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High-speed low-power modular arithmetic for elliptic curve cryptosystems based on the residue number system

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posted on 28.03.2022, 18:20 authored by Shahzad Asif
This thesis presents designs and hardware implementations of modular arithmetic for elliptic curve point multiplication (ECPM). The aim is to speed up elliptic curve cryptography (ECC) architectures and optimise their power consumption. Improvements are made in existing algorithms, and conventional number systems are replaced by residue number systems (RNS) to achieve a high speed for basic arithmetic operations. The proposed ECPM architectures are generic and can be scaled for different key sizes; the hardware implementations in this work are for 256-bit ECPM over prime field Fp. ECPM architectures are optimised in two ways. Firstly, three different hardware architectures are developed for the implementation of an efficient modular multiplier (MM). These architectures, named parallel, serial, and serial-parallel, offer a trade-off between area and delay. The performance of the proposed MM architectures is compared, based on their ASIC (Application Specific Integrated Circuit) and FPGA (Field Programmable Gate Array) implementation results. Moreover, the feasibility of serial MM architecture for practical implementation is proved by its ASIC fabrication using 65 nm CMOS technology. The measurement results for the fabricated chip show that the proposed MM is better than other state-of-the-art MM architectures. Secondly, two ECPM architectures, named multi-key ECPM and single-key ECPM, are proposed; they differ in terms of throughput and hardware complexity. Multi-key ECPM provides a high throughput by processing twenty one keys simultaneously within deep pipeline stages. Single-key ECPM attempts to optimise the hardware cost by resource sharing. Power optimisation techniques are employed to reduce the power consumption of the single-key ECPM. The proposed architectures are implemented on FPGA and ASIC platforms and the results are analysed to discuss the suitability of the proposed ECPM architectures for different applications.


Table of Contents

1. Introduction -- 2. Background -- 3. Counter-based Wallace multipliers -- 4. Modular multiplier using sum of residues in RNS -- 5. Chip fabrication for RNS-based modular multiplier -- 6. Elliptic curve point multiplication -- 7. Conclusions and future work.


Bibliography: pages 261-284 Empirical thesis.

Awarding Institution

Macquarie University

Degree Type

Thesis PhD


PhD, Macquarie University, Faculty of Science and Engineering, Department of Engineering

Department, Centre or School

Department of Engineering

Year of Award


Principal Supervisor

Yinan Kong


Copyright Shahzad Asif 2016. Copyright disclaimer: http://mq.edu.au/library/copyright




1 online resource (xxviii, 284 pages) colour illustrations

Former Identifiers

mq:69329 http://hdl.handle.net/1959.14/1253315