| Finite fields are widely used in numerous areas like cryptography, error-correcting codes or computer algebra. Dot products are ubiquitous in all computations especially when dealing with linear algebra. Developing fast libraries for computing dot products in finite fields is a key tool to tackle various problems in scientific computing. In this paper, our aim is to present several possibilities to use fast floating-point units for computing dot products in finite fields. The main concern is then to properly manage rounding errors that may appear during the computation. To solve this problem, we use error-free transformations (EFT). Using these EFT on recent processors (with an FMA), we show that it is possible to deal with large finite fields. We also compare our approach with Residue Number Systems (RNS), which is a modular approach. The RNS approach is presented using either integer arithmetic or floating-point arithmetic. Numerical experiments make it possible to compare the performances of these different approaches. |
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