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This paper is devoted to the design and the physical security of a parallel dual-core flexible cryptoprocessor for computing pairings over Barreto-Naehrig (BN) curves. The proposed design is specifically optimized for field-programmable gate-array (FPGA) platforms. The design explores the in-built features of an FPGA device for achieving an efficient cryptoprocessor for computing 128-bit secure pairings. The work further pinpoints the vulnerability of those pairing computations against side-channel attacks and demonstrates experimentally that power consumptions of such devices can be used to attack these ciphers. Finally, we suggest a suitable countermeasure to overcome the respective weaknesses. The proposed secure cryptoprocessor needs 1 730 000, 1 206 000, and 821 000 cycles for the computation of Tate, ate, and optimal-ate pairings, respectively. The implementation results on a Virtex-6 FPGA device shows that it consumes 23 k Slices and computes the respective pairings in 11.93, 8.32, and 5.66 ms.