M-ary Precomputation-Based Accelerated Scalar Multiplication Algorithms for Enhanced Elliptic Curve Cryptography

TL;DR

This work addresses the high cost of scalar multiplication in ECC by introducing an M-ary precomputation-based approach that decomposes scalars and precomputes a structured table to accelerate computation. By optimizing the base $B$ and depth $d$, and by employing a space-efficient binary storage variant, the method achieves a time complexity of $\Theta\left(\frac{Q \log p}{\log Q}\right)$ with memory $\Theta\left(\frac{Q \log p}{\log^2 Q}\right)$, demonstrating strong theoretical and practical gains. Empirical validation across secp256k1, secp384r1, and secp521r1 shows substantial reductions in encryption time (up to 59% on secp256k1) and memory usage (up to 30%), along with NS3-based network simulations indicating reduced communication and total simulation times. The results suggest the approach is scalable, hardware-friendly, and broadly applicable to secure communication and high-volume ECC workloads; future work includes extending to MSM, parallelization, and hardware implementations.

Abstract

Efficient scalar multiplication is critical for enhancing the performance of elliptic curve cryptography (ECC), especially in applications requiring large-scale or real-time cryptographic operations. This paper proposes an M-ary precomputation-based scalar multiplication algorithm, aiming to optimize both computational efficiency and memory usage. The method reduces the time complexity from $Θ(Q \log p)$ to $Θ\left(\frac{Q \log p}{\log Q}\right)$ and achieves a memory complexity of $Θ\left(\frac{Q \log p}{\log^2 Q}\right)$. Experiments on ElGamal encryption and NS3-based communication simulations validate its effectiveness. On secp256k1, the proposed method achieves up to a 59\% reduction in encryption time and 30\% memory savings. In network simulations, the binary-optimized variant reduces communication time by 22.1\% on secp384r1 and simulation time by 25.4\% on secp521r1. The results demonstrate the scalability, efficiency, and practical applicability of the proposed algorithm. The source code will be publicly released upon acceptance.

Figures (7)

• Figure 1: Steps for M-ary Precomputation-Based Algorithm, illustrating the process from input initialization, precomputation, scalar decomposition, scalar multiplication, and the final result computation.
• Figure 2: Evaluation on the time consumed by various optimized algorithms for scalar multiplication with increasing number of computations $Q$. Specifically, \ref{['fig:exp1a']}, \ref{['fig:exp1b']}, and \ref{['fig:exp1c']} represent the runtime for the elliptic curves secp256k1, secp384r1, and secp521r1, respectively.
• Figure 3: Evaluation on the proportion of time consumed by various optimized algorithms for scalar multiplication relative to the double-and-add algorithm as the number of computations $Q$ increases. Specifically, \ref{['fig:exp3a']}, \ref{['fig:exp3b']}, and \ref{['fig:exp3c']} represent the proportion for the elliptic curves secp256k1, secp384r1, and secp521r1, respectively.
• Figure 4: Evaluation of the proportion of time consumed by ElGamal Encryption using various optimized scalar multiplication algorithms compared to the double-and-add algorithm, as the plaintext length $L$ increases. Specifically, \ref{['fig:plaintext1']}, \ref{['fig:plaintext2']}, and \ref{['fig:plaintext3']} represent the respective proportions for the elliptic curves secp256k1, secp384r1, and secp521r1.
• Figure 5: NS3 simulation communication model. \ref{['fig:iot1']} demonstrates the overall network topology with point-to-point links and their associated communication metrics, such as transmission delays and data rates. \ref{['fig:iot2']} illustrates the internal architecture of nodes in the NS3 simulation, showing the application layer, protocol stack, and net device components.