Modular Inverse Calculation via Factorization for Smart Cards
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Solution Overview
Problem
Existing methods for modular inversion, such as those used in cryptographic applications like the RSA method, are computationally expensive and inefficient, particularly for low-power processors like those in smart cards, due to their high computing time requirements.
Innovation Solution
The method involves breaking down the module into at least two factors to calculate auxiliary values, which are then used to determine the modular inverse, reducing the computational load by splitting the calculation into smaller, more manageable parts.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the extended Euclidian algorithm is used for modular inversion, then the calculation can be performed, but the computing time is several times longer than other elementary modular computing operations
Solution Approach 1:
The patent applies segmentation by breaking down the modular inversion problem into multiple smaller sub-problems. The modulus M is factorized into M = M1 * M2, and the inversion is computed separately for each factor using auxiliary values R1 and R2, then combined to obtain the final result. This divides one expensive computation into several cheaper computations.
Solution Approach 2:
The patent transforms the single-module inversion problem into a multi-dimensional solution space by introducing the factorization dimension. Instead of solving E*R ≡ 1 (mod M) directly, it solves two separate equations E*R1 ≡ 1 (mod M1) and E*R2 ≡ 1 (mod M2), then combines them using the Chinese remainder theorem approach, adding a dimensional aspect to the solution.
2Use of energy by moving object
If modular inversion is executed on a low-power processor such as a smart card processor, then portability is maintained, but the high computing time requirement becomes problematic
Solution Approach 1:
The patent segments the computationally intensive modular inversion into smaller tasks that can be executed on low-power processors. By factorizing M into M1 and M2, the processor performs two smaller inversions instead of one large inversion, reducing peak computational demands and energy consumption while maintaining correctness.
3Productivity
If the module is broken down into factors and auxiliary values are calculated, then the computational load is reduced, but the device complexity increases
Solution Approach 1:
The patent introduces segmentation which inherently increases algorithmic steps (factorization, multiple inversions, combination), but this controlled complexity increase yields disproportionate productivity gains. The structured segmentation into well-defined phases makes the complexity manageable despite the additional steps.
Data Source
AI summary
A method for calculating the modular inverse of a value in relation to a module is used for cryptographic calculations on a portable data carrier. The method includes determining a breakdown of the module into at least two factors, calculating a respective auxiliary value for each of the factors, wherein each auxiliary value is the modular inverse of the value in relation to the respective factor as module, and calculating the modular inverse of the value in relation to the module using the calculated auxiliary values. The method offers an increase in efficiency, with greater efficiency obtained the stronger the computing outlay depends on the length of the module in the inversion method. The method is suitable for execution by relatively low-power processors, and security of the calculation against spying attacks is not impaired. If security requirements are high, combining the method with suitable measures against spying presents no problems.