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Fast polynomial arithmetic for Somewhat Homomorphic Encryption operations in hardware with Karatsuba algorithm

Vincent Migliore 1 Maria Mendez Real 1 Vianney Lapotre 1 Arnaud Tisserand 1 Caroline Fontaine 2, 3 Guy Gogniat 1
1 Lab-STICC_UBS_CACS_MOCS
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
2 Lab-STICC_TB_CID_SFIIS
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
Abstract : Most practical Somewhat Homomorphic Encryption (SHE) schemes require the implementation of fast polynomial arithmetic in the ring Zq[X]/f (X), for a given modulus q and an irreducible polynomial f (X). That is why hardware accelerators usually target the FFT/NTT algorithm, which has the smallest complexity asymptotically. Unlike standard approaches, this paper proposes a Karatsuba-based accelerator. Karatsuba implementation requires 3 steps: Pre-recursions producing several subpolynomials, a term by term multiplication of sub-polynomials, and post-computations to reconstruct the output polynomial. Compared to FFT/NTT, Karatsuba can address various size of polynomials, and is sufficiently flexible to be adapted to specific operations required by SHE schemes. In this paper, we propose a hardware/software co-design where several Karatsuba recursions are made in software, and the remaining ones plus the subpolynomial multiplication are made in hardware. We provide 3 different hardware approaches: An area efficient approach with 3 Karatsuba recursions in hardware, an intermediate design with 4 recursions, and a performance-oriented one with 5 recursions. The study evaluates proposed hardware accelerators for 3 FPGA platforms, the SoCkit and the DE5-net platforms from Terasic, and the Catapult platform from Microsoft. The area efficient approach can evaluate a degree-2559 polynomial multiplication in 2.44 ms and a relinearization/key switching evaluation in 2.29 ms, with an important save of hardware resources compared to FFT/NTT implementations. Compared to [1], our lightweight approach saves 57% of ALM resources, 46% of registers, 99.95% of embedded memory and 30% of DSPs. For the performanceoriented design, the accelerator can evaluate a degree-2559 polynomial multiplication in 1.24 ms and a relinearization/key switching evaluation in 1.1 ms.
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https://hal.archives-ouvertes.fr/hal-01427642
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Submitted on : Wednesday, March 3, 2021 - 5:23:23 PM
Last modification on : Wednesday, November 3, 2021 - 7:54:30 AM

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Vincent Migliore, Maria Mendez Real, Vianney Lapotre, Arnaud Tisserand, Caroline Fontaine, et al.. Fast polynomial arithmetic for Somewhat Homomorphic Encryption operations in hardware with Karatsuba algorithm. International Conference on Field-Programmable Technology (FPT), Dec 2016, Xi’an, China. ⟨10.1109/FPT.2016.7929535⟩. ⟨hal-01427642⟩

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