Accelerating residue-to-binary conversion of very high cardinality moduli set for fully homomorphic encryption

Recent hardware implementations of fully homomorphic encryption (FHE) exploit very high cardinality arbitrary moduli sets to parallelize large integer arithmetic. However, the benefit they gained are heavily offset by the slow residue-to-binary conversion due to the large modulo operations and limited number theoretic properties of arbitrary moduli. This paper presents a fast residue-to-binary (R2B) conversion method by transforming a large number of residues of arbitrary moduli into only three residues before the conversion. It exploits the smaller and parallel operations of base extension for the mapping to {2ⁿ-1, 2ⁿ, 2ⁿ +1} and the highly efficient adder-based R2B converter of the latter to achieve 2.25 ~ 6.8 times speedup over recent R2B converters, which translates to approximately 10.2% to 32.08% of overall speed improvement for their FHE implementations.