Incremental Integer Dataset Assessment via Product and Remainder Trees
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Solution Overview
Problem
Existing methods for assessing datasets of integers, particularly in the context of asymmetric encryption, are computationally intensive and time-consuming, especially when detecting common factors among large numbers of keys, due to the need to recompute product and remainder trees for every possible combination, which is burdensome in terms of processing and time.
Innovation Solution
The method involves splitting the dataset into subsets, generating product and remainder data structures for each subset, and only recomputing these structures for newly added integers, allowing for incremental assessment by reusing existing data structures and computing only distinct ordered pairs, thereby reducing the computational load and time required.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the Batch-GCD algorithm is used to compute the GCD of many integers at once, then the assessment of common factors can be performed, but the processing time becomes excessively long (many days) and the computational complexity increases significantly
Solution Approach 1:
The patent pre-computes and stores product trees and remainder trees for subsets of the dataset before new integers are added. When new integers are introduced, only incremental updates are performed rather than complete recomputation. This preliminary preparation of data structures enables rapid incremental assessment, reducing processing time from many days to minutes or hours while maintaining reliable detection of common factors.
Solution Approach 2:
The patent divides the dataset into smaller subsets and computes product trees and remainder trees for each subset independently. This segmentation allows parallel processing and enables incremental updates when new integers are added - only the affected subsets need to be reprocessed. The segmented approach reduces computational complexity and processing time compared to treating the entire dataset as a single unit.
2Reliability
If the Batch-GCD algorithm processes all integers in the dataset, then complete assessment is achieved, but the computational load and data transfer requirements become burdensome
Solution Approach 1:
Product trees and remainder trees are pre-computed and cached for all subsets before new integers are added to the dataset. This preliminary computation stores intermediate results that can be reused during incremental assessments. When new integers arrive, the system only needs to perform limited updates using the pre-computed structures, dramatically reducing computational complexity and data transfer requirements while maintaining complete assessment coverage.
Solution Approach 2:
The patent implements a dynamic assessment approach where the computational scope adapts based on changes in the dataset. Instead of statically processing all integers every time, the system dynamically identifies only the subsets affected by new integers and processes only those. This dynamic adaptation reduces computational complexity from O(n²) for complete reassessment to a much smaller fraction, while still ensuring complete assessment through systematic coverage of affected regions.
3Reliability
If traditional Batch-GCD algorithms are used, then all key pairs are assessed, but the method cannot efficiently handle incremental additions to the dataset
Solution Approach 1:
The system pre-computes product trees and remainder trees for all initial subsets and stores them for reuse. When new integers are incrementally added to the dataset, the pre-computed structures serve as a foundation, requiring only incremental updates rather than complete reassessment. This maintains full coverage of all key pairs (including new ones) while dramatically improving processing efficiency from days to minutes or hours.
Solution Approach 2:
The patent enables continuous assessment capability where the system maintains ready-to-use product trees and remainder trees that can immediately process new integers as they arrive. Unlike traditional Batch-GCD that requires stopping and complete reassessment, this approach allows continuous incremental updates, maintaining both complete coverage of all key pairs and high processing efficiency through uninterrupted useful action.
Data Source
AI summary
A testing method for verifying keys uses a dataset of integers, the dataset being previously split into subsets of the integers, each subset of the integers having a product data structure for a product of the integers in the subset. Each ordered pair of subsets in the dataset has a remainder data structure for factors of the integers in the subsets of the ordered pair. The method includes creating a subset including integers to be added to the dataset of integers, and generating a product data structure for the created subset, the product data structure based on computing a product of the integers in the created subset. The method also includes identifying distinct ordered pairs of subsets, each distinct ordered pair of subsets including a subset from the dataset and the created subset.