Elliptic Curve Cofactor Determination Without Square Roots
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Solution Overview
Problem
Determining the cofactor of an elliptic curve within electronic components like microcontrollers in smart cards is costly due to complex computational operations, especially when using existing methods that require square root calculations.
Innovation Solution
A method to determine the cofactor by comparing n and 6√q through simple operations such as squaring, multiplication, and subtraction, eliminating the need for square root calculations, and using the formula floor((q+2ceil(b/2)+1+1)/n when n>6√q.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If existing methods for determining the cofactor are used (requiring square root calculations), then the cofactor can be determined, but the computational complexity and cost increase significantly
Solution Approach 1:
The patent extracts and removes the computationally expensive square root calculation step from the cofactor determination process. By using the inequality n > 6√q and the formula h = floor((q + 2·ceil(b/2) + 1 + 1)/n), the method determines the cofactor through simple operations (squaring, multiplication, subtraction) without requiring square root calculations, thus eliminating the complexity while maintaining accuracy
Solution Approach 2:
The patent changes the parameter representation by working with n² and 36q instead of n and 6√q. This transformation allows comparison and calculation using only integer arithmetic operations (squaring, multiplication, subtraction) rather than requiring floating-point square root operations, significantly reducing computational complexity for electronic components
2Measurement precision
If square root calculation techniques are implemented in electronic components, then the cofactor can be determined, but modifications to the electronic component are required
Solution Approach 1:
The patent removes the requirement for square root calculation capability from the electronic component. By reformulating the cofactor determination to use only basic arithmetic operations (squaring, multiplication, subtraction), the method can be implemented in standard microcontrollers and smart cards without requiring specialized hardware modifications or additional instruction set implementations
Solution Approach 2:
The patent replaces expensive, complex square root calculation routines with inexpensive, simple arithmetic operations. The new method uses only basic integer arithmetic that is already available in standard electronic components, eliminating the need for costly hardware modifications or specialized mathematical co-processors
Data Source
AI summary
A method and apparatus are proposed for cryptographic computations implemented in an electronic component. The method includes determining the cofactor of an elliptic curve E defined over a finite field Fq with q elements, the elliptic curve comprising a base point P having an order equal to n. The step of determining includes determining a value of floor((q+2ceil(b/2)+1+1)/n) when n>6√q, where the function ceil corresponds to the ceiling function, floor corresponds to the floor function, and b corresponds to the size q in number of bits of q.