Gaussian Integer Reduction Using Shift-Based Modular Arithmetic
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Solution Overview
Problem
The inefficient implementation of modulo reduction for Gaussian integers requires computationally intensive operations, such as complex divisions and multiplications, which are not optimized for cryptographic and error-correction applications.
Innovation Solution
A two-part method is introduced for Gaussian integer reduction, involving the determination of a Gaussian integer congruent to the given integer with reduced norm, followed by a final reduction using computationally efficient digit shifts and comparisons.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If straightforward modulo reduction of Gaussian integers is implemented using complex division and rounding, then the reduction can be performed, but the computational complexity is high and efficiency is poor
Solution Approach 1:
The patent segments the Gaussian integer into real and imaginary parts, and further divides the reduction process into multiple steps involving separate calculations for real and imaginary components. This segmentation allows the use of simpler integer arithmetic operations instead of complex division, significantly reducing computational complexity while maintaining reduction efficiency
Solution Approach 2:
The patent replaces the mechanical complex division operation with a combination of simpler arithmetic operations including multiplication by precomputed values, bit shifting, and addition/subtraction. This substitution eliminates the need for complex division and rounding operations, reducing both computational complexity and improving productivity
2Reliability
If congruent solutions with smaller norm are computed for cryptographic and error-correction applications, then the solutions are more useful, but the computation is relatively inefficient
Solution Approach 1:
The patent precomputes values such as the inverse of the Gaussian integer modulus and other auxiliary quantities before the actual reduction process. These precomputed values are stored and reused during the reduction, ensuring that the final congruent solution has the desired smaller norm while avoiding repeated complex calculations, thus improving computation efficiency without sacrificing solution quality
Solution Approach 2:
The patent transforms the problem from working with complex Gaussian integer arithmetic to working with real integer parameters. By changing the representation and using parameter-based calculations with precomputed values, the method achieves both the required solution quality (smaller norm congruent solutions) and improved computation efficiency
Data Source
AI summary
Computer-implemented method for determining a Gaussian integer congruent to a given Gaussian integer modulo a Gaussian integer modulus, wherein the norm of the Gaussian integer is smaller than the norm of the square of the Gaussian integer modulus, wherein a real integer base raised to a first integer exponent having a norm larger than that of the real and larger than that of the imaginary part of the Gaussian integer modulus is considered, wherein a second integer exponent is considered that is equal to or smaller than -2 and wherein a third integer exponent is considered, that is equal to or larger than the first integer exponent raised by one, and wherein a variable value candidate for the Gaussian integer congruent is considered that is first initialized with the given Gaussian integer and wherein the Gaussian integer is, either fully or in truncated form, decremented by a multiple of the Gaussian integer modulus, wherein the multiple of the Gaussian integer modulus is evaluated by calculating an auxiliary product of a component-wisely down rounded quotient of the current value of the variable value candidate for the Gaussian integer congruent and the real integer base raised to the sum of the first integer exponent and the second integer exponent with a prefactor and by calculating a component-wisely down rounded quotient of this auxiliary product and the real integer base raised to the difference of the third integer exponent and the second integer exponent and multiplying this latter quotient with the Gaussian integer modulus. In the Computer-implemented method for determining a reduction of a given Gaussian integer modulo a Gaussian integer modulus first a Gaussian integer congruent to a modulo reduction of a given Gaussian integer modulo a Gaussian integer modulus is determined with the method described before and the Gaussian integer congruent is further reduced with a final reduction. These methods are used in computer-implemented cryptographic methods and error-correction methods.