of large composite numbers
• ECM relies on the fact that the order of a random elliptic curve modulo a composite number is likely to have small prime factors
• Point doubling is used in ECM to compute scalar multiples of points on elliptic curves, which is a key step in the factorization process

Elliptic curve primality proving

• Elliptic curve primality proving (ECPP) is a method for proving the primality of large integers using elliptic curves
• ECPP is based on the idea that if a certain condition holds for an elliptic curve modulo a given integer, then that integer must be prime
• Point doubling is used in ECPP to compute the order of points on elliptic curves, which is necessary for verifying the primality condition

Challenges in point doubling

• While point doubling is a well-defined operation in elliptic curve arithmetic, there are some challenges that need to be addressed when implementing point doubling in practice
• These challenges include handling special cases, such as the point at infinity and singularities, and ensuring the security of point doubling against side-channel attacks

Handling point at infinity

• The point at infinity, denoted as $\mathcal{O}$, is a special point on an elliptic curve that serves as the identity element for point addition
• When doubling the point at infinity, the result is the point at infinity itself: $2\mathcal{O} = \mathcal{O}$
• Implementations of point doubling must handle this special case correctly to ensure the consistency and correctness of elliptic curve arithmetic

Dealing with singularities

• Singularities are points on an elliptic curve where the tangent line is not well-defined, which can occur when the denominator of the point doubling formula becomes zero
• In the short Weierstrass form, singularities can happen when the $y$-coordinate of the point being doubled is zero
• To handle singularities, implementations of point doubling must check for these special cases and apply appropriate logic to compute the correct result

Resistance to side-channel attacks

• Side-channel attacks are a type of cryptanalytic attack that exploits information leaked by the physical implementation of a cryptographic algorithm, such as timing, power consumption, or electromagnetic radiation
• Point doubling, being a fundamental operation in elliptic curve cryptography, can be a target for side-channel attacks
• To resist these attacks, implementations of point doubling must employ countermeasures, such as constant-time execution, randomization, and masking, to minimize the leakage of sensitive information