8.1 Elliptic curve point multiplication algorithms

Elliptic curve point multiplication is a crucial operation in cryptography. It involves computing multiples of points on elliptic curves, which forms the basis for key generation, encryption, and digital signatures in elliptic curve cryptography (ECC).

Efficient algorithms for point multiplication are essential for practical ECC implementations. This topic covers various methods to optimize point multiplication, including binary, window-based, and endomorphism-based approaches, as well as techniques to protect against side-channel attacks.

Basics of elliptic curve point multiplication

• Point multiplication is a fundamental operation in elliptic curve cryptography that involves computing the multiple of a point on an elliptic curve by a scalar value
• The operation is denoted as $[k]P$ where $k$ is the scalar multiplier and $P$ is a point on the elliptic curve
• Point multiplication is used in key generation, encryption, decryption, and digital signature algorithms in ECC
• The security of ECC relies on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP), which involves finding the scalar multiplier given a point and its multiple

Efficient algorithms for point multiplication

• Efficient algorithms for point multiplication are essential for practical implementation of elliptic curve cryptography
• Naive approaches, such as repeated point addition, have a linear time complexity and are not suitable for real-world applications
• Several optimization techniques have been developed to reduce the number of point additions and improve the performance of point multiplication

Binary method for point multiplication

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• The binary method is a simple and widely used algorithm for point multiplication
• It represents the scalar multiplier $k$ in binary form and performs point doubling and addition based on the binary digits
• For each binary digit, the algorithm performs a point doubling, and if the digit is 1, it also performs a point addition
• The binary method has a time complexity of $O(\log k)$ point operations

Window-based methods for faster computation

• Window-based methods are an optimization of the binary method that processes multiple bits of the scalar multiplier at a time
• The scalar multiplier is divided into windows of a fixed or variable size, and pre-computed points are used to reduce the number of point additions
• Examples of window-based methods include the fixed-window method and the sliding-window method
• Window-based methods can significantly reduce the number of point operations compared to the binary method

Sliding window method vs fixed window method

• The sliding window method is an improvement over the fixed window method
• In the fixed window method, the window size is constant throughout the computation, while in the sliding window method, the window size varies based on the binary representation of the scalar multiplier
• The sliding window method skips consecutive zeros in the binary representation, reducing the number of point doublings
• The sliding window method requires more pre-computed points than the fixed window method but generally achieves better performance

Scalar multiplication in elliptic curve cryptography

• Scalar multiplication is the cornerstone of elliptic curve cryptography (ECC) and is used in various cryptographic protocols
• In ECC, the private key is a scalar value, and the corresponding public key is a point on the elliptic curve obtained by multiplying the base point by the private key
• Scalar multiplication is used in key exchange protocols (ECDH), digital signature algorithms (ECDSA), and encryption schemes (ECIES)

Importance of scalar multiplication in ECC

• The security of ECC relies on the hardness of the elliptic curve discrete logarithm problem (ECDLP)
• Given a point $P$ on an elliptic curve and its multiple $[k]P$, it is computationally infeasible to determine the scalar multiplier $k$
• The hardness of ECDLP enables smaller key sizes in ECC compared to other public-key cryptosystems (RSA, DSA) while maintaining the same level of security
• Efficient and secure implementations of scalar multiplication are crucial for the performance and security of ECC-based systems

Hardness of elliptic curve discrete logarithm problem

• The elliptic curve discrete logarithm problem (ECDLP) is the problem of finding the scalar multiplier $k$ given a point $P$ and its multiple $[k]P$ on an elliptic curve
• The hardness of ECDLP is the foundation of security in elliptic curve cryptography
• The best known algorithms for solving ECDLP have a subexponential time complexity, making it infeasible to solve for properly chosen elliptic curves and key sizes
• The hardness of ECDLP enables ECC to achieve the same level of security as other public-key cryptosystems with smaller key sizes, leading to more efficient implementations

Projective coordinates for point multiplication

• Projective coordinates are an alternative representation of points on an elliptic curve that can improve the efficiency of point multiplication
• In affine coordinates, a point is represented by a pair of coordinates $(x, y)$ satisfying the elliptic curve equation
• Projective coordinates introduce an additional coordinate (e.g., $Z$) to represent points, allowing for more efficient point arithmetic

Advantages of projective coordinates over affine

• Projective coordinates eliminate the need for expensive field inversions during point addition and doubling operations
• Field inversions are replaced by less expensive field multiplications, which can significantly speed up point multiplication
• Projective coordinates also provide a unified representation for both affine points and the point at infinity, simplifying the implementation of point arithmetic

Jacobian coordinates for efficient implementation

• Jacobian coordinates are a popular choice of projective coordinates for elliptic curve point multiplication
• In Jacobian coordinates, a point $(X, Y, Z)$ corresponds to the affine point $(X/Z^2, Y/Z^3)$ on the elliptic curve
• Jacobian coordinates allow for efficient point doubling and addition formulas that minimize the number of field multiplications and squarings
• Many efficient implementations of ECC use Jacobian coordinates to achieve better performance compared to affine coordinates

López-Dahab coordinates for binary curves

• López-Dahab coordinates are a type of projective coordinates specifically designed for elliptic curves over binary fields
• In López-Dahab coordinates, a point $(X, Y, Z)$ corresponds to the affine point $(X/Z, Y/Z^2)$ on the elliptic curve
• López-Dahab coordinates provide efficient point doubling and addition formulas for binary curves, which are commonly used in embedded systems and hardware implementations
• The use of López-Dahab coordinates can lead to faster point multiplication on binary curves compared to affine coordinates

Endomorphisms for accelerating point multiplication

• Endomorphisms are mappings from an elliptic curve to itself that preserve the group structure
• Certain classes of elliptic curves possess efficient endomorphisms that can be used to accelerate point multiplication
• Endomorphisms allow for the decomposition of the scalar multiplier into smaller components, reducing the number of point operations required for point multiplication

Frobenius endomorphism in characteristic p curves

• The Frobenius endomorphism is a special endomorphism available on elliptic curves defined over finite fields of characteristic $p$
• The Frobenius endomorphism maps a point $(x, y)$ to $(x^p, y^p)$, where $p$ is the characteristic of the field
• The Frobenius endomorphism can be efficiently computed and has a low cost compared to point addition and doubling operations
• Elliptic curves with efficient Frobenius endomorphisms, such as Koblitz curves, can benefit from faster point multiplication

Gallant-Lambert-Vanstone (GLV) method using endomorphisms

• The Gallant-Lambert-Vanstone (GLV) method is a technique that uses endomorphisms to accelerate point multiplication on certain classes of elliptic curves
• The GLV method decomposes the scalar multiplier $k$ into two smaller components $k_1$ and $k_2$ using the endomorphism
• The point multiplication $[k]P$ is then computed as $[k_1]P + [k_2]\phi(P)$, where $\phi$ is the endomorphism
• The GLV method can significantly reduce the number of point operations required for point multiplication, leading to faster execution times

Guillevic-Ionica method for fast endomorphisms

• The Guillevic-Ionica method is an extension of the GLV method that allows for the use of endomorphisms on a wider range of elliptic curves
• The method uses a combination of the Frobenius endomorphism and a non-trivial endomorphism to decompose the scalar multiplier into four smaller components
• The point multiplication is then computed using a combination of the smaller scalar multiplications and the endomorphisms
• The Guillevic-Ionica method can achieve even faster point multiplication compared to the GLV method on suitable elliptic curves

Side-channel attacks on point multiplication

• Side-channel attacks are a class of attacks that exploit information leakage from the physical implementation of cryptographic algorithms
• In the context of elliptic curve cryptography, side-channel attacks can target the point multiplication operation to recover the secret scalar multiplier
• Common side-channel attacks on point multiplication include timing attacks, power analysis attacks, and electromagnetic emanation attacks

Timing attacks on scalar multiplication algorithms

• Timing attacks exploit the differences in execution time of point multiplication algorithms depending on the value of the scalar multiplier
• If the execution time of point multiplication depends on the bits of the scalar multiplier, an attacker can observe the timing variations and deduce information about the secret key
• Timing attacks can be mitigated by ensuring that the execution time of point multiplication is independent of the scalar multiplier, using techniques such as fixed-time algorithms or blinding

Power analysis attacks on ECC implementations

• Power analysis attacks measure the power consumption of a device during the execution of point multiplication to extract information about the scalar multiplier
• Simple power analysis (SPA) attacks observe a single power trace to deduce the sequence of point operations and recover the scalar multiplier
• Differential power analysis (DPA) attacks use statistical analysis of multiple power traces to recover the scalar multiplier, even in the presence of noise
• Power analysis attacks can be mitigated using techniques such as randomization, masking, or using constant-time algorithms

Countermeasures for side-channel attack resistance

• Countermeasures are techniques used to protect point multiplication implementations against side-channel attacks
• Randomization countermeasures introduce random values into the point multiplication algorithm to make the side-channel information unpredictable to the attacker
• Masking countermeasures split the sensitive data (scalar multiplier) into multiple shares and perform computations on the shares to prevent direct leakage of the sensitive data
• Constant-time algorithms ensure that the execution time and power consumption of point multiplication are independent of the scalar multiplier, making it difficult for an attacker to extract useful information
• A combination of countermeasures is often used to provide comprehensive protection against various side-channel attacks

Batch point multiplication techniques

• Batch point multiplication refers to the simultaneous computation of multiple point multiplications on an elliptic curve
• Batch point multiplication techniques aim to optimize the computation of $[k_1]P_1 + [k_2]P_2 + ... + [k_n]P_n$, where $k_i$ are scalar multipliers and $P_i$ are points on the elliptic curve
• Batch point multiplication is used in scenarios where multiple point multiplications need to be computed efficiently, such as in signature verification or multi-party protocols

Straus-Shamir method for simultaneous point multiplication

• The Straus-Shamir method, also known as Shamir's trick, is a technique for computing the sum of multiple point multiplications
• The method is based on the observation that the sum of point multiplications can be computed more efficiently than individual point multiplications
• The Straus-Shamir method represents the scalar multipliers in binary form and performs point additions and doublings based on the binary representation
• The method can achieve a significant speedup compared to computing individual point multiplications and then adding the results

Lim-Lee method for fixed-base point multiplication

• The Lim-Lee method is a technique for efficiently computing multiple point multiplications with a fixed base point
• In the Lim-Lee method, the base point is pre-computed and stored in a table, allowing for faster computation of point multiplications
• The method splits the scalar multipliers into smaller windows and uses the pre-computed table to perform the point multiplications
• The Lim-Lee method is particularly useful in scenarios where multiple point multiplications with the same base point need to be computed, such as in signature verification

Bernstein's Batch Edwards point multiplication method

• Bernstein's Batch Edwards point multiplication method is a technique for efficient batch point multiplication on Edwards curves
• Edwards curves are a class of elliptic curves that provide fast and complete point addition formulas, making them suitable for efficient implementation
• Bernstein's method uses a combination of point addition and doubling formulas specific to Edwards curves to optimize the computation of batch point multiplication
• The method achieves high performance by exploiting the properties of Edwards curves and using optimized algorithms for point arithmetic
• Bernstein's Batch Edwards point multiplication method is widely used in modern elliptic curve cryptography implementations for its efficiency and security