Abstract
On the basis of a software implementation of Kummer based HECC over \(\mathbb {F}_p\) presented in 2016, we propose new hardware architectures. Our main objectives are: definition of architecture parameters (type, size and number of units for arithmetic operations, memory and internal communications); architecture style optimization to exploit internal parallelism. Several architectures have been designed and implemented on FPGAs for scalar multiplication acceleration in embedded systems. Our results show significant area reduction for similar computation time than best state of the art hardware implementations of curve based solutions.
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Notes
- 1.
Our CCABA model is inspired by Transaction Level Modeling (TLM) with full cycle accuracy for all control signals at the architecture level but not inside the units (when there is no input/output impact).
References
Ahmadi, H.-R., Afzali-Kusha, A., Pedram, M., Mosaffa, M.: Flexible prime-field genus 2 hyperelliptic curve cryptography processor with low power consumption and uniform power draw. ETRI J. 37(1), 107–117 (2015)
Alrimeih, H., Rakhmatov, D.: Fast and flexible hardware support for ECC over multiple standard prime fields. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 22(12), 2661–2674 (2014)
Batina, L., Mentens, N., Preneel, B., Verbauwhede, I.: Flexible hardware architectures for curve-based cryptography. In: Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS), pp. 4839–4842. IEEE, May 2006
Bernstein, D.J., Lange, T.: Explicit-formulas database. http://hyperelliptic.org/EFD/
Bos, J.W., Costello, C., Hisil, H., Lauter, K.: Fast cryptography in genus 2. J. Cryptol. 29(1), 28–60 (2016)
Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Maths and Applications. Chapman & Hall/CRC, London (2005)
Elias, G., Miri, A., Yeap, T.-H.: On efficient implementation of FPGA-based hyperelliptic curve cryptosystems. Comput. Electr. Eng. 33(5), 349–366 (2007)
Fan, J., Batina, L., Verbauwhede, I.: HECC goes embedded: an area-efficient implementation of HECC. In: Avanzi, R.M., Keliher, L., Sica, F. (eds.) SAC 2008. LNCS, vol. 5381, pp. 387–400. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04159-4_25
Gallin, G., Tisserand, A.: Hyper-threaded multiplier for HECC. In: Proceedings of 51st Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA. IEEE, October 2017
Gaudry, P.: Fast genus 2 arithmetic based on theta functions. J. Math. Cryptol. 1(3), 243–265 (2007)
Güneysu, T., Paar, C.: Ultra high performance ECC over NIST primes on commercial FPGAs. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 62–78. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85053-3_5
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004). https://doi.org/10.1007/b97644
Kim, H.W., Wollinger, T., Choi, Y.J., Chung, K.I., Paar, C.: Hyperelliptic curve coprocessors on a FPGA. In: Lim, C.H., Yung, M. (eds.) WISA 2004. LNCS, vol. 3325, pp. 360–374. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31815-6_29
Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptol. 1(3), 139–150 (1989)
Koc, C.K., Acar, T., Kaliski, B.S.: Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro 16(3), 26–33 (1996)
Satoh Laboratory and Morita Tech: Side-channel attack user reference architecture (SAKURA) (2013)
Lai, J.-Y., Wang, Y.-S., Huang, C.-T.: High-performance architecture for elliptic curve cryptography over prime fields on FPGAs. Interdiscip. Inf. Sci. 18(2), 167–173 (2012)
Lange, T.: Formulae for arithmetic on genus 2 hyperelliptic curves. Appl. Algebra Eng. Commun. Comput. 15(5), 295–328 (2005)
Ma, Y., Liu, Z., Pan, W., Jing, J.: A high-speed elliptic curve cryptographic processor for generic curves over \(\rm GF(p)\). In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 421–437. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43414-7_21
Mangard, S., Oswald, E., Popp, T.: Power Analysis Attacks: Revealing the Secrets of Smart Cards. Springer, Heidelberg (2007). https://doi.org/10.1007/978-0-387-38162-6
Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)
Montgomery, P.L.: Speeding the pollar and elliptic curves methods of factorisation. Math. Comput. 48(177), 243–264 (1987)
Orup, H.: Simplifying quotient determination in high-radix modular multiplication. In: Proceedings of 12th IEEE Symposium on Computer Arithmetic (ARITH), pp. 193–199, Bath, UK. IEEE, July 1995
Renes, J., Schwabe, P., Smith, B., Batina, L.: \(\mu \)Kummer: efficient hyperelliptic signatures and key exchange on microcontrollers. In: Gierlichs, B., Poschmann, A.Y. (eds.) CHES 2016. LNCS, vol. 9813, pp. 301–320. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53140-2_15
Sakiyama, K., Batina, L., Preneel, B., Verbauwhede, I.: Superscalar coprocessor for high-speed curve-based cryptography. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 415–429. Springer, Heidelberg (2006). https://doi.org/10.1007/11894063_33
Sghaier, A., Massoud, C., Zeghid, M., Machhout, M.: Flexible hardware implementation of hyperelliptic curves cryptosystem. Int. J. Comput. Sci. Inf. Secur. (IJCSIS) 14(4), 1–7 (2016)
Wollinger, T.: Software and hardware implementation of hyperelliptic curve cryptosystems. Ruhr University Bochum (2004)
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This work was done in the HAH project http://h-a-h.inria.fr/ partially funded by Labex CominLab, Labex Lebesgue and Brittany Region.
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Gallin, G., Celik, T.O., Tisserand, A. (2017). Architecture Level Optimizations for Kummer Based HECC on FPGAs. In: Patra, A., Smart, N. (eds) Progress in Cryptology – INDOCRYPT 2017. INDOCRYPT 2017. Lecture Notes in Computer Science(), vol 10698. Springer, Cham. https://doi.org/10.1007/978-3-319-71667-1_3
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