Optimizations in computing the algebraic normal form transform of Boolean functions

The Reed-Muller transform is widely used in discrete mathematics and cryptography, in particular for computing the algebraic normal form of Boolean functions. This is a good reason to look for ways to optimize the implementation of the algorithm. Here we present different ways for optimization based on the bitwise representation of the true table vector of a Boolean function. We compare the implementation of the standard algorithm with the implementation using AVX2 and AVX512 instruction sets and the corresponding extended registers and parallel implementation using GPUs with CUDA and OpenMP. The experimental results show that various degree of speedup can be achieved depending on the used platforms and approaches.