Multiplying boolean Polynomials with Frobenius Partitions in Additive Fast Fourier Transform

Abstract: We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order element under Frobenius map of $\mathbb{F}_{2}$. With the high order element, we can derive more values of the polynomials under Frobenius map. Besides, we also adapt the additive FFT to efficiently evaluate polynomials at the set with an encoding process. For the implementation, we reorder the computations in the additive FFT for reducing the number of memory writes and hiding the latency for reads. The algebraic operations, including field multiplication, bit-matrix transpose, and bit-matrix multiplication, are implemented with efficient SIMD instructions. As a result, we effect a software of best known efficiency, shown in our experiments.