Abstract
Applications such as error detection, cryptography, and data encoding for digital communications employ field polynomial division. For instance, it is the heart of the traditional Extended Euclidean Algorithm (EEA), which performs field inversion for public cryptosystems. The alignment of variables for each cycle of polynomial division requires revaluation of their degrees. Moreover, the unpredictability of this iterative process increases its area-time complexity. In order to make polynomial division over finite fields suitable for VLSI implementations, there were several implicit attempts to implement them in systolic architectures. This paper revisits polynomial division over ternary fields to derive its iterative equations and develop novel hardware architectures based on a former systolic arrays methodology. Finally, the area-time complexity of the resulted designs are analyzed and compared.
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References
Galbraith, S., Harrison, K., Soldera, D.: Implementing the tate pairing. In: ANTS 2002, pp. 324–337. Springer, Heidelberg (2002)
Gebali, F.: Algorithms and Parallel Computing. Wiley, Hoboken (2011)
Guo, J.-H., Wang, C.-L.: Hardware-efficient systolic architecture for inversion and division in GF(\(2^m\)). IEE Proc. Comput. Digit. Tech. 145(4), 272–278 (1998)
Harrison, K., Page, D., Smart, N.P.: Software implementation of finite fields of characteristic three, for use in pairing-based cryptosystems. LMS J. Comput. Math. 5, 181–193 (2002)
Hazmi, I.H., Gebali, F.: Systolic design space exploration of polynomial division over GF(\(2^m\)). In: IEEE 30th CCECE (2017)
Horng, Y.-T., Wei, S.-W.: Fast inverters and dividers for finite field GF(\(2^m\)). In: IEEE APCCAS, pp. 206–211 (1994)
Ibrahim, A., Al-Somani, T., Gebali, F.: New systolic array architecture for finite field inversion. CJECE 40, 23–30 (2017)
Ibrahim, A., Gebali, F.: Scalable and unified digit-serial processor array architecture for multiplication & inversion over GF(\(2^m\)). IEEE Trans. Circuits Syst. I(64), 2894–2906 (2017)
Jeong, Y., Burleson, W.: VLSI array synthesis for polynomial GCD computation. In: ASAP 1993, pp. 536–547. IEEE (1993)
Kawahara, Y., Aoki, K., Takagi, T.: Faster implementation of \(\eta \)T pairing over GF (\(3^m\)) using minimum number of logical instructions for GF(3)-addition. In: ICPB 2008, pp. 282–296. Springer, Heidelberg (2008)
Kung, H.T.: Use of VLSI in algebraic computation: some suggestions. In: 4th SYMSAC 1981, pp. 218–222. ACM (1981)
Rashidi, B., Farashahi, R.R., Sayedi, S.M.: High-performance and high-speed implementation of polynomial basis Itoh-Tsujii inversion algorithm over GF(\(2^m\)). IET Inf. Secur. 11(2), 66–77 (2017)
Sun, Y., Kim, M.S.: A table-based algorithm for pipelined CRC calculation. In: ICC 2010, pp. 1–5. IEEE (2010)
Yavuz, I., Yalçin, S.B. Ö., Koç, C.K.: FPGA implementation of an elliptic curve cryptosystem over GF(\(3^m\)). In: ReConFig 2008, pp. 397–402. IEEE (2008)
Zak, S.H., Hwang, K.: Polynomial division on systolic arrays. IEEE Trans. Comput. 100(6), 577–578 (1985)
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Hazmi, I., Gebali, F., Ibrahim, A. (2019). Systolic Design Space Exploration of Polynomial Division over \(GF(3^m)\). In: Arai, K., Bhatia, R., Kapoor, S. (eds) Proceedings of the Future Technologies Conference (FTC) 2018. FTC 2018. Advances in Intelligent Systems and Computing, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-02683-7_68
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