Low-Cost Hardware Implementation of Elliptic Curve Cryptography for General Prime Fields
In resource-constrained applications, elliptic curve cryptography (ECC) is preferable for the property of shorter key size with comparable security. Binary extension fields are usually used for area-optimized implementations, since the complex carry-propagation logics are avoided over these fields. However, efficient ECC implementations over (general) prime fields are still challenging for low-area constraint. As a popular implementation platform for cryptographic algorithms, Field Programmable Gate Array (FPGA) attracts more and more attentions for these applications due to its nice properties of flexibility and short development cycle. In this paper, we propose a compact and efficient arithmetic logical unit (ALU) by highly integrating the functions of Montgomery modular multiplications, additions and subtractions over general prime fields. Then we design a low-cost hardware architecture for generic elliptic curve point multiplications for FPGA platforms. Experimental results indicate that the implementation only occupies 105 Slices, 2 DSP blocks and 2 BRAMs in Spartan-6 FPGA. To the best of our knowledge, our implementation is the smallest for general prime fields in FPGAs.
KeywordsElliptic curve cryptography Low-cost FPGA Implementation
We thank the anonymous reviewers of SAC 2016 and ICICS 2016 for their invaluable suggestions and comments. This work was partially supported by National Basic Research Program of China (973 Program No. 2013CB338001), National Natural Science Foundation of China (No. 61602476, No. 61402470) and Strategy Pilot Project of Chinese Academy of Sciences (No. XDA06010702).
- 1.Bosmans, J., Roy, S.S., Järvinen, K., Verbauwhede, I.: A tiny coprocessor for elliptic curve cryptography over the 256-bit NIST prime field. In: 29th International Conference on VLSI Design and 15th International Conference on Embedded Systems, VLSID 2016, Kolkata, India, 4–8 January 2016, pp. 523–528 (2016)Google Scholar
- 3.McIvor, C., McLoone, M., McCanny, J.: An FPGA elliptic curve cryptographic accelerator over GF(p). In: Irish Signals and Systems Conference (ISSC), pp. 589–594 (2004)Google Scholar
- 4.Cericom Research: Standards for Efficient Cryptography - SEC-1: Elliptic curve cryptography (2000). www.secg.org/sec1-v2.pdf
- 5.Cericom Research: Standards for Efficient Cryptography - SEC-2: Recommended Elliptic Curve Domain Parameters (2000). www.secg.org/SEC2-Ver-1.0.pdf
- 6.Furbass, F., Wolkerstorfer, J.: ECC processor with low die size for rfid applications. In: International Symposium on Circuits and Systems (ISCAS) 2007, pp. 1835–1838. IEEE (2007)Google Scholar
- 11.Kern, T., Feldhofer, M.: Low-resource ECDSA implementation for passive RFID tags. In: 17th IEEE International Conference on Electronics, Circuits, and Systems, ICECS 2010, Athens, Greece, 12–15. pp. 1236–1239, December 2010Google Scholar
- 16.Office of State Commercial Cryptography Administration: Public Key Cryptographic Algorithm SM2 Based on Elliptic Curves (2012, in Chinese). http://www.oscca.gov.cn/UpFile/2010122214822692.pdf
- 17.Orup, H.: Simplifying quotient determination in high-radix modular multiplication. In: IEEE Symposium on Computer Arithmetic, pp. 193–199 (1995)Google Scholar
- 18.Pessl, P., Hutter, M.: Curved tags - a low-resource ECDSA implementation tailored for RFID. Radio Freq. Ident.: Secur. Priv. Issues (RFIDSec) 2014, 156–172 (2014)Google Scholar
- 20.Ghosh, S., Alam, M., Chowdhury, D.R., Gupta, I.S.: Parallel crypto-devices for GF(p) elliptic curve multiplication resistant against side channel attacks. In: Comput. Electr. Eng. pp. 329–338 (2009)Google Scholar
- 21.Varchola, M., Guneysu, T., Mischke, O.: Microecc: a lightweight reconfigurable elliptic curve crypto-processor. In: International Conference on Reconfigurable Computing and FPGAs (ReConFig) 2011, pp. 204–210. IEEE (2011)Google Scholar
- 22.Vliegen, J., Mentens, N., Genoe, J., Braeken, A., Kubera, S., Touhafi, A., Verbauwhede, I.: A compact fpga-based architecture for elliptic curve cryptography over prime fields. In: International Conference on Application-Specific Systems Architectures and Processors (ASAP) 2010, pp. 313–316. IEEE (2010)Google Scholar