The objective of this work is to establish a mathematical framework for the study of symmetric shift registers over the field GF(2). The present paper gives a new approach where the symmetric shift registers are represented by associated systems of nonlinear difference equations. Arithmetical progressions will play a central part. This approach clarifies the underlying structures and makes it easier to determine the minimal periods of the sequences generated by the symmetric shift registers. Key words: Shift registers, nonlinear difference equations, periods, arithmetical progressions, GF(2). An open-source implementation of the algorithms presented in this paper is available on GitHub (https://github.com/paalsoreng/symmetric-shift-register). In addition, an interactive web application is provided for experimenting with and evaluating the algorithms in practice (https://paalsoreng.github.io).