Reciprocal Calculation in Residue Field Modulo Power of 2
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Solution Overview
Problem
In embedded devices with low computing power, particularly in IoT applications, calculating reciprocals for multiplied values with a larger number of bits than the original value is computationally intensive, especially in residue field modulo operations like RSA cryptography, leading to performance issues.
Innovation Solution
The method limits the multiplying value to odd numbers on a residue field modulo a power of 2, allowing the reciprocal to be calculated efficiently by expressing the multiplied value with the same bit number as the original, using a process that updates coefficients to ensure the least significant bit is zero, thereby reducing the calculation load to O(n).
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If reciprocal calculation is performed on multiplied values with larger bit numbers in residue field modulo operations, then cryptographic security is maintained, but computational load increases significantly
Solution Approach 1:
The patent changes the parameter of the residue field from modulo a prime number to modulo a power of 2. This parameter change enables the use of optimized reciprocal calculation algorithms that exploit the special structure of powers of 2, reducing the computational complexity from O(n^2) to O(n) while maintaining cryptographic security through proper selection of odd multipliers in the residue field.
Solution Approach 2:
The patent segments the reciprocal calculation process into iterative steps that update coefficients sequentially. Instead of computing the full reciprocal at once, the method divides the calculation into stages where each stage refines the approximation, allowing intermediate results to be used and reducing the overall computational burden on embedded devices.
2Measurement precision
If reciprocal calculation is performed on multiplied values with larger bit numbers, then accurate cryptographic operations are achieved, but calculation time increases
Solution Approach 1:
The patent performs preliminary actions by pre-selecting odd multipliers and pre-establishing the residue field modulo power of 2 structure before cryptographic operations. This preliminary setup enables faster reciprocal calculations during actual cryptographic operations, as the computational framework is already in place and optimized for the specific modulus structure.
Solution Approach 2:
The patent employs periodic iterative updates in the reciprocal calculation process, where coefficients are updated in sequential stages. Each iteration refines the reciprocal approximation periodically, balancing accuracy requirements with time constraints by stopping after sufficient iterations rather than computing to full precision.
3Adaptability or versatility
If conventional reciprocal calculation methods are used in embedded devices, then general-purpose computation is maintained, but performance deteriorates due to computational intensity
Solution Approach 1:
The patent adapts the computational parameters to match the capabilities of embedded devices by changing from general-purpose modulo prime operations to specialized modulo power-of-2 operations. This parameter specialization maintains sufficient cryptographic adaptability while dramatically improving performance on resource-constrained devices with limited computational power.
Data Source
AI summary
With respect to a method for execution by an information processing apparatus, the method includes calculating a reciprocal in multiplication on a residue field modulo a power of 2.