He, Y., Ling, S., Shi, Y., Tan, B. H. M., Wang, H., & Yang, A. S. (2026). Fully Homomorphic Encryption on the Ring of Gaussian Periods. In (Editor), Topics in Cryptology – CT-RSAC 2026 (pp. 3–31). Springer Nature Switzerland. https://doi.org/10.1007/978-3-032-22931-1_1
Abstract:
In Geelen and Vercauteren (Eurocrypt 2025), a Generalized BFV (GBFV) fully homomorphic encryption scheme was proposed. Here, a plaintext space of form
was utilized to reduce the number of Single Instruction Multiple Data (SIMD) slots within the initial BFV plaintext space. This lowered its dimension and thus enabled lower latencies as well as greater flexibility in parameter selection. However, to obtain slots of degree 1, the methods of Geelen and Vercauteren limit the choice of plaintext modulus to that of large primes, which can be unnecessary for various use cases.
To resolve this, we propose a generalized method to perform FHE based on a subring of the plaintext polynomial ring. We utilize the decomposition ring
, with which when taking quotient with a rational prime p, already factors into residual fields of dimension 1. From here, we develop methods to perform FHE on subrings of the decomposition ring
, which we refer to as the decomposition subring
. We introduce novel methods to enable both encoding and decoding maps within the decomposition subring
. By utilizing
, we further lower the dimension of the underlying ring, improving upon efficiency while retaining sufficient security. In experiments, we provide a proof-of-concept implementation, demonstrating up to a
improvement in the latency of operations for selected parameters. This approach offers enhanced flexibility in the selection of parameters for FHE with the subring dimension being any suitable divisor of r. This direction also represents the first generalization of the subring approach for FHE.
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Publisher Copyright
Funding Info:
This research / project is supported by the National Research Foundation, Singapore and Infocomm Media Development Authority - Trust Tech Funding Initiative
Grant Reference no. : NA
This research / project is supported by the National Research Foundation, Singapore and Infocomm Media Development Authority - Trust Tech Funding Initiative
Grant Reference no. : DTC-IGC-01
This research / project is supported by the Singapore Ministry of Education - Academic Research Fund Tier 2 Grant
Grant Reference no. : MOE-T2EP20223-0016
Description:
This is a post-peer-review, pre-copyedit version of an article published in Topics in Cryptology – CT-RSAC 2026. The final authenticated version is available online at: http://dx.doi.org/https://doi.org/10.1007/978-3-032-22931-1_1.