Fully Homomorphic Encryption for Small Batches of Complex Numbers

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We propose a new approximate homomorphic encryption scheme to process small batches of complex numbers. Leveled operations are based on a new shaped rounding, whose error is a multiple of a fixed public ring element $g$, small on the data slots and larger on the others. The unused slots form a guard band that is computed on but never decoded, and the polynomial scaling factor $g^{-1}\Delta$ generalizes the CKKS scaling factor, with $g=1$ recovering CKKS. Guard-domain bounds determine the scaling