2.3 Elliptic curves over binary fields

Elliptic curves over binary fields are a fascinating area of study in cryptography. These curves, defined over finite fields of characteristic 2, offer unique properties that make them efficient and secure for various cryptographic applications.

The arithmetic of these curves involves point addition, doubling, and scalar multiplication. Understanding these operations and their efficient implementation is crucial for developing robust elliptic curve cryptosystems in binary fields.

Definitions of elliptic curves over binary fields

• Elliptic curves over binary fields are a type of algebraic curve defined over a finite field of characteristic 2
• Binary fields, denoted as $\mathbb{F}_{2^m}$, consist of elements that can be represented using $m$ bits
• Elliptic curves over binary fields have unique properties and are widely used in cryptography due to their efficiency and security

Weierstrass equation for binary fields

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• The Weierstrass equation for an elliptic curve over a binary field $\mathbb{F}_{2^m}$ is given by $y^2 + xy = x^3 + ax^2 + b$, where $a, b \in \mathbb{F}_{2^m}$
• The coefficients $a$ and $b$ must satisfy the condition $b \neq 0$ to ensure the curve is non-singular
• Example: $y^2 + xy = x^3 + x + 1$ over $\mathbb{F}_{2^3}$

Group law for elliptic curves over binary fields

• The set of points on an elliptic curve over a binary field, together with a special point called the "point at infinity" denoted as $\mathcal{O}$, form an abelian group under the group law
• The group law defines the rules for point addition and point doubling on the curve
• The group law for binary fields differs from that of prime fields due to the characteristic of the field
• Example: Given points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$, the sum $P + Q = (x_3, y_3)$ is computed using specific formulas involving the coordinates of $P$ and $Q$

Arithmetic of elliptic curves over binary fields

• Arithmetic operations on elliptic curves over binary fields involve point addition, point doubling, and scalar multiplication
• These operations form the basis for various cryptographic protocols and algorithms based on elliptic curves
• Efficient implementation of these operations is crucial for the performance and security of elliptic curve cryptosystems

Point addition in binary fields

• Point addition is the operation of adding two distinct points on an elliptic curve over a binary field to obtain a third point on the curve
• The formula for point addition in binary fields is derived from the group law and involves the coordinates of the two points being added
• Example: Given points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$, the sum $P + Q = (x_3, y_3)$ is computed as:
  • $x_3 = \lambda^2 + \lambda + x_1 + x_2 + a$
  • $y_3 = \lambda(x_1 + x_3) + x_3 + y_1$
  • where $\lambda = \frac{y_1 + y_2}{x_1 + x_2}$

Point doubling in binary fields

• Point doubling is the operation of adding a point to itself on an elliptic curve over a binary field
• The formula for point doubling in binary fields is a special case of the point addition formula when the two points being added are the same
• Example: Given a point $P = (x_1, y_1)$, the doubled point $2P = (x_3, y_3)$ is computed as:
  • $x_3 = \lambda^2 + \lambda + a$
  • $y_3 = x_1^2 + (\lambda + 1)x_3$
  • where $\lambda = x_1 + \frac{y_1}{x_1}$

Scalar multiplication in binary fields

• Scalar multiplication is the operation of multiplying a point on an elliptic curve over a binary field by a scalar (an integer)
• Scalar multiplication is computed using a combination of point addition and point doubling operations
• Efficient algorithms for scalar multiplication, such as the double-and-add method or the Montgomery ladder, are used to optimize the computation
• Example: Given a point $P$ and a scalar $k$, the scalar multiple $kP$ is computed by repeatedly applying point addition and point doubling operations based on the binary representation of $k$

Isomorphism classes of elliptic curves over binary fields

• Isomorphism classes of elliptic curves over binary fields classify curves that have the same structure and properties
• Two elliptic curves over a binary field are isomorphic if there exists an isomorphism (a bijective rational map) between them that preserves the group structure
• Understanding isomorphism classes is important for identifying equivalent curves and optimizing implementations

j-invariant for binary fields

• The j-invariant is a numerical invariant associated with an elliptic curve that characterizes its isomorphism class
• For an elliptic curve over a binary field with Weierstrass equation $y^2 + xy = x^3 + ax^2 + b$, the j-invariant is given by $j = \frac{a^{12}}{b^{2}}$
• Curves with the same j-invariant are isomorphic, while curves with different j-invariants belong to different isomorphism classes
• Example: The curves $y^2 + xy = x^3 + x + 1$ and $y^2 + xy = x^3 + x$ over $\mathbb{F}_{2^3}$ have the same j-invariant and are isomorphic

Isomorphisms of elliptic curves over binary fields

• An isomorphism between two elliptic curves over a binary field is a bijective rational map that preserves the group structure
• Isomorphisms can be used to map points from one curve to another, enabling the transfer of cryptographic protocols between isomorphic curves
• Example: The map $(x, y) \mapsto (u^2x + r, u^3y + u^2sx + t)$, where $u \neq 0$, $r$, $s$, $t$ are constants in the binary field, defines an isomorphism between elliptic curves over binary fields

Discrete logarithm problem in binary fields

• The discrete logarithm problem (DLP) in binary fields is the problem of finding an integer $k$ such that $Q = kP$, given points $P$ and $Q$ on an elliptic curve over a binary field
• The DLP forms the basis for the security of many elliptic curve cryptosystems, as it is believed to be computationally infeasible for properly chosen curves and field sizes
• Algorithms for solving the DLP in binary fields, such as Pollard's rho and Baby-step giant-step, have a time complexity that is exponential in the field size

Pollard's rho algorithm for binary fields

• Pollard's rho algorithm is a probabilistic algorithm for solving the DLP on elliptic curves over binary fields
• The algorithm uses a pseudo-random walk on the elliptic curve to find a collision, which leads to the solution of the DLP
• Pollard's rho has an expected running time of $O(\sqrt{\pi n/2})$, where $n$ is the order of the base point $P$
• Example: Given points $P$ and $Q$ on an elliptic curve over a binary field, Pollard's rho algorithm finds an integer $k$ such that $Q = kP$ by iteratively computing pseudo-random points until a collision occurs

Baby-step giant-step algorithm for binary fields

• The Baby-step giant-step (BSGS) algorithm is a deterministic algorithm for solving the DLP on elliptic curves over binary fields
• BSGS is based on the idea of precomputing a table of baby steps $(jP)$ and then taking giant steps $(Q - iP)$ to find a collision
• The algorithm has a time complexity of $O(\sqrt{n})$ and a space complexity of $O(\sqrt{n})$, where $n$ is the order of the base point $P$
• Example: Given points $P$ and $Q$ on an elliptic curve over a binary field, BSGS computes a table of baby steps $(jP)$ for $j = 0, 1, ..., \lfloor\sqrt{n}\rfloor$ and then searches for a collision with giant steps $(Q - iP)$ for $i = 0, 1, ..., \lfloor\sqrt{n}\rfloor$

Supersingular vs ordinary curves

• Elliptic curves over binary fields can be classified as either supersingular or ordinary based on their group structure and security properties
• The distinction between supersingular and ordinary curves is important for selecting appropriate curves for cryptographic applications
• Supersingular curves have additional structure that can be exploited for certain cryptographic protocols, but they also have potential vulnerabilities

Definitions of supersingular and ordinary curves

• An elliptic curve over a binary field $\mathbb{F}_{2^m}$ is called supersingular if the trace of the Frobenius endomorphism is divisible by the characteristic of the field (2)
• Equivalently, an elliptic curve over $\mathbb{F}_{2^m}$ is supersingular if the number of points on the curve is equal to $2^m + 1$
• Curves that are not supersingular are called ordinary curves
• Example: The curve $y^2 + y = x^3 + x$ over $\mathbb{F}_{2^m}$ is supersingular for all $m$, while the curve $y^2 + xy = x^3 + x + 1$ over $\mathbb{F}_{2^m}$ is ordinary for all $m$

Differences in group structure and security

• Supersingular curves have a unique group structure that allows for efficient pairings, such as the Weil pairing and Tate pairing
• The additional structure of supersingular curves can be exploited for certain cryptographic protocols, such as identity-based encryption and short signatures
• However, supersingular curves are more susceptible to certain attacks, such as the MOV attack, which reduces the DLP on the curve to the DLP in an extension field
• Ordinary curves, on the other hand, do not have the same additional structure and are generally considered more secure for traditional elliptic curve cryptography
• Example: The MOV attack can reduce the DLP on a supersingular curve over $\mathbb{F}_{2^m}$ to the DLP in $\mathbb{F}_{2^{6m}}$, while ordinary curves are not vulnerable to this attack

Pairings on elliptic curves over binary fields

• Pairings are bilinear maps that take two points on an elliptic curve over a binary field and map them to an element in an extension field
• Pairings, such as the Weil pairing and Tate pairing, have found numerous applications in cryptography, including identity-based encryption, short signatures, and other advanced protocols
• Efficient computation of pairings is crucial for the practicality of pairing-based cryptographic schemes

Weil pairing for binary fields

• The Weil pairing is a bilinear map that takes two points of order $r$ on an elliptic curve over a binary field and maps them to an element in the $r$-th roots of unity in an extension field
• The Weil pairing is defined using the divisor class group of the elliptic curve and has various properties, such as bilinearity, non-degeneracy, and antisymmetry
• Example: Given points $P$ and $Q$ of order $r$ on an elliptic curve over $\mathbb{F}_{2^m}$, the Weil pairing $e_r(P, Q)$ maps to an element in the $r$-th roots of unity in $\mathbb{F}_{2^{km}}$, where $k$ is the embedding degree

Tate pairing for binary fields

• The Tate pairing is another bilinear map that takes a point of order $r$ and a divisor class on an elliptic curve over a binary field and maps them to an element in an extension field
• The Tate pairing is defined using the evaluation of functions on divisors and has similar properties to the Weil pairing, such as bilinearity and non-degeneracy
• The Tate pairing is generally more efficient to compute than the Weil pairing and is often preferred in practical implementations
• Example: Given a point $P$ of order $r$ and a divisor class $D$ on an elliptic curve over $\mathbb{F}_{2^m}$, the Tate pairing $e_r(P, D)$ maps to an element in $\mathbb{F}_{2^{km}}^*/({\mathbb{F}_{2^{km}}^*})^r$, where $k$ is the embedding degree

Applications of pairings in cryptography

• Pairings on elliptic curves over binary fields have enabled the construction of various advanced cryptographic protocols
• Identity-based encryption (IBE) schemes, such as the Boneh-Franklin IBE, use pairings to allow for public keys to be derived from identities, simplifying key management
• Short signature schemes, like the Boneh-Lynn-Shacham (BLS) signature, use pairings to produce compact signatures that are verifiable using the signer's public key
• Other applications include attribute-based encryption, searchable encryption, and anonymous credentials
• Example: The Boneh-Franklin IBE scheme uses the Weil pairing on supersingular curves over binary fields to construct an efficient and secure identity-based encryption system

Efficient implementations of elliptic curves over binary fields

• Implementing elliptic curve cryptography over binary fields requires careful consideration of various factors, such as field representation, curve parameters, and optimization techniques
• Efficient implementations are crucial for the performance and practicality of elliptic curve cryptosystems in real-world applications
• Hardware and software optimizations can significantly improve the speed and scalability of elliptic curve operations over binary fields

Polynomial basis vs normal basis representations

• Binary fields can be represented using either a polynomial basis or a normal basis
• In a polynomial basis representation, field elements are represented as polynomials over $\mathbb{F}_2$ modulo an irreducible polynomial of degree $m$
• In a normal basis representation, field elements are represented as linear combinations of a normal element and its conjugates
• The choice of basis representation can impact the efficiency of field arithmetic and elliptic curve operations
• Example: The polynomial basis representation of $\mathbb{F}_{2^3}$ using the irreducible polynomial $x^3 + x + 1$ represents field elements as polynomials $a_0 + a_1x + a_2x^2$, while a normal basis representation uses a normal element $\beta$ and its conjugates $\beta^2$ and $\beta^4$

Optimizations for point arithmetic in binary fields

• Point arithmetic on elliptic curves over binary fields can be optimized using various techniques to improve performance
• Projective coordinates, such as Lopez-Dahab coordinates, can be used to avoid expensive field inversions by representing points using additional coordinates
• Efficient formulas for point addition and point doubling can be derived by exploiting the properties of binary fields and the curve equation
• Precomputation techniques, like window methods or comb methods, can be used to speed up scalar multiplication by precomputing and storing multiples of the base point
• Example: Using Lopez-Dahab projective coordinates, point addition on an elliptic curve over a binary field can be performed using only field multiplications and squarings, avoiding the need for field inversions

Hardware implementations for binary fields

• Hardware implementations of elliptic curve cryptography over binary fields can provide significant performance improvements compared to software implementations
• Field arithmetic operations, such as multiplication and squaring, can be efficiently implemented using dedicated hardware circuits
• Specialized hardware architectures, like systolic arrays or finite field processors, can be used to parallelize and accelerate elliptic curve operations
• Hardware implementations can also benefit from side-channel countermeasures to protect against attacks like power analysis or timing attacks
• Example: A hardware implementation of elliptic curve scalar multiplication over $\mathbb{F}_{2^{163}}$ using a systolic array architecture can achieve a throughput of several million point multiplications per second while occupying a small area on an FPGA or ASIC