Some New Methods to Generate Short Addition Chains

TL;DR

This paper introduces novel algorithms for generating shorter addition chains, significantly improving efficiency in modular exponentiation and scalar multiplication for cryptography.

Contribution

It proposes a Simplified Power-tree method and an improved Cross Window method with the Addition Sequence algorithm, reducing computation time and chain length.

Findings

Simplified Power-tree reduces complexity to O(k^2)

Cross Window method achieves up to 9.5% shorter chains

New methods outperform existing techniques in experiments

Abstract

Modular exponentiation and scalar multiplication are important operations of most public key cryptosystems, and their fast calculation is essential to improve the system efficiency. The shortest addition chain is one of the most important mathematical concepts to realize the optimization. However, finding a shortest addition chain of length k is an NP-hard problem, whose time complexity is comparable to O($k!$). This paper proposes some novel methods to generate short addition chains. We firstly present a Simplified Power-tree method by deeply deleting the power-tree, whose time complexity is reduced to O($k^2$) sacrificing some increasing of the addition chain length. Moreover, a Cross Window method and its variant are introduced by improving the Window method. More precisely, the Cross Window method uses the cross correlation to deal with the windows and its pre-computation is optimized by the Addition Sequence algorithm. The theoretical analysis is conducted to show the correctness and effectiveness. Meanwhile, the experiment shows that the new methods can obtain shorter addition chains compared to the existing methods. The Cross Window method with the Addition Sequence algorithm can attain 9.5% reduction of the addition chain length, in the best case, compared to the Window method.