Fast architectures for the $\eta_T$ pairing over small-characteristic supersingular elliptic curves

Abstract : This paper is devoted to the design of fast parallel accelerators for the cryptographic $\eta_T$ pairing on supersingular elliptic curves over finite fields of characteristics two and three. We propose here a novel hardware implementation of Miller's algorithm based on a parallel pipelined Karatsuba multiplier. After a short description of the strategies we considered to design our multiplier, we point out the intrinsic parallelism of Miller's loop and outline the architecture of coprocessors for the $\eta_T$ pairing over $\F_{2^m}$ and $\F_{3^m}$. Thanks to a careful choice of algorithms for the tower field arithmetic associated with the $\eta_T$ pairing, we manage to keep the pipelined multiplier at the heart of each coprocessor busy. A final exponentiation is still required to obtain a unique value, which is desirable in most cryptographic protocols. We supplement our pairing accelerators with a coprocessor responsible for this task. An improved exponentiation algorithm allows us to save hardware resources. According to our place-and-route results on Xilinx FPGAs, our designs improve both the computation time and the area-time trade-off compared to previously published coprocessors.
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IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2011, Special Section on Computer Arithmetic, 60 (2), pp.266-281. 〈10.1109/TC.2010.163〉
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Jean-Luc Beuchat, Jérémie Detrey, Nicolas Estibals, Eiji Okamoto, Francisco Rodríguez-Henríquez. Fast architectures for the $\eta_T$ pairing over small-characteristic supersingular elliptic curves. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2011, Special Section on Computer Arithmetic, 60 (2), pp.266-281. 〈10.1109/TC.2010.163〉. 〈inria-00424016v2〉

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