Calculation of Fourier-Galois transforms in reduced binary number systems

The paper proposes a new method for calculating Fourier-Galois transforms (number-theoretical transforms), which are a modular analog of the discrete Fourier transform. A number of specific problems related to the calculation of transforms in a finite field can be solved by representing the elements of these fields in “exotic” number systems, which are reductions of the canonical number systems proposed by I. Katai when mapping the corresponding ring of an integer quadratic field into a field of the prime residue classes modulo. The case of binary reduced number systems is studied in detail. It is proved that such number systems exist for any prime number.