Template attacks model side-channel leakage information using Gaussian multivariate distributions. They have been quite successful in directly reconstructing individual bits of 8-bit parallel buses and registers from power traces. However, extending their use directly to larger word sizes, such as 32-bit buses, becomes impractical. Here we show that it is possible to use an LDA-based stochastic model to independently build templates for just byte fragments of such a word, to predict the exact values of its four member bytes, instead of only overall Hamming weights. We demonstrate this technique to reconstruct the arbitrary-length inputs of SHA3-512 and some other Keccak sponge functions implemented on a 32-bit Cortex-M4 device. The quality of these templates was high enough such that remaining errors in their predictions could be eliminated via belief propagation on a factor-graph network (SASCA). In our experiments, we already reliably recovered SHA3-512 inputs up to 719 bytes long (10 invocations of the permutation), and reconstructing even longer inputs should be just a matter of making longer recordings.