Optimal Representation for Right-to-Left Parallel Scalar Point Multiplication

2025-08-10

AI Summary

Existing right-to-left (RTL) parallel scalar multiplication on elliptic curves lacks an optimized digit representation framework, leading to suboptimal computational efficiency. Method: We propose the first optimization framework tailored for RTL parallel scenarios, establishing an accurate timing model that jointly accounts for point-doubling and point-addition latencies; we design a representation-generation algorithm minimizing total execution time and analyze it in conjunction with a simplified Robert’s method. Contribution/Results: Theoretical analysis and experimental evaluation demonstrate that the Non-Adjacent Form (NAF) is nearly optimal under RTL parallelism—any alternative representation yields at most a 1% speedup. This work fills a critical theoretical gap in digit representation optimization for RTL parallel scalar multiplication, rigorously confirming NAF’s robustness and providing foundational guidance for high-performance elliptic curve cryptography implementations.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper introduces an optimal representation for a right-to-left parallel elliptic curve scalar point multiplication. The right-to-left approach is easier to parallelize than the conventional left-to-right approach. However, unlike the left-to-right approach, there is still no work considering number representations for the right-to-left parallel calculation. By simplifying the implementation by Robert, we devise a mathematical model to capture the computation time of the calculation. Then, for any arbitrary amount of doubling time and addition time, we propose algorithms to generate representations which minimize the time in that model. As a result, we can show a negative result that a conventional representation like NAF is almost optimal. The parallel computation time obtained from any representation cannot be better than NAF by more than 1%.

Problem

• Optimal representation for right-to-left parallel scalar multiplication
• Mathematical model to minimize computation time for any doubling and addition
• Negative result showing NAF is nearly optimal for parallel computation

Innovation

• Optimal right-to-left parallel representation
• Mathematical model for computation time
• Algorithms minimize doubling and addition time