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E\({^2}\)CSM: efficient FPGA implementation of elliptic curve scalar multiplication over generic prime field GF(p)

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Abstract

Elliptic curve scalar multiplication (ECSM) is the primitive operation that is also the main computational hurdle in almost all protocols based on elliptic curve cryptography (ECC). This work proposes a novel ECSM hardware architecture by adopting several optimization strategies at circuit and system levels. On the circuit level, it is based on an efficient finite field multiplier that takes fewer clock cycles, produces low latency, and consumes fewer hardware resources. On the system level, Jacobian coordinates with the Montgomery laddering algorithm and a fast scheduling mechanism to execute group operations are adopted. The proposed ECSM design is synthesized and implemented targeting different FPGAs using Xilinx ISE Design Suite. It takes 1.01 ms on the Virtex-7 FPGA to compute a single ECSM operation, occupies 7.1K slices, and achieves 187 MHz frequency. This provides a 30% improvement in computational time with a significantly lower area-time product with better efficiency. Therefore, the proposed ECSM design is better optimized in terms of speed, area-time product, and throughput per slice and hence is suitable for many ECC applications.

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Funding

This work was supported by Science Foundation Ireland under Grant No. 13/RC/2094_P2, co-funded under the European Regional Development Fund through the Southern & Eastern Regional Operational Programme to Lero (Science Foundation Ireland Research Centre for Software), and has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No 754489.

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Authors and Affiliations

  1. Department of Computer Engineering, University of Sharjah, Sharjah, UAE

    Khalid Javeed & Ali El-Moursy

  2. School of Computer Science, Trinity College Dublin, Dublin, Ireland

    David Gregg

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  1. Khalid Javeed
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  2. Ali El-Moursy
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  3. David Gregg
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Contributions

KJ was responsible for implementation and manuscript preparation, AM helped in manuscript preparation and implementation, while DG pointed out several optimization strategies along with manuscript writing and proofreading.

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Correspondence to Khalid Javeed.

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Javeed, K., El-Moursy, A. & Gregg, D. E\({^2}\)CSM: efficient FPGA implementation of elliptic curve scalar multiplication over generic prime field GF(p). J Supercomput 80, 50–74 (2024). https://doi.org/10.1007/s11227-023-05428-4

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