In this memory, we introduce a vectorial design of feedback with carry registers introduced by Goresky and Klapper denominated VFCSR. Through the ring of Witt vectors, we develop an analysis of these registers that establishes the essential properties of output sequences as the existence of sequences of maximal period, the pseudorandomnes in terms of the arithmetic and statistical correlation, the behavior of the memory, etc..We study different ways of designing these registers (Fibonacci, Galois and Ring modes). As an application, we propose a cryptographic random number generator as a "stream cipher" based on a VFCSR quadratic register.
