Most homomorphic encryption (FHE) schemes exploit a technique called single-instruction multiple-data (SIMD) to process several messages in parallel. However, they base their security in somehow strong assumptions, such as the hardness of approximate lattice problems with superpolynomial approximation factor. On the other extreme of the spectrum, there are lightweight FHE schemes that have much faster bootstrapping but no SIMD capabilities. On the positive side, the security of these schemes is based on lattice problems with (low-degree) polynomial approximation factor only, which is a much weaker security assumption. Aiming the best of those two options, Micciancio and Sorrell (ICALP'18) proposed a new amortized bootstrapping that can process many messages at once, yielding sublinear time complexity per message, and allowing one to construct FHE based on lattice problems with polynomial approximation factor. Some subsequent works on this line achieve near-optimal asymptotic performance, nevertheless, concrete efficiency remains mostly an open problem. The only existing implementation to date (GPV23, Asiacrypt 2023) requires keys of up to a hundred gigabytes while only providing gains for relatively large messages. In this paper, we introduce a new method for amortized bootstrapping where the number of homomorphic operations required per message is O(h) and the noise overhead is O(hλlogλ), where h is the Hamming weight of the LWE secret key and λ is the security parameter. This allows us to use much smaller parameters and to obtain faster running time. Our method is based on a new efficient homomorphic evaluation of sparse polynomial multiplication. We bootstrap 2 to 8-bit messages in 1.1 ms to 26.5 ms, respectively. Compared to TFHE-rs, this represents a performance improvement of 3.9 to 41.5 times while requiring bootstrapping keys up to 50.4 times smaller.