Gaussian Integer Modulo Reduction via Montgomery Method
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Solution Overview
Problem
Existing methods for reducing Gaussian integers modulo a Gaussian integer modulus are inefficient, requiring divisions and rounding operations, which are computationally costly, especially in cryptographic and error-correction applications.
Innovation Solution
A two-part method is employed to efficiently reduce Gaussian integers, involving the determination of a Gaussian integer congruent to the given integer with reduced norm, followed by a final reduction using Montgomery's method, utilizing bit shifts and truncations to minimize computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a straightforward implementation of modulo reduction is used for Gaussian integers, then the reduction can be performed using standard division and rounding operations, but the computational efficiency is poor due to the complexity of division and rounding operations
Solution Approach 1:
The modulo reduction process is divided into two separate phases: a first phase that computes an intermediate Gaussian integer congruent to the input, and a second phase (Montgomery reduction) that computes the final reduced result. This segmentation allows each phase to use optimized operations suitable for its specific purpose, avoiding the need for complex division and rounding in both phases simultaneously.
Solution Approach 2:
The first phase performs preliminary computation to transform the input Gaussian integer into an intermediate form that is congruent to the original but has properties suitable for efficient Montgomery reduction. This preliminary action eliminates the need for division and rounding during the final reduction phase, as the intermediate form can be reduced using only multiplication, addition, and bit shifting.
2Manufacturing precision
If congruent solutions with smaller norm are computed in cryptographic and error-correction applications, then the solutions are more useful and compact, but the computation becomes relatively inefficient using traditional methods
Solution Approach 1:
The patent replaces the traditional mechanical division and rounding operations with a combination of Gaussian integer multiplication, addition, and bit shifting operations. Specifically, Montgomery reduction substitutes the division-based approach with an algorithm that uses only multiplication by pre-computed constants and conditional addition, which are significantly faster on digital computers.
Solution Approach 2:
The patent changes the parameter representation by working with Gaussian integers in a transformed domain. By selecting a Gaussian integer base and representing numbers in this base, the patent enables efficient computation of congruent solutions with smaller norm through parameterized algorithms that adapt to the specific base chosen, improving both accuracy and speed.
Data Source
AI summary
In the Computer-implemented method for determining a Gaussian integer congruent to a given Gaussian integer modulo a Gaussian integer modulus a Gaussian integer base raised to an integer exponent having a norm smaller than or equal to that of the Gaussian integer modulus and larger than the norm of the difference of the Gaussian integer base raised to the integer exponent and the Gaussian integer modulus is considered, wherein a variable value candidate for the Gaussian integer congruent is considered that is first initialized with the given Gaussian integer and then iteratively decremented by a product of the Gaussian integer modulus and a component-wisely down rounded quotient of the current value of the variable value candidate for the Gaussian integer congruent and the Gaussian integer base raised to the integer exponent, as long as the quotient of the resulting quotient is not vanishing, whereafter the resulting variable value candidate for the Gaussian integer congruent is determined as the Gaussian integer congruent. 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.