Algorithms for Finding Almost Irreducible
and Almost Primitive Trinomials

Irreducible and primitive polynomials over finite fields have many applications in cryptography, coding theory, random number generation, etc. See, for example, [13, 15, 17, 20, 21].

For simplicity we restrict our attention to the finite field ${\mathbb{Z}}_{2}=\operatorname{GF}(2)$; the generalization to other finite fields is straightforward. All polynomials are assumed to be in ${\mathbb{Z}}_{2}[x]$, and computations on polynomials are performed in ${\mathbb{Z}}_{2}[x]$ or in a specified quotient ring. A polynomial $P(x)\in{\mathbb{Z}}_{2}[x]$ may be written as $P$ if the argument $x$ is clear from the context. We recall some standard definitions.

If $P(x)$ is primitive, then $x$ is a generator for the multiplicative group of the field ${\mathbb{Z}}_{2}[x]/P(x)$, giving a concrete representation of $\operatorname{GF}(2^{n})$. See Lidl and Niederreiter [20] or Menezes et al. [21] for background information.

There is an interest in discovering primitive polynomials of high degree $n$ for applications in random number generation [4, 7] and cryptography [21]. In such applications it is often desirable to use primitive polynomials with a small number of nonzero terms, i.e. a small weight. In particular, we are interested in trinomials of the form $x^{n}+x^{s}+1$, where $n>s>0$ (so there are exactly three nonzero terms).

If $P(x)$ is irreducible and $\deg(P)=n>1$, then the order of $x$ in ${\mathbb{Z}}_{2}[x]/P(x)$ is a divisor of $2^{n}-1$. To test if $P(x)$ is primitive, we must test if the order of $x$ is exactly $2^{n}-1$. To do this efficiently¹¹1Here “efficiently” means in time polynomial in the degree $n$. it appears that we need to know the complete prime factorization of $2^{n}-1$. At the time of writing these factorizations are known for $n<713$ and certain larger $n$, see [8].

We say that $n$ is a Mersenne exponent  if $2^{n}-1$ is prime. In this case the factorization of $2^{n}-1$ is trivial and an irreducible polynomial of degree $n$ is necessarily primitive. Large Mersenne exponents are known [14], so there is a possibility of finding primitive trinomials of high degree. To test if a trinomial of prime degree $n$ is reducible takes time $O(n^{2})$, so to test all trinomials of degree $n$ takes time $O(n^{3})$.

Several authors [16, 18, 19, 27] have computed primitive trinomials whose degree is a Mersenne exponent, up to some bound imposed by the computing resources available. Recently Brent, Larvala and Zimmermann [6] gave a new algorithm, more efficient than those used previously, and computed all the primitive trinomials of Mersenne exponent $n\leq 3021377$ (subsequently extended to $n\leq 6972593$).

For some $n\geq 2$, irreducible trinomials of given degree $n$ do not exist. Swan’s theorem (see §2) rules out $n=0{\;\bmod\;}8$ and also most $n=\pm 3{\;\bmod\;}8$. Since about half of the known Mersenne exponents are $\pm 3{\;\bmod\;}8$, we can only hope to find primitive trinomials of degree $n$ for about half the Mersenne exponents $n$.

In the cases where primitive trinomials are ruled out by Swan’s theorem, the conventional approach is to use primitive polynomials with more than three nonzero terms. A polynomial with an even number of nonzero terms is divisible by $x+1$, so we must use polynomials with five or more nonzero terms [18, 19, 21]. This is considerably more expensive in applications because the number of operations required for multiplication or division by a sparse polynomial is approximately proportional to the number of nonzero terms.

In §2 we discuss Swan’s theorem, then in §3 we introduce “almost irreducible” and “almost primitive” polynomials as a way of circumventing the implications of Swan’s theorem. An algorithm (AIT) for computing almost irreducible trinomials, and an extension (APT) for almost primitive trinomials, are described in §4. Algorithm APT has been used to find almost primitive trinomials with high degree in cases where Swan’s theorem shows that primitive trinomials do not exist. Computational results and examples are given in §§5–6. In §7 we give some computational results on almost primitive trinomials that are useful for representing the finite fields $\operatorname{GF}(2^{2^{k}})$, $k\leq 12$. In §8 we explain how to use almost irreducible/primitive trinomials efficiently in applications. Finally, in §9, we conclude with some theoretical results on the density of almost irreducible and almost primitive polynomials, and some computational results on the density of the corresponding trinomials. We thank Shuhong Gao and the referees for their comments on a draft of this paper.

Swan’s theorem is a rediscovery of results of Pellet [23], Stickelberger [24], Dickson [10] and Dalen [9] – see Swan [25, p. 1099] and von zur Gathen [12]. Let $\nu(P)$ denote the number of irreducible factors (counted according to their multiplicity) of a polynomial $P\in{\mathbb{Z}}_{2}[x]$.

If both $n$ and $s$ are odd, we can replace $s$ by $n-s$ (leaving the number of irreducible factors unchanged, since $\nu(x^{n}+x^{s}+1)=\nu(x^{n}+x^{n-s}+1)$) and apply Swan’s theorem. If $n$ and $s$ are both even, then $T=x^{n}+x^{s}+1$ is a square and $\nu(T)$ is even. Thus, in all cases we can determine $\nu(T){\;\bmod\;}2$ using Swan’s theorem.

Since a polynomial that has an even number of irreducible factors is reducible, we have:

Corollary 2.2 shows that there are no irreducible trinomials with degree a Mersenne exponent $n=\pm 3{\;\bmod\;}8$ (except possibly for $s=2$ or $n-2$). This appears to prevent us from using trinomials with periods $2^{n}-1$ in these cases. Fortunately, there is a way to circumvent Swan’s theorem and avoid paying a significant speed penalty in most applications of irreducible/primitive trinomials. We describe this in the following section.

Tromp, Zhang and Zhao [26] asked the following question: given an integer $r>1$, do there exist integers $n,s$ such that

is a primitive polynomial of degree $r$? They verified that the answer is “yes” for $r\leq 171$, and conjectured that the answer is always “yes”.

Blake, Gao and Lambert [3] confirmed the conjecture for $r\leq 500$. They also relaxed the condition slightly and asked: do there exist integers $n,s$ such that $G$ has a primitive factor of degree $r$? Motivated by [3], we make some definitions.

For example, the trinomial $x^{16}+x^{3}+1$ is almost primitive with exponent 13 and increment 3, because

where

is primitive. From a computational viewpoint it is more efficient to work in the ring ${\mathbb{Z}}_{2}[x]/(x^{16}+x^{3}+1)$ than in the field $F={\mathbb{Z}}_{2}[x]/D_{13}(x)$. In §8 we outline how it is possible to work in the field $F$, while performing most arithmetic in the ring ${\mathbb{Z}}_{2}[x]/(x^{16}+x^{3}+1)$, and without explicitly computing the dense primitive polynomial $D_{13}(x)$.

Note that, according to Definition 3.1, a primitive polynomial is a fortiori an almost primitive polynomial (the case $r=n$). Similarly, an irreducible polynomial other than $1$ or $x$ is almost irreducible. The restriction $r>n/2$ in Definition 3.1 ensures that polynomials such as $(x^{3}+x+1)^{2}$ are not regarded as almost irreducible.

In practice we choose the smallest possible increment $\delta$ for given exponent $r$, e.g. in Tables 1–2 we have $\delta\leq 16$. For most practical purposes, almost primitive trinomials of exponent $r$ and small increment are almost as useful as primitive trinomials of degree $r$ (see §8).

In this section we outline algorithms for finding almost irreducible or almost primitive trinomials with large exponent $r$. In the latter case we assume that the complete factorization of $2^{r}-1$ is known. The algorithms are generalizations of those given in [6, 13, 21], which handle the case $\delta=0$.

We conducted a search for almost primitive trinomials whose exponent $r$ is also a Mersenne exponent. For all Mersenne exponents $r=\pm 1{\;\bmod\;}8$ with $r\leq 6972593$, primitive trinomials of degree $r$ are known, see [6]. Here we consider the cases $r=\pm 3{\;\bmod\;}8$, where the existence of irreducible trinomials $x^{r}+x^{s}+1$ is ruled out by Swan’s theorem (except for $s=2$ or $r-2$, but the only known cases are $r=3,5$). For each exponent $r$ we searched for all almost primitive trinomials with the minimal increment $\delta$ for which at least one almost primitive trinomial exists. The search has been completed for all Mersenne exponents $r<10^{7}$.

In all cases of Mersenne exponent $r=\pm 3{\;\bmod\;}8$, where $5<r<10^{7}$, we have found at least one almost primitive trinomial with exponent $r$ and some increment $\delta\in[2,12]$. The results are summarized in Table 1. The first four entries are from [3, Table 4]; the other entries are new. They were computed using Algorithm APT, with some simplifications that are possible because $r$ is a Mersenne exponent (see §4.3.3 above).

For all but two of the almost primitive trinomials $x^{r+\delta}+x^{s}+1$ given in Table 1, the period $\rho=(2^{r}-1)f$ satisfies $\rho>2^{r+\delta-1}$. Thus, the period is greater than the greatest period ($2^{r+\delta-1}-1$) that can be obtained for any polynomial of degree less than $r+\delta$. In the two exceptional cases the small factors of degree $8$ are not primitive, having period $85=255/3$.

For the largest known Mersenne exponent, $r=13466917$, we have not yet started an extensive computation, but we have shown that $\delta\geq 3$ and $\delta\neq 4$ (see Theorem 6.3).

We considered the almost primitive trinomial $x^{16}+x^{3}+1$ in §3. Here we give an example with much higher degree: $r=216091$, $\delta=12$. We have

where

and $D(x)$ is a (dense) primitive polynomial of degree 216091. The factor $S(x)$ of degree 12 splits into a product of two primitive polynomials, $x^{5}+x^{4}+x^{3}+x+1$ and $x^{7}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$. The contribution to the period from these factors is $f=\operatorname{lcm}(2^{5}-1,2^{7}-1)=3937$.

If we have found an almost irreducible trinomial $T=x^{n}+x^{s}+1$ with exponent $r=n-\delta$, then to check if $T$ is almost primitive we need the complete factorization of $2^{r}-1$. In §5 we chose $r$ so the factorization was trivial, because $2^{r}-1$ was a Mersenne prime. Another case of interest, considered in this section, is when $r$ is a power of two, say $r=2^{k}$. Then

where $F_{j}=2^{2^{j}}+1$ is the $j$-th Fermat number. The complete factorizations of these $F_{j}$ are known for $j\leq 11$ (see [5]) so we can factor $2^{2^{k}}-1$ for $k\leq 12$.

In Table 2 we give almost primitive trinomials $T=x^{r+\delta}+x^{s}+1$ with exponent $r=2^{k}$ for $3\leq k\leq 12$. Thus $T=SD$ where $D$ is primitive and has degree $2^{k}$. We also give $S$ in factored form. The irreducible factors of $S$ are not always primitive. The period of $T$ is $\operatorname{lcm}(2^{r}-1,\operatorname{period}(S))=(2^{r}-1)f$.

By Swan’s theorem, a primitive trinomial of degree $2^{k}$ does not exist for $k\geq 3$. However, we can work efficiently in the finite fields $\operatorname{GF}(2^{2^{k}})$, $k\in[3,12]$, using the trinomials listed in Table 2 and the implicit algorithms of §8.

Suppose we wish to work in the finite field $\operatorname{GF}(2^{r})$ where $r$ is the exponent of an almost primitive trinomial $T$. We can write $T=SD$, where $\deg(S)=\delta$, $\deg(D)=r$. Thus

but because $D$ is dense we wish to avoid working directly with $D$, or even explicitly computing $D$. We show that it is possible to work modulo the trinomial $T$.

We can regard ${\mathbb{Z}}_{2}[x]/T(x)$ as a redundant representation of ${\mathbb{Z}}_{2}[x]/D(x)$. Each element $A\in{\mathbb{Z}}_{2}[x]/T(x)$ can be represented as

where $A_{c}\in{\mathbb{Z}}_{2}[x]/D(x)$ is the “canonical representation” that would be obtained if we worked in ${\mathbb{Z}}_{2}[x]/D(x)$, and $A_{d}\in{\mathbb{Z}}_{2}[x]$ is some polynomial of degree less than $\delta$.

We can perform computations such as addition, multiplication and exponentiation in ${\mathbb{Z}}_{2}[x]/T(x)$, taking advantage of the sparsity of $T$ in the usual way.If $A\in{\mathbb{Z}}_{2}[x]/T(x)$ and we wish to map $A$ to its canonical representation $A_{c}$, we use the identity

where the division by the (small) polynomial $S$ is exact. A straightforward implementation requires only $O(\delta r)$ operations. We avoid computing $A_{c}=A{\;\bmod\;}D$ directly; in fact we never compute the (large and dense) polynomial $D$: it is sufficient that $D$ is determined by the trinomial $T$ and the small polynomial $S$.

In applications such as random number generation [7], where the trinomial $T=x^{n}+x^{s}+1$ is the denominator of the generating function for a linear recurrence $u_{k}=u_{k-s}+u_{k-n}$, it is possible (by choosing appropriate initial conditions that annihilate the unwanted component) to generate a sequence that satisfies the recurrence defined by the polynomial $D$. However, this is not necessary if all that matters is that the linear recurrence generates a sequence with period at least $2^{r}-1$.

In this section we state some theorems regarding the distribution of almost irreducible polynomials. The proofs are straightforward, and similar to the proof of Theorem 1.2 in [11], which generalizes our Corollary 9.2. We would like to prove similar theorems about almost irreducible trinomials, but this seems to be difficult.

Let $I_{n}$ denote the number of irreducible polynomials of degree $n$, excluding the polynomial $x$, and let $N_{r,\delta}$ denote the number of almost irreducible polynomials with exponent $r$ and increment $\delta$. Thus $\sum_{d|n}dI_{d}=2^{n}-1$ and, by Möbius inversion,