[Resource Topic] 2024/153: Revisiting the Slot-to-Coefficient Transformation for BGV and BFV

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Revisiting the Slot-to-Coefficient Transformation for BGV and BFV

Authors: Robin Geelen


Numerous algorithms in homomorphic encryption require an operation that moves the slots of a ciphertext to the coefficients of a different ciphertext. We describe an FFT-like method for decomposing this slot-to-coefficient transformation (and its inverse) for BGV and BFV. The proposed method is specific to power-of-two cyclotomic rings and can handle both fully and sparsely packed slots. Previously, such a method was only known for non-power-of-two cyclotomic rings.

Our algorithm admits more freedom in the complexity-depth trade-off than prior works. Moreover, it brings down the computational complexity of the slot-to-coefficient transformation from a linear to a logarithmic number of FHE operations in the best case, which is shown via a detailed complexity analysis. We also provide a proof-of-concept implementation in the Magma bootstrapping library.

ePrint: https://eprint.iacr.org/2024/153

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