Montgomery Value Conversion Using Dual Modular Addition Loops
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Solution Overview
Problem
The existing Montgomery algorithm for numerical value conversion in public key cryptography has limitations in flexibility due to a restricted integer range, requiring high computational complexity and efficiency loss when dealing with large values.
Innovation Solution
A method and system utilizing two modular addition loops followed by a Montgomery modular exponentiation function to generate a Montgomery conversion parameter without integer range constraints, employing low complexity hardware to convert values efficiently between integer and Montgomery domains.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If the current Montgomery algorithm is used for numerical value conversion, then the conversion can be performed within a limited integer range (2^k), but the flexibility and adaptability are insufficient when dealing with larger values
Solution Approach 1:
The patent segments the Montgomery conversion process into two distinct phases: first converting the input value to an intermediate representation using a first modular addition loop, then converting the intermediate value to the final Montgomery domain value using a second modular addition loop. This segmentation allows each loop to handle specific portions of the conversion, enabling support for larger integer ranges while keeping each individual loop's hardware complexity manageable.
Solution Approach 2:
The patent introduces an intermediate value dimension between the input value and the final Montgomery conversion result. Instead of directly converting from the original domain to Montgomery domain in a single step, the invention adds an intermediate representation layer that allows the system to handle larger values by breaking down the conversion process across multiple computational dimensions.
2Productivity
If a single modular addition loop is used for Montgomery conversion, then the hardware complexity is relatively low, but the operation efficiency decreases when handling large integer values
Solution Approach 1:
The conversion process is divided into two sequential modular addition loops, each handling a portion of the conversion task. The first loop processes the input value to generate an intermediate result, and the second loop processes the intermediate value to generate the final Montgomery conversion result. This segmentation improves operational efficiency for large integers by distributing the computational workload across two specialized loops rather than overloading a single loop.
3Adaptability or versatility
If the Montgomery conversion is performed without intermediate steps, then the conversion process is simple, but it cannot handle values beyond the limited integer range
Solution Approach 1:
The patent performs a preliminary conversion action by first transforming the input value into an intermediate representation before performing the final Montgomery conversion. This preliminary step (first modular addition loop) prepares the data in a form that enables the subsequent conversion to handle larger integer ranges, effectively extending the algorithm's capability beyond the original limited range.
Solution Approach 2:
The intermediate value acts as a mediator between the input value and the final Montgomery conversion result. This intermediate representation allows the system to bridge the gap between the limited integer range of traditional Montgomery conversion and the need to handle larger values, enabling extended range support while maintaining the mathematical correctness of the conversion process.
Data Source
AI summary
A numerical conversion method for a public key cryptography system includes accumulating a first value by using a first modular addition loop according to the first value for generating a second value after a first predetermined loop count is reached, accumulating the second value by using a second modular addition loop according to the second value for generating a third value after a second predetermined loop count is reached, inputting the third value to a Montgomery modular exponentiation function for generating a Montgomery conversion parameter, and converting a first conversion value in an integer domain into a second conversion value in a Montgomery domain.