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PairSwitch: Efficient Galois-Paired Key Switching for Gentry–Lee Matrix FHE

ePrint Report: PairSwitch: Efficient Galois-Paired Key Switching for Gentry–Lee Matrix FHE Zhenyu Xiong, Mingsheng Wang, Han Wang Matrix-native fully homomorphic encryption, introduced by Gentry and Lee, multiplies encrypted matrices with a Trace product and returns its four-component output to an ordinary ciphertext with a BigSwitch, whose keys over an extended ring dominate the key material of the scheme. Existing BigSwitches follow two routes: they either store one key per source, as in the original construction, or factor the secret through the trace and relinearize at run time, as in the Trace-Factored BigSwitch (TFB) of Park et al., which halves the keys at the price of $50\%$ more key switches and a whole switching error multiplied by the secret. Meanwhile, the Galois splitting of Park (Eurocrypt'25) decomposes such switches into base-ring diagonals, but it has not been used to reduce their keys. However, whether a BigSwitch can publish only the keys of TFB and still run at the speed of the original one has remained open. In this work, we first propose the pair-merge BigSwitch, which observes that the diagonals of a split BigSwitch are indexed by an inversion-closed coset $\kappa H$ and rewrites the target of a diagonal as $x\,g(s)+y\,s\,g(s)=g\big(g^{-1}(x)\,s+g^{-1}(y)\,s\,g^{-1}(s)\big)$, so that it borrows the product key of its inverse at the cost of one Galois step; it makes the $2n$ key switches of the original BigSwitch with $n/2+1$ instead of $n$ product keys, and we prove that no per-diagonal BigSwitch with $2n$ key switches holds fewer than $3n/2+1$ keys. Building on it, we propose public key rebasing, which turns Galois keys into product keys with one relinearization key placed one small modulus layer above the chain, so that the client publishes only the keys of TFB and a 10.6 MiB layer key, at the price of $1.5\times$ the resident keys of TFB on the server. Consequently, together with a Hermitian hop that finishes a partner diagonal of a Gram product with one Galois step, PairSwitch makes $2n$ key switches on general inputs and $3n/2+1$ on Gram products. On the parameters of Park et al. ($n=256$, $p=17$, $\log_2 PQ=200$), our implementation runs a BigSwitch $1.89$--$1.99\times$ faster than TFB and $1.28$--$1.38\times$ faster than the original BigSwitch, with $2.4$ bits less noise than TFB and $4.3$ bits more than the original, at a $128.3$-bit layer key; it is $2.45\times$ faster than TFB on Gram inputs. Our implementation and the raw logs are available at https://github.com/acprk/pairswitch-gl.

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