## Irreducible Polynomials of Given Forms (1999)

Citations: | 7 - 4 self |

### BibTeX

@MISC{Gao99irreduciblepolynomials,

author = {Shuhong Gao and Jason Howell and Daniel Panario},

title = {Irreducible Polynomials of Given Forms},

year = {1999}

}

### OpenURL

### Abstract

We survey under a unified approach on the number of irreducible polynomials of given forms: x + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little is know when q is small or fixed. We present computer experiments on several classes of polynomials over F 2 and compare our data with the results that hold for large q. We also mention some related applications and problems of (irreducible) polynomials with special forms.

### Citations

285 |
La conjecture de Weil
- DELIGNE
- 1980
(Show Context)
Citation Context ...h a given factorization pattern; the general result is described in the next section. In case (a) above, Stepanov [37] independently proves a formula for In(Vq) by using the deep Deligne-Weil theorem =-=[14]-=-. For an arbitary variety, the problem has been studied by Chatzidakis, van den Dries and Macintyre [10], Wan [38], and Fried, Haran and Jarden [15] in a more general setting. Theorem 1.4 ([10, 15, 38... |

96 |
Sur les courbes algébriques et les variétés qui s’en déduisent
- WEIL
- 1945
(Show Context)
Citation Context ...y that indeed the number is q/n + O(q1/2 ). They both use a function field analog of the Čebotarev density theorem, or Weil’s theorem on the Riemann hypothesis for function fields over a finite field =-=[41]-=-, and the fact that the Galois group of the polynomial xn + x + t over the function field Fq(t) is the symmetric group Sn of order n. The latter fact was previously determined by Birch and Swinnerton-... |

90 |
Fast Evaluation of Logarithms in Fields of Characteristic Two
- Coppersmith
- 1984
(Show Context)
Citation Context ... can be sped up using this type of polynomial, which is desirable in implementing pseudorandom number generators and several public-key cryptosystems. By exploring the low degree of g(x), Coppersmith =-=[13]-=- designs one of the fastest algorithms for computing discrete logarithms in F2n. Recently, Gao [16] constructs elements of provable high orders in finite fields by using irreducible factors of xn + g(... |

60 |
Shift Register Sequences. Aegean Park
- Golomb
- 1982
(Show Context)
Citation Context ...ors of xn + g(x) with deg g(x) small. Irreducible polynomials with a few nonzero terms are also important in efficient hardware implementation of feedback shift registers and finite field arithmetic (=-=[2, 21, 40]-=-). When the degree n is a power of 2, there is always an irreducible binomial or trinomial over Fq. For example, when q ≡ 1 mod 4, if a ∈ Fq is a quadratic nonresidue then x2k − a is irreducible over ... |

56 |
Quadratische Körper im Gebiete der höheren Kongruenzen
- Artin
- 1924
(Show Context)
Citation Context ...an’s paper [39] for more information. When V is a linear affine variety, In(Vq) has been studied by several people. Suppose that a(x) ∈ Fq[x] has degree r<n−1 and b(x) ∈ Fq[x] has degree ≤ n−1. Artin =-=[1]-=- studies In(Vq) for g0 = b(x) and gi(x) =a(x)x i−1 for 1 ≤ i ≤ n − r; here In(Vq) is the number of monic irreducible polynomials F (x) inFq[x] of degree n that are congruent to b(x) modulo a(x). Hayes... |

53 | Applications of Finite Fields - Menezes, Blake, et al. - 1993 |

28 |
Bit-serial Reed-Solomon encoder
- Berlekamp
- 1982
(Show Context)
Citation Context ...ors of xn + g(x) with deg g(x) small. Irreducible polynomials with a few nonzero terms are also important in efficient hardware implementation of feedback shift registers and finite field arithmetic (=-=[2, 21, 40]-=-). When the degree n is a power of 2, there is always an irreducible binomial or trinomial over Fq. For example, when q ≡ 1 mod 4, if a ∈ Fq is a quadratic nonresidue then x2k − a is irreducible over ... |

28 |
The distribution of polynomials over finite fields
- Cohen
- 1970
(Show Context)
Citation Context ...es. The special polynomial xn + x + a (i.e. m =1,g0 =xand g1 = 1 in (3)), has attracted much attention. Chowla [9] conjectures that the number of such irreducibles is asymptotically q/n. Later, Cohen =-=[11]-=- and Ree [32] prove independently that indeed the number is q/n + O(q1/2 ). They both use a function field analog of the Čebotarev density theorem, or Weil’s theorem on the Riemann hypothesis for func... |

25 |
Definable sets over finite fields
- Dries
- 1992
(Show Context)
Citation Context ... Stepanov [37] independently proves a formula for In(Vq) by using the deep Deligne-Weil theorem [14]. For an arbitary variety, the problem has been studied by Chatzidakis, van den Dries and Macintyre =-=[10]-=-, Wan [38], and Fried, Haran and Jarden [15] in a more general setting. Theorem 1.4 ([10, 15, 38]). Let V be an affine variety of dimension m over Fq. Then, for large q, there is a constant d ≥ 0 such... |

24 | Absolute irreducibility of polynomials via Newton polytopes
- Gao
(Show Context)
Citation Context ...ere cij ∈ F and the sum is over all pairs (i, j) such that in the real Euclidean plane the point (i, j) is inside the triangle determined by the points (m, 0), (0,n) and (u, v) (so un + vm �= mn). In =-=[17]-=-, it is proved that if gcd(m, n, u, v) = 1 then F (x, y) is absolutely irreducible over F. In particular, let F = Fq and F (x, y, z1,... ,zk)=x m +y n +x u y v + � x i y j cij(z1,... ,zk) where cij(z1... |

24 |
Discrete logarithms and their cryptographic significance
- Odlyzko
- 1985
(Show Context)
Citation Context ...m 2.1. Let V be an affine variety over Fq and r ≤ n. Determine Sr(Vq). When Vq = Fn q , Sr(Vq) is well studied. Let Nq(n, r) =Sr(Fn q), the number of r-smooth polynomials of degree n over Fq. Odlyzko =-=[30]-=- gives estimates when q =2 that easily generalize to any q (see [26]). Using the saddle point method when n →∞and n1/100 ≤ r ≤ n99/100 , one has Nq(n, r) =q n � r � n (1+o(1)) r . n Car [7] shows that... |

23 | Optimal normal bases
- Gao, Lenstra
- 1992
(Show Context)
Citation Context ...has a small primitive prime divisor r, i.e., r|(qn − 1) but r ∤ (qk − 1) for 1 ≤ k<n.sIRREDUCIBLE POLYNOMIALS 5 The condition (a) is equivalent to the existence of an optimal normal basis in Fqn; see =-=[29, 20]-=-. In case (a), the running time analysis of Semaev’s algorithm relies on the assumption that polynomials of the following forms behave like random polynomials: d� d� ckDk(x), ckφik (x) (8) k=d−m k=d−m... |

23 |
Optimal normal bases
- Mullin, Onyszchuk, et al.
- 1988
(Show Context)
Citation Context ...has a small primitive prime divisor r, i.e., r|(qn − 1) but r ∤ (qk − 1) for 1 ≤ k<n.sIRREDUCIBLE POLYNOMIALS 5 The condition (a) is equivalent to the existence of an optimal normal basis in Fqn; see =-=[29, 20]-=-. In case (a), the running time analysis of Semaev’s algorithm relies on the assumption that polynomials of the following forms behave like random polynomials: d� d� ckDk(x), ckφik (x) (8) k=d−m k=d−m... |

20 |
Primitive Polynomials over Finite Fields
- Hansen, Mullen
- 1992
(Show Context)
Citation Context ...constructions of more general sparse irreducible polynomials appear in [35, Theorem 1], and [19, Theorem 5.1]. Irreducible trinomials have been extensively studied and tabulated (see [27, Chapter 3], =-=[4, 5, 22, 43, 44]-=-). The factorization “behaviour” of polynomials of special forms are important in algorithm designs. This is particularly true for index-calculus methods for computing discrete logarithms in Fqn for s... |

19 |
Constructive problems for irreducible polynomials over finite fields
- Blake, Gao, et al.
- 1994
(Show Context)
Citation Context ...constructions of more general sparse irreducible polynomials appear in [35, Theorem 1], and [19, Theorem 5.1]. Irreducible trinomials have been extensively studied and tabulated (see [27, Chapter 3], =-=[4, 5, 22, 43, 44]-=-). The factorization “behaviour” of polynomials of special forms are important in algorithm designs. This is particularly true for index-calculus methods for computing discrete logarithms in Fqn for s... |

19 |
Rigorous discrete logarithm computations in finite fields via smooth polynomials
- Bender, Pomerance
- 1998
(Show Context)
Citation Context ...). When Vq = Fn q , Sr(Vq) is well studied. Let Nq(n, r) =Sr(Fn q), the number of r-smooth polynomials of degree n over Fq. Odlyzko [30] gives estimates when q =2 that easily generalize to any q (see =-=[26]-=-). Using the saddle point method when n →∞and n1/100 ≤ r ≤ n99/100 , one has Nq(n, r) =q n � r � n (1+o(1)) r . n Car [7] shows that for large values of r, sayr > cnlog log n/ log n, the smooth polyno... |

18 | Construction and distribution problems for irreducible trinomials over finite fields
- Blake, Gao, et al.
- 1996
(Show Context)
Citation Context ...constructions of more general sparse irreducible polynomials appear in [35, Theorem 1], and [19, Theorem 5.1]. Irreducible trinomials have been extensively studied and tabulated (see [27, Chapter 3], =-=[4, 5, 22, 43, 44]-=-). The factorization “behaviour” of polynomials of special forms are important in algorithm designs. This is particularly true for index-calculus methods for computing discrete logarithms in Fqn for s... |

14 | Computing zeta functions over finite fields
- Wan
- 1999
(Show Context)
Citation Context ...ases. Finite fields, irreducible polynomials, affine algebraic varieties, smooth polynomials. 1s2 GAO, HOWELL, AND PANARIO itself is already a difficult problem; the reader is referred to Wan’s paper =-=[39]-=- for more information. When V is a linear affine variety, In(Vq) has been studied by several people. Suppose that a(x) ∈ Fq[x] has degree r<n−1 and b(x) ∈ Fq[x] has degree ≤ n−1. Artin [1] studies In(... |

13 | An analytic approach to smooth polynomials over finite fields
- Panario, Gourdon, et al.
- 1998
(Show Context)
Citation Context ...smooth polynomials behave like the well–known number theoretic Dickman function. Later, Soundararajan [36] obtained estimates for the full range of q, r and n. Recently, Panario, Gourdon and Flajolet =-=[31]-=- used an analytic approach to show that the smooth polynomials also behave like the Dickman function for r>(log n) 1/k for k a positive integer constant. Nothing is known about Sr(Vq) when Vq �= Fn q ... |

12 | Elements of provable high orders in finite fields
- Gao
- 1999
(Show Context)
Citation Context ...er generators and several public-key cryptosystems. By exploring the low degree of g(x), Coppersmith [13] designs one of the fastest algorithms for computing discrete logarithms in F2n. Recently, Gao =-=[16]-=- constructs elements of provable high orders in finite fields by using irreducible factors of xn + g(x) with deg g(x) small. Irreducible polynomials with a few nonzero terms are also important in effi... |

9 |
Note on a problem of Chowla
- Birch, Swinnerton-Dyer
- 1959
(Show Context)
Citation Context ... the fact that the Galois group of the polynomial xn + x + t over the function field Fq(t) is the symmetric group Sn of order n. The latter fact was previously determined by Birch and Swinnerton-Dyer =-=[3]-=- and a simple proof of it is given by Hayes [24]. In fact, Cohen considers the more general polynomials (3) for m = 1 in [11] and for an arbitrary m in [12]. InsIRREDUCIBLE POLYNOMIALS 3 other words, ... |

8 | Tests and constructions of irreducible polynomials over finite fields - Gao, Panario - 1997 |

8 |
An algorithm for evaluation of discrete logarithms in some nonprime finite fields
- Semaev
- 1998
(Show Context)
Citation Context ...olynomials of the form (7) u1(x)h(x)+u2(x) behave like random polynomials of the same degree where h(x) ∈ F2[x] is fixed and u1(x),u2(x)∈F2[x] are chosen at random of certain degrees. Recently Semaev =-=[33]-=- designs another fast algorithm for computing discrete logarithms in Fqn when q and n satisfy one of the two conditions: (a) if r =2n+ 1 is a prime and Z × r =< q,−1>; (b) if qn − 1 has a small primit... |

8 | Finding irreducible and primitive polynomials - Shparlinski - 1993 |

7 |
Explicit factorization of x2k + 1 over Fp with prime p ≡ 3 (mod 4
- Blake, Gao, et al.
- 1993
(Show Context)
Citation Context ...q ≡ 1 mod 4, if a ∈ Fq is a quadratic nonresidue then x2k − a is irreducible over Fq for all k ≥ 0. When q ≡ 3mod4, there is no irreducible binomial of degree 2k for k ≥ 2. In this case, we have from =-=[6]-=- the following construction of irreducible trinomials. Suppose that q = pm where m is odd and p ≡ 3 mod 4 is a prime. Let 2v |(p + 1), 2v+1 ∤ (p + 1). Then v ≥ 2. Compute u ∈ Fp iteratively as follows... |

7 |
A theorem of Dickson on irreducible polynomials
- Carlitz
- 1952
(Show Context)
Citation Context ...nfinite sequence of r such that e ɛ� 1 1 + logq r ≤ κ(xr − 1) ≤ e0.83 (1 + logq r) for some constant ɛ depending only on q. Theorem 1.2 improves previous work of Uchiyama [42] for b(x) =xrand Carlitz =-=[8]-=- for b(x) =xrand s = r = 1. By using Theorem 1.2, Hsu [25] proves that there is always an irreducible polynomial of degree n in Fq[x] with the lower or higher half of the coefficients fixed at any val... |

6 |
Effective counting of the points of definable sets over finite fields
- Fried, Haran, et al.
(Show Context)
Citation Context ...a for In(Vq) by using the deep Deligne-Weil theorem [14]. For an arbitary variety, the problem has been studied by Chatzidakis, van den Dries and Macintyre [10], Wan [38], and Fried, Haran and Jarden =-=[15]-=- in a more general setting. Theorem 1.4 ([10, 15, 38]). Let V be an affine variety of dimension m over Fq. Then, for large q, there is a constant d ≥ 0 such that (6) In(Vq) =d· qm n +O(qm−12). In the ... |

4 |
Density of normal elements in finite fields
- Gao, Panario
- 1997
(Show Context)
Citation Context ...n for some 1/2 ≤ v<1where κ(a) =ϕ(a)/qr and ϕ(a) is the number of units in Fq[x]/(a(x)). The estimate (4) is nontrivial only if m>n/2. Lower bounds for κ(a) are known. By Theorem 2.1 and its proof in =-=[18]-=-, ⎧ � ⎨ 1 − 1 ≥ κ(a) ≥ ⎩ 1 �r � q 1 − if r ≤ q, 1 � 1 q e0.83 (1+logq r) if r>q, where r is the degree of a(x). Hence 1 ≤ 1 κ(a) ≤ ⎧ � ⎨ 1+ ⎩ 1 �r � q−1 1+ if r ≤ q, 1 � e0.83 (1 + logq r) if r>q. q−1... |

4 | Open problems and conjectures in finite fields
- Mullen, Shparlinski
- 1996
(Show Context)
Citation Context ...8 − x, or x16 1 − x, the corresponding d is expected to be κ(a) =4,5.33, 5.22, and 6.47, respectively. This is indeed verified by our computation. These polynomials provide examples for Problem 27 in =-=[28]-=-. We also did an experiment on the existence of irreducible polynomials of the form xn + g(x) ∈ Fq[x] with deg g(x) = log n + O(1). For q = 2 and n ≤ 2000, it turns out that such irreducibles always e... |

4 |
Asymptotic formulae for the counting function of smooth polynomials
- Soundararajan
(Show Context)
Citation Context ...n � r � n (1+o(1)) r . n Car [7] shows that for large values of r, sayr > cnlog log n/ log n, the smooth polynomials behave like the well–known number theoretic Dickman function. Later, Soundararajan =-=[36]-=- obtained estimates for the full range of q, r and n. Recently, Panario, Gourdon and Flajolet [31] used an analytic approach to show that the smooth polynomials also behave like the Dickman function f... |

3 |
A note on the construction of finite Galois fields GF(p n
- Chowla
- 1966
(Show Context)
Citation Context ... of degree n in Fq[x] with the lower or higher half of the coefficients fixed at any values. The special polynomial xn + x + a (i.e. m =1,g0 =xand g1 = 1 in (3)), has attracted much attention. Chowla =-=[9]-=- conjectures that the number of such irreducibles is asymptotically q/n. Later, Cohen [11] and Ree [32] prove independently that indeed the number is q/n + O(q1/2 ). They both use a function field ana... |

3 |
Proof of a conjecture of S
- Ree
- 1971
(Show Context)
Citation Context ...al polynomial xn + x + a (i.e. m =1,g0 =xand g1 = 1 in (3)), has attracted much attention. Chowla [9] conjectures that the number of such irreducibles is asymptotically q/n. Later, Cohen [11] and Ree =-=[32]-=- prove independently that indeed the number is q/n + O(q1/2 ). They both use a function field analog of the Čebotarev density theorem, or Weil’s theorem on the Riemann hypothesis for function fields o... |

3 |
Bit-serial multiplication in finite fields
- Wang, Blake
- 1990
(Show Context)
Citation Context ...ors of xn + g(x) with deg g(x) small. Irreducible polynomials with a few nonzero terms are also important in efficient hardware implementation of feedback shift registers and finite field arithmetic (=-=[2, 21, 40]-=-). When the degree n is a power of 2, there is always an irreducible binomial or trinomial over Fq. For example, when q ≡ 1 mod 4, if a ∈ Fq is a quadratic nonresidue then x2k − a is irreducible over ... |

2 |
Hilbert sets and zeta functions over finite fields
- Wan
- 1992
(Show Context)
Citation Context ...[37] independently proves a formula for In(Vq) by using the deep Deligne-Weil theorem [14]. For an arbitary variety, the problem has been studied by Chatzidakis, van den Dries and Macintyre [10], Wan =-=[38]-=-, and Fried, Haran and Jarden [15] in a more general setting. Theorem 1.4 ([10, 15, 38]). Let V be an affine variety of dimension m over Fq. Then, for large q, there is a constant d ≥ 0 such that (6) ... |

2 |
les polynomes irréductibles dans un corps fini
- Uchiyama, Sur
- 1955
(Show Context)
Citation Context ...[18] shows that there is an infinite sequence of r such that e ɛ� 1 1 + logq r ≤ κ(xr − 1) ≤ e0.83 (1 + logq r) for some constant ɛ depending only on q. Theorem 1.2 improves previous work of Uchiyama =-=[42]-=- for b(x) =xrand Carlitz [8] for b(x) =xrand s = r = 1. By using Theorem 1.2, Hsu [25] proves that there is always an irreducible polynomial of degree n in Fq[x] with the lower or higher half of the c... |

2 |
Table of Primitive Binary Polynomials
- unknown authors
- 1994
(Show Context)
Citation Context |

2 | Theoremes de densite dans Fq [X - Car - 1987 |

1 |
Théorèmes de densité dansFq[X
- Car
- 1987
(Show Context)
Citation Context .... Odlyzko [30] gives estimates when q =2 that easily generalize to any q (see [26]). Using the saddle point method when n →∞and n1/100 ≤ r ≤ n99/100 , one has Nq(n, r) =q n � r � n (1+o(1)) r . n Car =-=[7]-=- shows that for large values of r, sayr > cnlog log n/ log n, the smooth polynomials behave like the well–known number theoretic Dickman function. Later, Soundararajan [36] obtained estimates for the ... |

1 |
The distribution of irreducibles in Fq[x
- Hayes
- 1965
(Show Context)
Citation Context ...studies In(Vq) for g0 = b(x) and gi(x) =a(x)x i−1 for 1 ≤ i ≤ n − r; here In(Vq) is the number of monic irreducible polynomials F (x) inFq[x] of degree n that are congruent to b(x) modulo a(x). Hayes =-=[23]-=- generalizes Artin’s result to the case where g0 = b(x) and gi(x) =a(x)x i−1 ,1≤i≤n−r−s, where s is fixed with 0 ≤ s ≤ n − r − 1 (that is, the first s coefficients t1,... ,ts of F (x) are fixed). Theo... |

1 |
The Galois group of x n + x - t
- Hayes
- 1973
(Show Context)
Citation Context ...al xn + x + t over the function field Fq(t) is the symmetric group Sn of order n. The latter fact was previously determined by Birch and Swinnerton-Dyer [3] and a simple proof of it is given by Hayes =-=[24]-=-. In fact, Cohen considers the more general polynomials (3) for m = 1 in [11] and for an arbitrary m in [12]. InsIRREDUCIBLE POLYNOMIALS 3 other words, Cohen determines In(Vq) for a linear affine vari... |

1 |
The distribution of irreducibles in Fq[t
- Hsu
- 1996
(Show Context)
Citation Context ...− 1) ≤ e0.83 (1 + logq r) for some constant ɛ depending only on q. Theorem 1.2 improves previous work of Uchiyama [42] for b(x) =xrand Carlitz [8] for b(x) =xrand s = r = 1. By using Theorem 1.2, Hsu =-=[25]-=- proves that there is always an irreducible polynomial of degree n in Fq[x] with the lower or higher half of the coefficients fixed at any values. The special polynomial xn + x + a (i.e. m =1,g0 =xand... |

1 |
The number of irreducible polynomials of a given form over a finite field
- Stepanov
- 1987
(Show Context)
Citation Context ...siders the more general problem of determining the number of polynomials in P (Vq) with a given factorization pattern; the general result is described in the next section. In case (a) above, Stepanov =-=[37]-=- independently proves a formula for In(Vq) by using the deep Deligne-Weil theorem [14]. For an arbitary variety, the problem has been studied by Chatzidakis, van den Dries and Macintyre [10], Wan [38]... |

1 | Explicit factorization of x k + 1 over Fp with prime p # 3(mod 4 - Blake, Gao, et al. - 1993 |

1 | The distribution of irreducibles in Fq [x - Hayes - 1965 |

1 | The distribution of irreducibles in Fq [t - Hsu - 1996 |

1 | c, A table of primitive binary polynomials - Zivkovi - 1994 |