Efficient linear feedback shift registers with maximal period

Linear feedback shift registers (LFSR’s) are fundamental primitives in the theory and practice of pseudorandom number generation and coding theory (see, e.g., [1], [2], [3], [4], [6], [7], and references therein). Figure 1 describes a typical LFSR over the two-element field F2 = {0, 1}, where each step consists of adding some of the state bits (we follow the convention that the elements of F2 are called bits), and the result is inserted to the register in a FIFO manner. Such a construction is slow in the sense that it produces only one new bit per step. Moreover, it is difficult to implement in software, since many bit manipulations are required. In certain cases (but not always [10]), it is