Elliptic Curve Construction for Cryptographic Efficiency
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Solution Overview
Problem
Current cryptographic processes using supersingular elliptic curves are hindered by complex calculations, making them unsuitable for applications with limited computational resources.
Innovation Solution
A method for constructing non-supersingular elliptic curves through a step-by-step process using a single reduction rule, where parameters are generated by randomly selecting prime numbers and applying complex multiplication techniques to simplify calculations, ensuring the elliptic curve has q*q-order points and aligns with the equation y^2=x^3+a*x+b, thereby reducing computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If supersingular elliptic curves are used for cryptographic processes, then security based on bilinear pairing is achieved, but calculation complexity increases making it unsuitable for resource-constrained environments
Solution Approach 1:
The patent changes the type of elliptic curve from supersingular to non-supersingular, and modifies the reduction rules from multiple rules to a single reduction rule. This parameter change maintains the cryptographic security based on bilinear pairing while significantly reducing calculation complexity, making it suitable for resource-constrained environments like chip cards.
Solution Approach 2:
The patent segments the curve construction process into distinct steps: first constructing the curve over a prime field with a single reduction rule, then systematically generating q-order points through scalar multiplication. This segmentation simplifies the overall process by breaking down the complex curve generation into manageable, computationally efficient steps.
2Reliability
If multiple reduction rules are implemented in the finite group construction, then mathematical completeness is maintained, but computational burden increases
Solution Approach 1:
The patent extracts and eliminates unnecessary reduction rules from the finite group construction, retaining only the essential single reduction rule modulo p. This extraction maintains the mathematical correctness needed for cryptographic operations while removing the computational burden of implementing multiple reduction rules, thereby improving calculation efficiency.
3Reliability
If complex multiplication techniques are applied to generate curve parameters, then non-supersingular curves are obtained, but parameter generation complexity increases
Solution Approach 1:
The patent performs preliminary actions by first establishing the prime field and single reduction rule, then systematically generating curve parameters through controlled scalar multiplications. This preliminary structuring of the mathematical framework simplifies subsequent parameter generation, making the overall process more manageable despite the complexity of generating non-supersingular curves.
Data Source
AI summary
The invention relates, mainly to a cryptographic process using an elliptic curve represented by means of an equation containing first and second parameters (a, b), a bilinear matching, and calculations in a finite group of integers constructed around at least one first reduction rule reducing each integer to its remainder in a whole division by a first prime number (p) that constitutes a third parameter, the elements of the finite group being in bijection with points selected on the elliptic curve, and the number of which is linked to a fourth parameter (q), where this process uses public and private keys, each of which is represented by a given point of the elliptic curve or by a multiplication factor between two points of this curve.According to the invention, the first reduction rule is the only reduction rule implemented, and the elliptic curve is obtained through a step-by-step construction process, directly allocating to the finite group q*q q-order points in the elliptic curve.