I was reading a text on cryptology by Wayne Patterson and came across the $L^3$ algorithm which reduces integer lattices with respect to their base. I've also read on the NIST CFP A8 that attacks based on Grover's algorithm aren't really seen as viable. There is already research covering vulnerabilities in the group rings of NTRU variants, and additional research showing approximate closest vectors return approximate shortest vectors in polynomial time.
Page 70 from "Mathematical Cryptology for Computer Scientists and Mathematicians" by Wayne Patterson states: "The central algorithm developed (and proved to run in polynomial time) by Lenstra, Lenstra, and Lovasz is the reduced basis algorithm. The algorithm begins with any basis $\beta_{i}$ for an integer lattice, and eventually computes a reduced basis in the sense of the previous section." Page 72 then demonstrates how to apply the LLL algorithm in a Shamir attack against knapsacks.
Quoting again from Patterson, "These two theorems (6.1 and 6.2) taken together give an estimate on how close a vector of the reduced basis must be to the shortest (non-zero) vector in the entire lattice." pg. 70.
My question: Why would it be infeasible to encode the LLL reduction and group structures of NTRU variants as a search space in a Grover search to return an output that would break the NTRU schemes? This would be in line with the Coppersmith and Shamir lattice attack from section 4.2 in the linked article of NTRU Variants.
This seems in line with oracle access to a subroutine based on the linked research by Goldreich, et al.
Phrased differently, Can we know with certainty that LLL reductions and lattice attacks will not be made stronger if supplemented with Grover searches?