A Table of Primitive Binary Polynomials

Abstract

For those n < 5000, for which the factorization of 2n− 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5– and 7–nomial of degree n in GF(2) are given. A primitive polynomial of degree n over GF(2) is useful for generating a pseudo–random sequence of n–tuples of zeros and ones, see [8]. If the polynomial has a small number k of terms, then the sequence is easily computed. But for cryptological applications (correlation attack, see [5]) it is often necessary to have the primitive polynomials with k larger than one can find in the existing tables. For example, Zierler and Brillhart [10, 11] have calculated all irreducible trinomials of degree n ≤ 1000, with the period for some for which the factorization of 2n−1 is known; Stahnke [7] has listed one example of a trinomial or pentanomial of degree n ≤ 168; Zierler [12] has listed all primitive trinomials whose degree is a Mersenne exponent ≤ 11213 = M23 (here Mj denotes the jth Mersenne exponent); Rodemich and Rumsey [6] have listed all primitive trinomials of degree Mj, 12 ≤ j ≤ 17; Kurita and Matsumoto [2] have listed all primitive trinomials of degree Mj, 24 ≤ j ≤ 28, and one example of primitive pentanomials of degree Mj, 8 ≤ j ≤ 27. ∗1980 Mathematics Subject Classification (1985 Revision). Primary 11T06, 11T71. †