Fast Dot Product over Finite Field

Finite fields have great applications in various areas as cryptography, that is why it is important to have fast ways of computation to manipulate them. A first approach developed in this report lies in representing integers of the field using floating-point numbers, which lead to efficient computations. Operations in our case are done by restricting the characteristic p of the field to a floating-point mantissa: p − 1 < 2M−1. Taking advantage of error-free transformations on modern architectures, one can manage quite large finite fields exactly with floating-point arithmetic. After returning back to the basic of floating-point numbers, we introduce slightly different approaches to compute the dot product in an efficient way. In a second part, we have the same calculations done in a Residue Number System (RNS) over both integer and floating-point numbers. We show how this system can be efficient for well-chosen basis and present experimental results. Finally, we discuss how we parallelized our algorithms on a GPU card. Skalärprodukt i ändliga kroppar