Elliptic Curve Cryptography Scalar Multiplication Using Jacobian Coordinates
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Solution Overview
Problem
Conventional elliptic curve cryptography (ECC) optimizations are not sufficiently general and often focus on specific steps like signature verification, failing to optimize overall system performance due to increased computational cost from side-channel attack countermeasures.
Innovation Solution
A protocol for determining linear combinations of points in ECC schemes using Jacobian projective coordinates, involving scalar multiplications and mixed additions, to enhance computational efficiency and security.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If side-channel attack countermeasures are implemented in ECC schemes, then security is improved, but computational cost increases
Solution Approach 1:
The patent changes the coordinate system parameter from affine coordinates to Jacobian projective coordinates for point representation. This parameter change allows countermeasures to be implemented with reduced computational overhead because the coordinate transformation properties of Jacobian coordinates enable more efficient scalar multiplication operations while maintaining security requirements
Solution Approach 2:
The patent segments the scalar multiplication operation into multiple iterations of doubling and mixed addition operations. By breaking down the scalar multiplication into discrete steps that operate on Jacobian coordinates, the countermeasures can be applied incrementally at each step rather than requiring complete reimplementation of the entire multiplication process, reducing overall computational cost
2Productivity
If conventional optimization techniques are applied to signature verification, then verification speed is improved, but overall system performance is not sufficiently optimized
Solution Approach 1:
The patent develops a universal optimization approach based on Jacobian projective coordinates that can be applied to multiple ECC operations including scalar multiplication, point addition, and signature verification. This universal coordinate system and associated algorithms provide multi-functionality, allowing the same optimized operations to serve different cryptographic purposes rather than requiring separate optimizations for each operation
Solution Approach 2:
The patent transitions from optimizing a single verification step to optimizing the entire cryptographic system by changing the dimensional approach from scalar-based operations to coordinate-based operations. By working in Jacobian projective coordinates, the optimization extends across multiple dimensions of the cryptographic protocol, improving overall system performance rather than just verification speed
Data Source
AI summary
A method and protocol for determining linear combinations of a first and second point for an elliptic curve cryptography scheme, including determining a first scalar multiplication of the first point with a first scalar, the first scalar multiplication including performing iteratively in relation to the value of the first scalar either one of: doubling of the first point in Jacobian projective coordinates; or mixed addition with the first point in affine coordinates; determining a combination point by adding the second point to the resultant of the first scalar multiplication; obtaining an affine coordinate representation of the combination point; determining a second scalar multiplication of the combination point with a second scalar, the second scalar multiplication including performing iteratively in relation to the value of the second scalar either one of: doubling of the combination point in Jacobian projective coordinates; or mixed addition with the combination point in affine coordinates.