Fixed-Base Multi-Precision Exponentiation Using Lookup Tables
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Solution Overview
Problem
Current computational systems face inefficiencies when performing arithmetic operations on very large numbers, particularly in encryption algorithms like Paillier, RSA, or ElGamal, as they exceed the capabilities of modern general-purpose hardware limited to single or double precision calculations.
Innovation Solution
Implementing a multi-precision exponentiation algorithm using a pre-generated lookup table to efficiently compute exponentiated values of a base number, allowing for faster calculations by pre-calculating and storing these values for later use, thereby enabling arbitrary-precision computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If multi-precision exponentiation is performed using traditional algorithms on very large numbers, then computational accuracy is maintained, but computational time and resource consumption increase significantly
Solution Approach 1:
The patent pre-calculates and stores exponentiated values of the base number in a lookup table before the actual exponentiation operation. This preliminary action allows the algorithm to retrieve pre-computed values during execution, significantly reducing computational time while maintaining accuracy for very large numbers
Solution Approach 2:
The patent divides the large exponent into multiple smaller segments or chunks. By processing the exponentiation in segmented portions and utilizing the lookup table for each segment, the algorithm reduces the computational complexity and time required while preserving the overall computational accuracy
2Productivity
If pre-generated lookup tables are used to store pre-calculated exponentiated values, then computational speed is improved, but memory requirements increase
Solution Approach 1:
The patent optimizes the lookup table by storing only specific pre-calculated values that are most frequently needed or most beneficial for the given base number. This selective approach ensures that memory is used efficiently, storing only the necessary data to achieve speed improvements without unnecessarily increasing memory requirements
Solution Approach 2:
The patent dynamically adjusts the size and content of the lookup table based on parameters such as the base number characteristics, the range of exponents, and available memory resources. This allows the system to optimize the balance between computational speed and memory usage for different operational contexts
3Adaptability or versatility
If arbitrary-precision computations are implemented to handle numbers larger than 64 bits, then computational capability is expanded, but device complexity increases
Solution Approach 1:
The patent performs complex multi-precision arithmetic operations in advance to build the lookup table. By pre-computing these complex operations during table generation, the actual exponentiation algorithm can handle arbitrary-precision numbers without repeatedly performing the complex multi-precision arithmetic during execution, thus managing device complexity
Data Source
AI summary
Systems and methods for efficient fixed-base multi-precision exponentiation are disclosed herein. An example method includes applying a multi-precision exponentiation algorithm to a base number, the multi-precision exponentiation algorithm comprises a pre-generated lookup table used to perform calculations on the base number, the pre-generated lookup table comprising pre-calculated exponentiated values of the base number.