Finite Field Calculation for Efficient Pairing Products
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing cryptographic methods are inefficient in calculating the product of multiple pairings over finite fields, as they perform separate calculations for each pairing and lack a method for cooperative calculation of pairing products.
Innovation Solution
A technique that calculates the product of multiple pairings by using a finite field calculation part to perform extension field calculations and polynomial multiplications, reducing unnecessary multiplications by utilizing polynomials with predetermined zero coefficients, and optimizing the calculation process through a loop process involving selected pairs of elements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If separate calculations are performed for each pairing, then each pairing can be calculated independently, but the overall calculation time and computational complexity increase significantly
Solution Approach 1:
The patent merges multiple separate pairing calculations into a single unified calculation framework. By combining the calculation of multiple pairings (e(P0,Q0), e(P1,Q1), ..., e(PN-1,QN-1)) into one integrated process using shared intermediate values and coordinated loop iterations, the system reduces total computation time while maintaining individual pairing accuracy through systematic computation of each component.
Solution Approach 2:
The patent performs preliminary calculations of intermediate values (such as u[i] = x1[i] + x2[i] + c and g[i] = -u[i]² + y1[i]·y2[i]·σ - u[i]·ρ - ρ²) before final pairing computations. These pre-computed intermediate results are stored and reused across multiple pairing calculations, reducing redundant computations and accelerating the overall process.
2Productivity
If the number of multiplications is reduced, then computational efficiency improves, but the complexity of the calculation method increases
Solution Approach 1:
The patent extracts and removes unnecessary multiplication operations from the standard pairing calculation process. By identifying and eliminating redundant multiplications through algebraic simplification and shared computation of intermediate values, the method reduces the total multiplication count from O(N·M) to approximately O(N+M) operations, significantly improving computational efficiency.
Solution Approach 2:
The patent creates universal intermediate values (u[i], g[i], f[i]) that serve multiple functions across different pairing calculations. These intermediate results are computed once and reused across multiple pairing operations, allowing a single calculation framework to handle multiple pairings efficiently without requiring separate dedicated computation paths for each pairing.
3Speed
If loop repetitions are reduced, then execution speed increases, but the precision of calculation may be compromised
Solution Approach 1:
The patent segments the pairing calculation process into distinct computational phases: preparation phase (computing intermediate values u[i], g[i], f[i] for all i), and final computation phase (computing the product of all pairing results). This segmentation allows the system to perform all necessary computations in an organized manner while minimizing redundant loop repetitions, maintaining precision through systematic computation.
Solution Approach 2:
The patent maintains continuous computation of intermediate values across all pairing indices in a single unified loop structure. By continuously computing u[i], g[i], and f[i] for all i from 0 to N-1 in sequence and then computing the product continuously, the system eliminates idle loop termination determinations while maintaining calculation precision through uninterrupted systematic computation.
Data Source
AI summary
A calculation Πi=0N−1e(R(i, 0), . . . , R(i, K−1)) where a calculation of K elements R(i, k)εGF(pm) of a finite field GF(pm) over a finite field is expressed as e(R(i, 0), . . . , R(i, K−1)) is efficiently performed. Polynomials poly(R(i, 0), . . . , R(i, K−1)) that express a d-th-order extension field of the finite field GF(pm), which are obtained by the calculations e(R(i, 0), . . . , R(i, K−1)) for different values of i, are multiplied by each other, and a cumulative multiplication of the products is performed, for example. The polynomial poly(R(i, 0), . . . , R(i, K−1)) is a mapping from the input elements of the finite field GF(pm), and the coefficients of at least some of the terms thereof are 0. The same process is performed for different sets of values of i, and the calculation Πi=0N−1e(R(i, 0), . . . , R(i, K−1)) is performed using the result.