# Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions
## 1 Introduction  
The convolution operation is widely used in various sciences (optics [1, 2], electronics [3, 4], etc.). Digital convolution is the basis of convolutional neural networks (CNN), [5, 6]. By digital convolution, hereinafter, we will understand convolution, under the sign of which are functions whose values correspond to a certain finite set of discrete quantities. It is appropriate to emphasize that CNN [5, 6] represent one of the most common varieties of artificial neural networks (ANNs). The last wave of ANN development, demonstrating the ability of CNN to show extraordinary results in the field of pattern classification, is exactly associated with such networks [7]. An important area of application of convolutional neural networks is pattern recognition and computer vision [8, 9, 10], natural language text processing [11, 12], speech recognition in noisy environments [13], etc. As it was noted in [14, 15], for digital signals, there is a wide enough space in the choice of the function serving as a signal model. In particular, in these reports, it was emphasized that functions taking values on the set of real or complex numbers actually represent some mathematical object, allowing for the creation of a signal model. The choice of such an object allowing for the creation of a signal model is ultimately a matter of agreement and convenience [14, 15]. In particular, if a signal corresponding to discrete levels lying within a finite range of amplitudes is considered, then any function that can serve as a model of this signal must actually represent a mapping of the current variable (time variable) onto some finite set. Both Galois fields [16; 17] and finite algebraic rings [18] can be chosen as such a set. It should be emphasized that such algebraic structures are nowadays more and more widely used in information technologies [19; 20; 21], and this also applies to non-associative algebras [22]. In this paper we show that it is possible to bring the computation of digital convolutions to computations in Galois fields, in which, in particular, the numerical analog of the classical convolution theorem [15] is valid. This approach is of interest, in particular, because from the functions taking the values within Galois fields, it is admissible to pass to functions whose values correspond to variables of multivalued (as well as fuzzy) logic [23; 24]. Specific computations that are performed in Galois fields or finite algebraic rings can in many cases be reduced to computations modulo integers. At present, circuits of various types allowing for the realization of such an operation are being developed quite actively [25; 26; 27; 28], with cryptography remaining the main field of application here. This question is also reflected in the patent literature [29; 30; 31; 32], where various variants of circuits of multipliers and adders modulo some numbers have been proposed. It is well known that computations modulo integers allows for operating independently on the components of an integer representation [33; 34; 35; 36]. It can also be used to represent a convolution operation as a set of partial convolutions, each of which corresponds to a particular Galois field. However, when using digital convolutions, there is a significant nuance. The sampling scales of the original function and the function resulting from the convolution operator are different. To bring them into a mutually unambiguous correspondence, it is necessary to use an operation that can be interpreted as a transition to the reduced digital convolution. By the term \"reduced digital convolution\", we mean the result of the convolution operation computation reduced to the same sampling scale as the original functions (functions under the convolution sign). Such operation can be implemented using the usual rounding operation for fractional numbers; however, it is of interest to ensure that the calculation reduced digital convolution in terms of operations modulo integers.