Putting Fürer Algorithm into Practice
نویسندگان
چکیده
The multiplication of polynomials is a primitive widely used in computer algebra. It appears to contribute to a large panel of mathematical functions. Consequently, the optimization of this primitive is critical in cryptography for instance. Fast Fourier Transform (FFT) is commonly used in order to multiply polynomials. This is why an improvement of the Fast Fourier Transform algorithm is important. The multiplication of polynomials is connected to the multiplications of big integers. For almost 35 years, the best asymptotical complexity of the best known algorithm to multiply big integers of size n was O(n log n log log n). The algorithm in itself was due to Schönhage-Strassen [Knu97]. In 2007, Martin Fürer published a paper [Für07] achieving a complexity of O(n log n2O(log ∗ n)). We still don’t know if we could reach the theoretical bound O(n log n log∗ n) or even O(n log n). A more practical problem is to be able to make the Fürer’s algorithm as efficient as the current implementations of Schnhage-Strassen algorithm.
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