On primitive trinomials (Mod 2)

Abstract

The present paper is a completion of a previous paper of the same title (Zierler and Brillhart, 1968). In our preceding work 187 of the irreducible trinomials T~.k(x) = ~ x k ~- 1 were left to be tested for primitivity at a later date, even though the requisite complete factorizations of 2 ~ - 1 were known (these trinomials were identified in (Zierler and Brillhart, 1968) by a superscript minus sign on n). This testing has now been done on the CDC 6600 at the Communications Research Division of the Institute for Defense Analyses, Princeton (the testing in Zierler and Brillhart (1968) was done on both this computer and the IBM 7094 at Bell Telephone Laboratories, ttolmdel, New Jersey). The results are given in Table 1. The italic entries in this table refer to primitive trinomials, while those that are not italicized refer to trinomials whose periods are less than 2 ~ - 1. For those imprimitive Tn,~(x) with (n, k) = 1 we have given in Table 2 the index (= (2 ~ - 1)/Period) rather than the period of T,~(x), since the periods are extremely large. For the remaining T. .k(x) with (n, k) > 1, we have entirely omitted giving their periods, because of the ease with which these can be calculated from the entries in Tables 1 and 2 and the information in (Zierler and Brillhart, 1968). Relevant to this calculation is the following theorem from (Berlekamp, 1968), p. 153: THEOaEM: Let f(x) be an irreducible polynomial with period n over GF (p~), and let t~ p be a pmme. If t [ n, then every irreducible factor of f(x ~) has period tn. If t ~ n, then one irreducible factor of f(x ~) has period n, while the other factors have period tn.