Final Exponentiation Computation Device Using Cyclotomic Polynomials
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Solution Overview
Problem
There is no general method for efficiently computing the hard part of final exponentiation in pairing computation, which depends on polynomial parameters of elliptic curves, requiring pre-computation for each curve.
Innovation Solution
A final exponentiation computation device decomposes the exponent portion into easy and hard parts using a polynomial Φk(p(x)) and factorizes the hard part with a homogeneous cyclotomic polynomial Ψn(x, p) to enable efficient computation applicable to many elliptic curves.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a general method for computing the hard part of final exponentiation is developed, then pairing computation efficiency is improved, but the method must work for all elliptic curves which increases algorithmic complexity
Solution Approach 1:
The invention changes the parameter representation by expressing the hard part computation in terms of polynomial coefficients λi(x) rather than curve-specific parameters. This allows the same algorithm to work across different elliptic curves by simply changing the polynomial inputs, achieving both efficiency and universality without increasing algorithmic complexity
Solution Approach 2:
The invention creates a universal computation method that works for all pairing-friendly elliptic curves by formulating the hard part computation using homogeneous cyclotomic polynomials and polynomial coefficients. This single method replaces the need for curve-specific pre-computation, making the algorithm multi-functional across different curve types while maintaining efficiency
2Productivity
If curve-specific pre-computation is performed for each elliptic curve, then final exponentiation efficiency is improved for that curve, but the overall system complexity increases due to multiple pre-computed means
Solution Approach 1:
The invention replaces multiple curve-specific pre-computed means with a single universal method based on homogeneous cyclotomic polynomials. This one method serves all elliptic curves, eliminating the need to maintain separate pre-computation data structures for each curve and thereby reducing system complexity while preserving efficiency
Solution Approach 2:
The invention transforms the approach from storing curve-specific computational parameters to using polynomial coefficient parameters λi(x) that can be universally applied. This parameter transformation allows the same computational framework to handle different curves without requiring separate pre-computation for each, reducing overall system complexity
Data Source
AI summary
A decomposition unit (211) decomposes an exponent portion of a final exponentiation computation portion of pairing computation in an elliptic curve into an easy part and a hard part, the elliptic curve being expressed by a polynomial r(x), a polynomial p(x), a polynomial t(x), an embedding degree k, and an integer u. A factorization unit (212) factorizes the hard part with using a homogeneous cyclotomic polynomial Ψn(x, p). An exponentiation computation unit (22) performs computation of final exponentiation with using the easy part and the factorized hard part.