Fast multiplication algorithms

Fast multiplication algorithms for both large numbers and large square matrices are considered here. When multiplying numbers, a group algebra is introduced, and the Fourier transform is expressed as a representation of the elements of the group algebra in another basis associated with characters. Next, bigroup algebra is introduced as an extension of group algebra operators using characters acting as diagonal matrices in the standard basis of group algebra. The analogue of multiplication of large numbers by the Fourier transform extends to group algebra, i.e. to the algebra of matrices.