Sign-Based Partial Reduction for Modular Arithmetic Units
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Solution Overview
Problem
Conventional modular arithmetic operations in cryptographic computing are resource-intensive, particularly on low-bit microprocessors, due to the need for frequent reduction steps within the interval [0, N-1], which burdens processing devices and is inefficient for complex algorithms.
Innovation Solution
Implementing sign-based partial reduction of modular arithmetic operations within an expanded interval [−N, N-1], where the processing device tracks the sign bit of numbers to determine if they belong to the negative or positive half of the interval, allowing for fewer computational operations by shifting results as needed to maintain the interval, thereby reducing the number of bits required to represent results.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional modular arithmetic operations are performed within the interval [0, N-1], then the results are mathematically correct, but the number of computational operations and resource consumption increase significantly
Solution Approach 1:
The patent changes the parameter space from the conventional interval [0, N-1] to an expanded interval [-N, N-1] by utilizing the sign bit. This parameter change allows numbers to be represented with the same number of bits while enabling fewer reduction operations, as results can remain in the expanded interval longer without requiring modulo reduction.
Solution Approach 2:
The patent segments the number representation by separating the sign bit from the magnitude, allowing the most significant bit to indicate the sign (negative or positive half) while the remaining bits represent the magnitude. This segmentation enables tracking of the sign state without requiring additional bits, and allows conditional reduction based on sign bit patterns.
2Reliability
If frequent reduction steps are performed to maintain results within [0, N-1], then mathematical correctness is ensured, but processing time and energy consumption increase
Solution Approach 1:
The patent applies partial reduction by allowing results to remain in the expanded interval [-N, N-1] rather than fully reducing to [0, N-1] after every operation. Reduction is performed selectively based on sign bit conditions, performing only the necessary portion of reduction needed to maintain correctness for subsequent operations, rather than excessive full reduction after each step.
Solution Approach 2:
The patent uses the sign bit as feedback to determine when reduction is needed. By monitoring the sign bits of operands and results, the system dynamically decides whether reduction operations are necessary, creating a feedback mechanism that optimizes the frequency of reduction steps while maintaining mathematical correctness.
3Productivity
If the interval is expanded to [-N, N-1], then fewer reduction operations are needed, but the number of bits required to represent numbers increases
Solution Approach 1:
The patent makes the most significant bit universal by giving it dual functionality: it serves as both the sign indicator and part of the magnitude representation. This multi-functionality allows the expanded interval [-N, N-1] to be represented using the same number of bits as the conventional interval [0, N-1], eliminating the need for additional bits while supporting the expanded range.
Data Source
AI summary
Aspects of the present disclosure involve a method and a system to execute the method to perform a cryptographic operation involving a modulo N computation, the method comprising loading a first integer number and a second integer number, wherein the first integer number and the second integer number are within an interval of 2N integer numbers, and performing an arithmetic operation involving the first integer number and the second integer number, wherein the arithmetic operation is to produce a third integer number, and wherein the arithmetic operation comprises a shifting operation to ensure that the third integer number is inside the interval of 2N integer numbers.