Paper 2026/1592

Sub-Cubic Homomorphic Matrix Multiplication via Self-Dual Normal Bases

Abstract

This paper analyzes the bilinear embedding of matrix algebras into commutative cyclotomic rings. We apply the Cohn-Umans method. This establishes that single-multiplication bilinear monomial embeddings require a ring degree of $\widetilde{\Omega}(N^3)$. We circumvent this bound by routing Strassen tensor rank decompositions through orthogonal Chinese Remainder Theorem ideals. This reduces the asymptotic complexity to $\widetilde{\Omega}(N^{\log_2 7})$. We optimize the coprime tensor decomposition. Embedding the inner tensor into the maximal real subfield satisfies the Lempel-Weinberger parity constraint. This guarantees the existence of a Self-Dual Normal Basis, reducing the required basis generators to a single element and mathematically halving the homomorphic trace depth. Canonical integer polynomial lifts ensure uniform norm bounds. Type I Optimal Normal Bases bound the trace dual expansions to an $O(1)$ constant. By invoking Kronecker's theorem, we prove that the polynomial power basis minimizes the canonical expansion for the non-evaluated tensor components. A towered evaluation over composite degrees controls noise propagation. This decouples key-switching errors into a logarithmic bound. We generalize the embedding to Galois rings via Hensel's and Nakayama's lemmas to support high-precision integer arithmetic. Furthermore, we extend the architecture to boundless matrices exceeding the fixed ring capacity via a multi-ciphertext block-Strassen decomposition. By deferring the homomorphic trace operator to post-Strassen recombination, we completely eliminate homomorphic basis-switching, achieving an asymptotic complexity of $O(N^{\log_2 7 - 1/\rho})$ multiplications and $\widetilde{O}(N^{2 - 2/(\rho \log_2 7)})$ automorphisms for matrices of arbitrary dimension. Empirical benchmarks over the BGV scheme validate the approach. A multi-threaded towered trace evaluates $32 \times 32$ matrices in $141.3$ milliseconds at a security level of $\lambda=148$ using one ciphertext-ciphertext multiplication. We achieve a speedup factor of $2.49$ over multi-threaded baselines.