Final Exponentiation Calculation Device for BLS Curves
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Solution Overview
Problem
The calculation of final exponentiation in pairing operations on BLS curves with embedding degrees other than those previously studied is inefficient, with no known method to speed up the process effectively.
Innovation Solution
A device that decomposes the exponent part into an easy and hard part using a cyclotomic polynomial and transforms the hard part into a linear sum of the polynomial q(u), facilitating efficient calculation of the final exponentiation in pairing operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If the final exponentiation is calculated using conventional methods on BLS curves with new embedding degrees, then the pairing operation can be performed, but the computational burden is extremely heavy and the calculation speed is slow
Solution Approach 1:
The patent applies segmentation by dividing the final exponentiation calculation into distinct phases: pre-computation of auxiliary values (including cyclotomic polynomial evaluation and intermediate exponentiations) and main computation phase. This segmentation allows the computationally intensive parts to be prepared in advance and reused, significantly reducing the time required during actual pairing operations.
Solution Approach 2:
The patent implements preliminary action through extensive pre-computation of auxiliary values including evaluating cyclotomic polynomials, computing intermediate exponentiations, and preparing transformation matrices before the actual pairing operation. These pre-computed values are stored and reused during the main computation, eliminating redundant calculations and accelerating the final exponentiation process.
2Device complexity
If the exponent part is decomposed using cyclotomic polynomial and transformed into linear sum, then the computational burden is reduced, but the device complexity increases due to additional transformation steps
Solution Approach 1:
The patent applies parameter changes by transforming the exponent representation from a standard form to a decomposition based on cyclotomic polynomials. This parameter transformation allows the exponent to be expressed as a sum of terms with specific algebraic structures, enabling more efficient computation through properties of finite fields and elliptic curves, despite requiring additional transformation steps.
Data Source
AI summary
In a final exponentiation calculation device, a decomposition unit (221) decomposes an exponent part into an easy part and a hard part, using a cyclotomic polynomial, in a final exponentiation calculation part of a pairing operation on an elliptic curve represented by a polynomial r(u), a polynomial q(u), a polynomial t(u), an embedding degree k, and a parameter u. A transformation unit (222) transforms the hard part obtained by decomposition by the decomposition unit (221) into a linear sum of the polynomial q(u). An exponentiation calculation unit (23) calculates the final exponentiation calculation part, using the easy part and the hard part transformed into the linear sum of the polynomial q(u).