Gaussian Integer Reduction Without Division or Rounding
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Solution Overview
Problem
Efficient reduction mechanisms for Gaussian integers modulo a Gaussian integer modulus are lacking, leading to inefficient computations in cryptography and error-correction codes.
Innovation Solution
A two-part method is introduced for determining a Gaussian integer congruent to a given Gaussian integer modulo a Gaussian integer modulus, involving a first part to reduce the norm of the Gaussian integer and a second part for final reduction, utilizing computationally efficient digit shifts and bit operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If straightforward modulo reduction of Gaussian integers is implemented, then correctness of the reduction is ensured, but computational efficiency deteriorates due to requiring division and rounding operations
Solution Approach 1:
The patent replaces the mechanical division and rounding operations with a digit-by-digit subtraction process. Instead of performing a single complex division followed by rounding, the algorithm processes the Gaussian integer modulus digit by digit from most significant to least significant, subtracting appropriate multiples of the modulus at each step. This substitution of the computational mechanism eliminates the need for complex division and rounding while maintaining correctness.
Solution Approach 2:
The patent segments the modulo reduction process into multiple digit-level operations. The Gaussian integer is processed digit by digit, with each digit position handled separately through subtraction of scaled multiples of the modulus. This segmentation transforms a single complex operation into a sequence of simpler, more efficient digit-level operations that can be executed without division or rounding.
2Reliability
If congruent solutions with smaller norm are computed for cryptographic applications, then security and correctness are improved, but computational complexity increases due to inefficient reduction mechanisms
Solution Approach 1:
The patent replaces inefficient division-based reduction with a digit-by-digit subtraction mechanism that naturally produces congruent solutions with smaller norm. By processing digits from most significant to least significant and subtracting appropriate multiples of the modulus at each step, the algorithm ensures the result has reduced norm without requiring complex final rounding or adjustment operations.
Solution Approach 2:
The patent performs preliminary scaling of the Gaussian integer by a factor related to the modulus norm before initiating the digit-by-digit reduction process. This preliminary action ensures that the subsequent subtraction operations will naturally produce a result with the desired smaller norm, eliminating the need for additional post-processing steps to achieve the correct congruence class representative.
Data Source
AI summary
Various embodiments of the teachings herein include a method for generating for encryption. An example includes: determining a Gaussian integer congruent to a given Gaussian integer modulo a Gaussian integer modulus. The norm of the integer is smaller than the norm of the square of the modulus. The method includes considering a real integer base raised to a first integer exponent having a norm larger than that of the real and imaginary parts of the modulus. A second integer exponent is considered equal to or smaller than −2. A third integer exponent is considered, equal to or larger than the first integer exponent incremented by one. The method includes considering a variable value candidate for the Gaussian integer congruent first initialized with the given integer; and decrementing the Gaussian integer by a multiple of the modulus.