Cryptology ePrint Archive: Report 2009/398

Fast Architectures for the $\eta_T$ Pairing over Small-Characteristic Supersingular Elliptic Curves

Jean-Luc Beuchat and Jérémie Detrey and Nicolas Estibals and Eiji Okamoto and Francisco Rodríguez-Henríquez

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 $\mathbb{F}_{2^m}$ and $\mathbb{F}_{3^m}$. 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.

Category / Keywords: implementation / Tate pairing, $\eta_T$ pairing, elliptic curve, finite field arithmetic, Karatsuba multiplier, hardware accelerator, FPGA

Date: received 13 Aug 2009, last revised 18 Aug 2009

Contact author: jeanluc beuchat at gmail com

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Version: 20090819:042515 (All versions of this report)

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