Feedback with carry shift registers synthesis with the Euclidean algorithm

Feedback with carry shift registers (FCSR) are very similar to the classical linear feedback shift registers (LFSR) used in many pseudorandom generators. The main difference is the fact that the elementary additions are not modulo 2 but with propagation of carries. In this paper we prove that it is possible to synthesize the FCSR with the extended Euclidean algorithm in the ring Z of integers. This algorithm is based on De Weger and Malher´s rational approximation. However, it seems simpler to understand, to implement and to prove.