Field Instruction Multiple Data

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Field Instruction Multiple Data
Field Instruction Multiple Data
Journal Title:
Advances in Cryptology – EUROCRYPT 2022
Publication Date:
25 May 2022
Aung, K. M. M., Lim, E., Sim, J. J., Tan, B. H. M., Wang, H., & Yeo, S. L. (2022). Field Instruction Multiple Data. Lecture Notes in Computer Science, 611–641.
Fully homomorphic encryption (FHE) has flourished since it was first constructed by Gentry (STOC 2009). Single instruction multiple data (SIMD) gave rise to efficient homomorphic operations on vectors in (Ftd)ℓ, for prime t. RLWE instantiated with cyclotomic polynomials of the form X2N+1 dominate implementations of FHE due to highly efficient fast Fourier transformations. However, this choice yields very short SIMD plaintext vectors and high degree extension fields, e.g. ℓ<100,d>100 for small primes (t=3,5,…). In this work, we describe a method to encode more data on top of SIMD, Field Instruction Multiple Data, applying reverse multiplication friendly embedding (RMFE) to FHE. With RMFE, length-k Ft vectors can be encoded into Ftd and multiplied once. The results have to be recoded (decoded and then re-encoded) before further multiplications can be done. We introduce an FHE-specific technique to additionally evaluate arbitrary linear transformations on encoded vectors for free during the FHE recode operation. On top of that, we present two optimizations to unlock high degree extension fields with small t for homomorphic computation: r-fold RMFE, which allows products of up to 2r encoded vectors before recoding, and a three-stage recode process for RMFEs obtained by composing two smaller RMFEs. Experiments were performed to evaluate the effectiveness of FIMD from various RMFEs compared to standard SIMD operations. Overall, we found that FIMD generally had >2× better (amortized) multiplication times compared to FHE for the same amount of data, while using almost k/2× fewer ciphertexts required.
License type:
Publisher Copyright
Funding Info:
This research / project is supported by the A*STAR - Advanced Manufacturing and Engineering (AME) Programmatic Programme
Grant Reference no. : A19E3b0099
This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at:
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