Abstract
It is known that we can obtain primitive pentanomials over GF (2) from primitive trinomials f(x)=xn+xp+1 by sampling every s digit of the M-sequence generated by f(x). In this paper, the authors first describe the relations among n, p and s where we can obtain a primitive pentanomial g(x) by this method. Then we show that the remainder polynomial r(x) modulo g(x) at one of the i/s phases (1≤i≤s-1) of the sampled M-sequence consists of a few terms in some cases, and that r(x) becomes a rational polynomial r(x)=P(x)/Q(x) in other cases, where P(x) and Q(x) are both polynomials consisting of two or three terms.
From these results, the authors show that the partial properties of the sampled M-sequence at i/s phases resemble each other. Therefore, when we use an M-sequence generated by g(x) for generating pseudo-random numbers, we must avoid, as an initial n-tuple, not only an unsuitable n-tuple and some of its cyclotomic phases, but also its i/s phases.
