A Swan-like note for a family of binary pentanomials

TL;DR

This paper uses classical techniques to analyze the irreducibility of a specific family of binary pentanomials, revealing conditions under which these polynomials are reducible based on their degree modulo 8.

Contribution

It extends Swan's method to a new family of binary pentanomials, providing criteria for their reducibility based on degree congruences.

Findings

If n ≡ ±1 mod 8, the polynomial may be irreducible.

If n ≠ ±1 mod 8, the polynomial is reducible.

The paper applies classical techniques to a new polynomial family.

Abstract

In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial $X^n+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_2[X]$, where $s$ is even and $n>3s$. Our results imply that if $n \not\equiv \pm 1 \pmod{8}$, then the polynomial in question is reducible.

Tables

Table 1. Table 1: Pairs ( n , s ) 𝑛 𝑠 (n,s) with 7 ≤ n < 3000 7 𝑛 3000 7\leq n<3000 and s 𝑠 s even, such that X n + X n − s + X n − 2 ​ s + X n − 3 ​ s + 1 ∈ 𝔽 2 ​ [ X ] superscript 𝑋 𝑛 superscript 𝑋 𝑛 𝑠 superscript 𝑋 𝑛 2 𝑠 superscript 𝑋 𝑛 3 𝑠 1 subscript 𝔽 2 delimited-[] 𝑋 X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{F}_{2}[X] is irreducible.

$n$	$s$	$n$	$s$	$n$	$s$	$n$	$s$	$n$	$s$
7	2	17	2, 4	23	6	25	6	31	2, 6, 8
47	14	49	4	55	8, 16	65	6	71	2, 6, 12
73	14, 16	79	20	95	26, 28	97	2, 4	103	10, 24, 30
113	10	121	6, 10	127	10, 40, 42	137	34	151	22, 28, 36, 40
161	6, 20	167	2, 30, 36, 44	169	14, 28	175	2, 6	185	8, 48
191	6, 40	193	36, 40	199	44	209	2, 54	215	38, 46, 64
217	22, 44	223	44	239	12	247	34	257	4, 16, 64, 72
265	14, 46	287	54, 72	289	12, 28	295	16	305	34
313	64, 78	319	12	329	18	337	66, 94	343	46
353	46, 60, 70, 86	377	112	383	30, 36	385	2, 8, 18	391	120
407	112	415	34, 84	425	4, 14, 22, 78	431	40	433	124
439	52, 98, 102, 130	449	94	457	70, 80, 132	463	56	481	46
487	120	497	26, 72, 76	505	52, 58	511	72, 160	521	16, 56
527	66, 96, 160	529	14, 38, 124	551	80	553	86, 148	559	70
569	70, 164	575	86	577	184	593	36, 158	599	10, 70
623	104, 124, 146	625	52, 164	631	108	641	12, 118, 182, 210	647	50, 104, 144, 214
649	192, 204	655	64	665	48, 64, 116, 132, 204	673	84, 100, 138	679	22, 72
689	112, 170	713	224	719	50, 58, 140, 154	721	90, 146	727	60, 170
737	244	743	30, 48, 70, 168, 178	745	86, 112, 114	751	6, 188	761	28, 46
767	56	769	40, 72	775	136, 186	785	198	791	10, 36, 180
793	180	799	258	809	70, 192, 224	815	112	817	154, 210
823	244	833	206, 228	839	18	841	48	847	92
857	86, 90, 134, 212, 214, 246	865	76, 162, 288	871	126	881	26, 28	887	112, 190
889	104, 240, 254	895	4	905	188	911	68, 126	913	158, 274
919	12, 112, 130	937	240	943	8, 150	953	56	959	104, 188, 272
961	6	967	12, 70	977	160	983	114	985	74
991	266	1001	18, 118, 328	1007	32	1009	318	1015	62, 86
1025	98, 102, 214	1031	220, 248	1033	36, 110, 166	1049	130, 252, 274	1055	8
1057	66, 146, 242	1063	56	1079	94, 114	1081	8, 106, 116, 282	1087	80, 202, 210, 214
1103	344, 346	1105	32, 222	1111	366	1121	306, 336, 338	1127	270
1129	342	1135	12	1145	306	1151	30, 342	1153	204, 218, 246, 304
1159	22	1169	38, 332	1177	42, 62, 96	1183	36	1193	340
1199	38	1201	120	1207	232	1223	196	1225	78, 226
1231	130, 238, 292, 322	1241	18	1247	30	1249	354	1255	222
1265	184, 192, 266, 298, 382	1271	150, 406, 418	1273	56	1279	72	1289	68
1297	66, 244, 320	1313	118, 308	1327	124, 316, 350	1337	34, 136	1343	116, 120
1345	264	1351	50, 262, 370	1361	126	1369	120, 314	1375	42, 228, 246, 326
1385	4, 242, 256	1391	28	1393	100, 114	1399	88, 180, 270, 380, 448	1415	94, 346
1417	114	1423	76, 264, 378	1447	114, 242, 382	1463	296	1465	174
1471	482	1481	390	1487	92	1489	84	1495	272, 418
1505	82, 122, 398	1511	96	1513	230, 234	1519	164, 262	1529	20, 62, 214, 322
1537	364, 460	1543	226, 366	1553	84, 306, 358	1559	324, 480	1561	186, 356
1567	480	1577	270, 292, 412	1583	138, 280, 300	1591	152	1607	146, 176
1615	504	1625	216, 408	1633	490	1639	310, 404	1649	220
1655	534	1663	448	1673	30, 356	1679	14, 68, 400	1681	436
1687	216, 432, 516, 552	1697	140, 256, 274	1703	514	1705	54, 248, 496	1721	100, 228
1729	72	1735	294, 336	1745	234	1753	326	1769	294
1775	162	1777	306, 520	1783	408, 448	1793	38	1799	104
1807	146	1817	446	1823	228, 310	1831	440	1841	22
1847	60, 256, 530	1849	600	1855	416	1865	246, 254, 448, 604	1873	370, 558
1879	58, 284, 422	1889	464, 620	1903	310	1913	154	1919	240
1921	156	1927	56, 592, 634	1937	310	1943	20, 176, 320, 644	1945	96, 98, 618
1951	610	1961	332	1967	386, 506, 590	1969	182, 598	1975	378
1985	224, 258, 558	1991	182	1993	254, 630	1999	244, 544	2009	18, 50
2015	14, 186, 238	2017	110, 180, 476	2023	424	2033	384	2039	280, 364, 628
2047	22, 410, 512	2057	604	2063	16, 190, 448, 570	2065	306, 656	2081	538
2087	450, 506	2089	50, 282, 412, 580	2095	468, 496, 546	2111	440, 560	2113	184, 576
2119	28, 144, 282	2129	178	2135	96, 696	2137	216, 602	2153	194, 496
2159	2, 260	2167	280, 432	2183	54, 410	2185	108	2191	218, 240, 298
2201	386, 532, 686	2207	146, 280	2209	272, 580	2225	726	2231	218, 246, 396
2233	204, 590	2239	40, 408, 544	2249	42, 272, 360	2255	332, 496	2257	476
2273	210, 358, 410	2279	92, 222, 490	2281	522	2287	486, 492	2303	34, 554
2305	396, 552	2311	408, 700	2321	588	2327	44, 524, 634	2329	112, 430
2335	594	2345	214, 772	2353	586	2359	272, 394	2369	98, 648, 764
2375	66, 202, 724	2383	494, 778	2393	778	2399	690, 764	2401	388, 600, 736, 740, 780
2407	450, 730, 772	2417	642	2423	34, 208, 370, 378	2425	226, 494	2431	210, 336, 696
2441	270	2447	262, 420	2449	378	2455	714	2465	224
2473	514	2479	24, 600	2495	232, 414, 666	2497	442	2513	804
2519	414, 490, 558	2521	142	2537	450, 646, 812, 820	2543	80	2551	490, 816
2561	620	2569	522	2585	228, 252, 774	2591	528	2593	454, 630, 644
2599	612, 770	2609	164, 728	2615	318	2617	154, 270, 644	2623	318, 484, 554, 738
2633	464	2639	48, 814	2641	402	2647	516, 530	2657	642
2665	504	2671	418, 580	2681	218, 416, 682	2687	380, 602, 802	2689	12
2695	684	2711	230	2713	280, 434	2719	34, 48, 140, 662, 778	2729	32, 280
2737	156, 344, 404, 514	2743	760	2753	326	2759	430, 474	2761	36, 490, 510, 630
2767	34, 206, 564	2783	56, 138, 734	2785	508, 812	2791	280	2807	90, 772
2815	28, 252	2825	96, 404	2831	62, 582	2833	280, 550, 808	2839	602
2849	38, 438	2855	278, 938	2857	252	2863	908	2879	734, 910
2881	560	2887	554, 806	2897	546	2903	410	2905	246
2911	308, 742	2921	160, 188, 332	2927	616	2945	504, 606, 792, 806, 868	2951	512
2953	890	2959	228, 938	2969	192, 212	2975	64, 334, 542	2977	42, 146
2983	110, 490, 566, 838	2993	680, 808	2999	350, 498, 554

Equations

$$X^n+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in {\mathbb{F}}_2[X],$$

$$D(F):=\prod _{1\le i<j\le n}({\alpha }_i-{\alpha }_j{)}^2,$$

$$D(F)=(-1{)}^{\frac{n(n-1)}{2}}\prod _{i=1}^{n}H({\alpha }_i).$$

$$F^{\prime }(X)=\sum _{i=1}^n\left(\prod _{j\ne i}(X-{\alpha }_j)\right),$$

$$\prod _{i=1}^n{F}^{\prime }({\alpha }_i)=\prod _{i=1}^n\prod _{\begin{array}{c}1\le j\le n\\ j\ne i\end{array}}({\alpha }_i-{\alpha }_j)=(-1{)}^{\frac{n(n-1)}{2}}\prod _{1\le i<j\le n}({\alpha }_i-{\alpha }_j{)}^2=(-1{)}^{\frac{n(n-1)}{2}}D(F).$$

$$\prod _{i=1}^{n}H({\alpha }_i)=(-1{)}^n{\alpha }_1\cdots {\alpha }_n\prod _{i=1}^n{F}^{\prime }({\alpha }_i)$$

$$X{F}^{\prime }(X)\sum _{i=0}^{\infty }(-1{)}^{i+1}(F(X)-1{)}^i=\sum _{i=1}^{\infty }{S}_{-i}^{(F)}{X}^i,$$

$$-\frac{X{F}^{\prime }(X)}{F(X)}=X\cdot \sum _{j=1}^k\frac{1}{{\alpha }_j}\cdot \frac{1}{1-X/{\alpha }_j}.$$

$$\frac{1}{1-X/{\alpha }_j}=\sum _{i=0}^{\infty }{\left(\frac{X}{{\alpha }_j}\right)}^i$$

$$\frac{1}{F(X)}=\frac{1}{1-(1-F(X))}=\sum _{i=0}^{\infty }(1-F(X){)}^i=\sum _{i=0}^{\infty }(-1{)}^i(F(X)-1{)}^i.$$

$$X{F}^{\prime }(X)\sum _{i=0}^{\infty }(-1{)}^{i+1}(F(X)-1{)}^i=\sum _{i=1}^{\infty }\left(\sum _{j=1}^n\frac{1}{{\alpha }_j^i}\right)X^i$$

$$S_m^{(F)}+F_{n-1}{S}_{m-1}^{(F)}+\cdots +F_{n-l+1}{S}_{m-l+1}^{(F)}+\frac{l}{m}{F}_{n-l}{S}_{m-l}^{(F)}=0,$$

$$S_m:=S_m^{(F)}=\sum _{j=1}^n{\alpha }_j^m\text{and}{T}_m:=\sum _{1\le i<j\le n}({\alpha }_i{\alpha }_j{)}^m,$$

$$H(X):=nF(X)-X{F}^{\prime }(X)=s{X}^{n-s}+2s{X}^{n-2s}+3s{X}^{n-3s}+n.$$

$$\begin{gathered} \prod _{i=1}^{n}H({\alpha }_i)=n^n+n^{n-1}s{S}_{n-s}+2n^{n-1}s{S}_{n-2s}+3n^{n-1}s{S}_{n-3s}+ \\ {n}^{n-2}{s}^2{T}_{n-s}+n^{n-2}{s}^2{T}_{n-3s}+3n^{n-2}{s}^2(S_{n-s}{S}_{n-3s}-S_{2n-4s})+8K, \end{gathered}$$

$$\begin{gathered} \prod _{i=1}^{n}H({\alpha }_i)\equiv n+s{S}_{n-s}+2s{S}_{n-2s}+3s{S}_{n-3s}+n{s}^2{T}_{n-s}+n{s}^2{T}_{n-3s} \\ +3n{s}^2(S_{n-s}{S}_{n-3s}-S_{2n-4s})(\mathrm{mod}8). \end{gathered}$$

$$(n{X}^n+s{X}^s+2s{X}^{2s}+3s{X}^{3s})\sum _{i=0}^{\infty }(-1{)}^{i+1}(X^n+X^s+X^{2s}+X^{3s}{)}^i=\sum _{i=1}^{\infty }{S}_i{X}^i.$$

$$S_{n-ks}=0.$$

$$\prod _{i=1}^{n}H({\alpha }_i)\equiv n+n{s}^2(T_{n-s}+T_{n-3s})-3n{s}^2{S}_{2n-4s}(\mathrm{mod}8).$$

$$S_{2n-4s}\equiv \sum _{i=1}^n{\alpha }_i^{2n-4s}\equiv {\left(\sum _{i=1}^n{\alpha }_i^{n-2s}\right)}^2\equiv (S_{n-2s}{)}^2\equiv 0(\mathrm{mod}2),$$

$$\{\begin{array}{ll}{S}_{2n-2s}+S_{2n-3s}+S_{2n-4s}+S_{2n-5s}+S_{n-2s}=0,& \text{and}\\ S_{2n-3s}+S_{2n-4s}+S_{2n-5s}+S_{2n-6s}+S_{n-3s}=0.& \end{array}$$

$$\prod _{i=1}^{n}H({\alpha }_i)\equiv n(\mathrm{mod}8).$$

$$D(F)\equiv (-1{)}^{n(n-1)/2}n\equiv \{\begin{array}{ll}1(\mathrm{mod}8),& \text{if }n\equiv \pm 1(\mathrm{mod}8),\\ 5(\mathrm{mod}8),& \text{if }n\equiv \pm 3(\mathrm{mod}8).\end{array}$$

$$t_f\equiv \{\begin{array}{ll}1(\mathrm{mod}2),& \text{if }n\equiv \pm 1(\mathrm{mod}8),\\ 0(\mathrm{mod}2),& \text{otherwise}.\end{array}$$

Full text

††journal: AAECC

1 Introduction

Let $\mathbb{F}_{2^{n}}$ be the extension of degree $n$ of $\mathbb{F}_{2}$, the binary field. This field has numerous applications in practical areas, like cryptography or coding theory. The most common way to represent such fields is to utilize a polynomial basis, in which case an irreducible polynomial of degree $n$ over $\mathbb{F}_{2}$ is required. In particular, there are obvious computational advantages in choosing low-weight polynomials, that is polynomials with as few non-zero coefficients as possible. Namely, it is advised, see ieee00 , to favor trinomials or pentanomials. It is then natural to study the irreducibility (or equivalently the lack of it) of binary trinomials and pentanomials. We refer the interested reader to (lidlniederreiter97, , Section 3.3) and (mullenpanario13, , Chapter 3) and the references therein.

The interest for studying pentanomials was renewed as a result of the computational advantages that certain families of pentanomials bear banegascustodiopanario18 ; reyhanimasoleshhasan04 ; rodriguezhenriquezkoc03 . More specifically, the usage of the pentanomial

[TABLE]

where $s<n/3$, has been proposed. In particular, several authors reyhanimasoleshhasan04 ; zhangparhi01 have proved the computational advantages that the usage of such polynomials (known as class 2 pentanomials), as the corresponding Mastrovito multipliers have low complexity. On the other hand, the number of irreducible polynomials within the family of class 2 pentanomials has been observed to be abnormally small, see (reyhanimasoleshhasan04, , Remark 4). In this work, we study class 2 pentanomials, when $s$ is even and we prove the following.

Theorem 1.1

Let $f(X)=X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{F}_{2}[X]$ and $g(X)=X^{n}+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_{2}[X]$, with $s$ even and $n>3s$. If $n\not\equiv\pm 1\pmod{8}$, then $f$ and $g$ are reducible.

In particular, our results explain the small number of irreducible polynomials within the family of class 2 pentanomials, as a large numbers of representatives of this family are a priori reducible.

Our method is based on Swan’s swan62 techniques, who studied the parity of the number of irreducible factors of binary trinomials and proved the theorem below.

Theorem 1.2 (Swan)

Let $n>k>0$. Assume exactly one of $n$, $k$ is odd. Then $X^{n}+X^{k}+1\in\mathbb{F}_{2}[X]$ has an even number of factors (and hence is reducible) in the following cases.

$n$* is even, $k$ is odd, $n\neq 2k$ and $nk/2\equiv 0$ or $1\pmod{4}$.* 2. 2.

$n$* is odd, $k$ is even, $k\nmid 2n$ and $n\equiv\pm 3\pmod{8}$.* 3. 3.

$n$* is odd, $k$ is even, $k\mid 2n$ and $n\equiv\pm 1\pmod{8}$.*

In all other cases $X^{n}+X^{k}+1$ has an odd number of factors over $\mathbb{F}_{2}$.

The above has been extended by several authors bluher06 ; fredricksenhalessweet86 ; halesnewhart06 ; hansonpanariothomson11 . In addition, several results are known for trinomials hansonpanariothomson11 , tetranomials halesnewhart06 and certain families of binary pentanomials ahmandi06 ; koepfkim09 .

We conclude this note with some observations about the distribution of the binary irreducible polynomials of the form $X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1$, when $s$ is even and $n$ is small.

2 Preliminaries

The discriminant of the monic polynomial $F$ over an integral domain is defined as

[TABLE]

where $\alpha_{1},\ldots,\alpha_{n}$ are the roots of $F$ counted with multiplicity and $\deg(F)=n$. By using standard properties of the discriminant, see halesnewhart06 , one can show the following alternative formula for $D(F)$.

Lemma 1

Let $F$ be as above, with derivative $F^{\prime}$ and the additional assumption that its constant term is equal to $1$. Further, let $H(X):=nF(X)-XF^{\prime}(X)$. Then

[TABLE]

Proof

Since $F(X)=\prod_{i=1}^{n}(X-\alpha_{i})$, hence

[TABLE]

that is $F^{\prime}(\alpha_{i})=\prod_{j\neq i}(\alpha_{i}-\alpha_{j})$, for every $1\leq i\leq n$. Then, we get that

[TABLE]

The result follows after observing that

[TABLE]

and that $(-1)^{n}\alpha_{1}\cdots\alpha_{n}=1$. ∎

Towards the proof of 1.2, Swan’s main tool was the following.

Lemma 2 (swan62 , Corollary 3)

Let $f\in\mathbb{F}_{2}[X]$ such that $f$ is square-free and let $t_{f}$ denote the number of irreducible factors of $f$ over $\mathbb{F}_{2}$. If $F$ is a lift of $f$ in $\mathbb{Z}$, then $t_{f}\equiv\deg(f)\pmod{2}$ if and only of $D(F)\equiv 1\pmod{8}$.

Motivated by the above, we calculate $D(F)$ modulo $8$. Following the work in halesnewhart06 , we begin with proving the lemma below.

Lemma 3

Let $F\in\mathbb{Z}[X]$ be a monic polynomial such that $F(0)=1$ with $\alpha_{1},\ldots,\alpha_{n}$ its roots counted with multiplicity. Then for every $X$, with absolute value small enough,

[TABLE]

where for every $j\in\mathbb{Z}$, $S_{j}^{(F)}:=\sum_{i=1}^{n}\alpha_{i}^{j}$.

Proof

Notice that for every $j\in\{1,\ldots,n\}$ we have that $\alpha_{j}\neq 0$. Also, notice that

[TABLE]

Further, notice that for every $j$ and $X$ with small enough absolute value,

[TABLE]

and that for $X$ with small enough absolute value,

[TABLE]

It follows that for $X$ with small enough absolute value,

[TABLE]

and the result follows. ∎

Another useful tool for computing the discriminant of polynomials is the following, see (lidlniederreiter97, , Theorem 1.75).

Theorem 2.1 (Newton’s formula)

Let $F(X)=X^{n}+F_{n-1}X^{n-1}+\cdots+F_{1}X+F_{0}$ be a polynomial over some field with roots $\alpha_{1},\ldots,\alpha_{n}$, counted with multiplicity and fix some $m\in\mathbb{Z}$. Further, for any integer $t$, define $S_{t}^{(F)}=\sum_{j=1}^{n}\alpha_{j}^{t}$, then

[TABLE]

where $l:=\min(m,n)$.

3 Proof of the main theorem

From now on let $s$ be an even positive integer and $n>3s$. Also, set $f(X):=X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{F}_{2}[X]$, the typical class 2 pentanomial, and $F(X):=X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{Z}[X]$. It is clear that $F$ is a lift of $f$ in $\mathbb{Z}[X]$. The case when $n$ is even is trivial, since then clearly $f$ is a square in $\mathbb{F}_{2}$, hence reducible. So from now on we will additionally assume that $n$ is odd.

Also, if $\alpha_{1},\ldots,\alpha_{n}$ are the roots of $F$ counted with multiplicity, then for every integer $m$, set

[TABLE]

while one easily verifies that the above, as well as similar expressions, are symmetric expressions of the roots the $F$. Furthermore, it is well-known that such expressions can be written as polynomials of the elementary symmetric functions with integer coefficients. This means that, since $F$ is monic and has integer coefficients, by Vieta’s formulas, the elementary symmetric functions have integer values, which implies that all symmetric expressions of the roots the $F$ with integer coefficients have integer values. Additionally, set

[TABLE]

It is clear from Lemma 2 that we are interested in computing $D(F)$ modulo $8$. Towards this end, we compute

[TABLE]

for some $K\in\mathbb{Z}$, where we note that $s$ is even, hence the terms that include $4s$, $2s^{2}$ or $s^{k}$ for $k\geq 3$ are divisible by $8$, so one can sum them up as $8K$. Since $n$ is odd, we have that $n^{2}\equiv 1\pmod{8}$, hence

[TABLE]

Next, we apply Lemma 3 on the reciprocal of $F$ and, for small enough $X$, we get:

[TABLE]

In the LHS of the above equation, we observe that the only non-zero coefficients of terms with degree smaller than $n$ have in fact a degree that is a multiple of $s$, i.e. are even since $s$ is even. This implies that all the terms of odd degree that are smaller than $n$ are zero. In particular, $n-ks$ is odd and strictly smaller than $n$, hence in the LHS of the equation, the coefficient of $X^{n-ks}$ is zero. It follows that the same holds for the RHS, that is

[TABLE]

Now, Eq. (1) yields

[TABLE]

Furthermore, since we are only interested in the value of $D(F)$ modulo $8$ and since $s$ is even, Eq. (3) implies that for our purposes it suffices to compute $T_{n-s}+T_{n-3s}$ and $S_{2n-4s}$ modulo $2$. First, we observe that

[TABLE]

from Eq. (2). Also, by applying Theorem 2.1 for the polynomial in question for $m=2n-2s$ and $m=2n-3s$, we get

[TABLE]

By subtracting the above equations, and with Eq. (2) in mind, we conclude that $S_{2n-2s}=S_{2n-6s}$. This combined with the identity $T_{k}=(S_{k}^{2}-S_{2k})/2$ implies $T_{n-s}=T_{n-3s}$, hence $T_{n-s}+T_{n-3s}\equiv 0\pmod{2}$. The latter, along with Eqs. (3) and (4) yields

[TABLE]

This, combined with Lemma. 1 gives

[TABLE]

The combination of the above with Lemma 2 implies the following.

Proposition 1

Set $f(X)=X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}\in\mathbb{F}_{2}[X]$ and let $t_{f}$ be the number of irreducible factors of $f$ in $\mathbb{F}_{2}[X]$. Then

[TABLE]

Theorem 1.1 is an immediate consequence of Proposition 1, once we notice that $X^{n}+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_{2}[X]$ and $X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{F}_{2}[X]$ are reciprocal to each other, that is they share reducibility and irreducibility.

4 Concluding remarks

In Theorem 1.1, we proved that all the irreducible polynomials of the form $X^{n}+X^{n-s}+X^{n-2s}+X^{n-3s}+1\in\mathbb{F}_{2}[X]$, with $s$ even, satisfy $n\equiv\pm 1\pmod{8}$. In order to obtain some insight about the distribution of irreducible polynomials of that form, a computer search was performed with the computer algebra system SageMath. Namely, we did an exhaustive search of all possible values for odd $n$ and even $s$, for $7\leq n<3000$ and a total of 374,250 polynomials were checked for irreducibility. The search revealed that 804 of them are irreducible111As a confirmation of our results, we verified our results with Magma for $7\leq n\leq 1000$ and found identical results and the pairs $(n,s)$ that yielded irreducible polynomials are presented in Table 1.

The results confirm that all 804 class 2 irreducible polynomials satisfy $n\equiv\pm 1\pmod{8}$. Out of these, roughly half, i.e. 401 out of 804, satisfy $n\equiv 1\pmod{8}$ and the other 403 satisfy $n\equiv-1\pmod{8}$. Another interesting observation is that the irreducible polynomials seem to be also uniformly distributed among the even values of $s$ modulo $8$. In particular, 214 polynomials satisfy $s\equiv 2\pmod{8}$, 188 satisfy $s\equiv 4\pmod{8}$, 198 satisfy $s\equiv 6\pmod{8}$ and the rest 204 satisfy $s\equiv 0\pmod{8}$.

Regarding the frequency of irreducibility within the class 2 pentanomials tested, we have observe that out of the 749 values of $n$ considered (i.e. such that $7\leq n\leq 3000$ and $n\equiv\pm 1\pmod{8}$), 408 of them yield irreducible polynomials for some $s$ and the other 341 do not. What is more interesting however, is that it seems to be more possible for a class 2 pentanomial with $n\equiv\pm 1\pmod{8}$ and $s$ even to be irreducible than an arbitrarily chosen binary polynomial of degree $n$. More precisely, out of the 187,125 class 2 pentanomials with those specifications we tested, 804 turned out to be irreducible, hence we had a frequency $\sim 0.43\%$. In contrast, the corresponding frequency for an arbitrary binary polynomial of the same degrees is $\sim 0.13\%$, even if we exclude the obviously reducible polynomials (i.e. those with roots in $\mathbb{F}_{2}$).

An extension of Theorem 1.1 for $s$ odd does not hold, as a quick computer search verifies. Namely, we performed an exhaustive search in the interval $7\leq n<3000$, for $s$ odd. We checked 749,996 polynomials for irreducibility and identified 1707 irreducible polynomials. We had a sample roughly double the size compared to the one from our previous test and we found roughly double the number of irreducible polynomials. This suggests that if one excludes the obviously reducible class 2 pentanomials (when $n$ and $s$ are even), then the possibility of the arbitrary class 2 pentanomial to be irreducible is almost the same for both $s$ odd and even. However, after also counting Theorem 1.1 in, we see that one such polynomial with $s$ even and $n\equiv\pm 3\pmod{8}$ looks more likely to be irreducible than one with $s$ odd, while it is worth mentioning that irreducibility seems to be close to uniformly distributed for different values of (odd) $s$ modulo 8, but we found zero pairs $(n,s)$ with $8\mid n$ and very few with $n\equiv 3,5\pmod{8}$.

We conclude this note with two final remarks. First, since reducibility implies the lack of primitivity, the pentanomials described in Theorem 1.1 are also non-primitive. A quick computer test suggests that among the irreducible pentanomials of Table 1, one finds a reasonable number of primitive polynomials without any obvious pattern. Second, we note that class 2 pentanomials share many similarities with equally spaced polynomials, that is polynomials of the form $f(X^{n})\in\mathbb{F}_{2}[X]$. Such polynomials can also be used to construct low complexity multipliers wuhasan98 . It is natural to wonder about connections between the two families or special properties of pentanomials that belong in both families.

Acknowledgements.

This work was initiated during the author’s visit to the Federal University of Santa Catarina. The author is grateful to the anonymous reviewers for their valuable comments.