We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Let
$r,n>1$
be integers and
$q$
be any prime power
$q$
such that
$r\mid q^{n}-1$
. We say that the extension
$\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$
possesses the line property for
$r$
-primitive elements property if, for every
$\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$
such that
$\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$
, there exists some
$x\in \mathbb{F}_{q}$
such that
$\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$
has multiplicative order
$(q^{n}-1)/r$
. We prove that, for sufficiently large prime powers
$q$
,
$\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$
possesses the line property for
$r$
-primitive elements. We also discuss the (weaker) translate property for extensions.
We obtain a new lower bound on the size of the value set
$\mathscr{V}(f)=f(\mathbb{F}_{p})$
of a sparse polynomial
$f\in \mathbb{F}_{p}[X]$
over a finite field of
$p$
elements when
$p$
is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of
$f$
and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in
$\mathscr{V}(f)$
.
A polynomial
$f$
over a finite field
$\mathbb{F}_{q}$
can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers
$f^{e}$
for each
$e$
from
$1$
to
$q-2$
, as well as the existence of a unique solution to
$f(x)=0$
in
$\mathbb{F}_{q}$
. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.
We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree
$d$
over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree
$d$
. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for
$d>6$
. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order
$q>8$
. Such PPs exist if and only if
$q\in \{11,13,19,23,27,29,31\}$
and are explicitly listed in normalised form.
Let
$K$
be a field that admits a cyclic Galois extension of degree
$n\geq 2$
. The symmetric group
$S_{n}$
acts on
$K^{n}$
by permutation of coordinates. Given a subgroup
$G$
of
$S_{n}$
and
$u\in K^{n}$
, let
$V_{G}(u)$
be the
$K$
-vector space spanned by the orbit of
$u$
under the action of
$G$
. In this paper we show that, for a special family of groups
$G$
of affine type, the dimension of
$V_{G}(u)$
can be computed via the greatest common divisor of certain polynomials in
$K[x]$
. We present some applications of our results to the cases
$K=\mathbb{Q}$
and
$K$
finite.
We prove the reciprocity law for the twisted second moments of Dirichlet
$L$
-functions over rational function fields, corresponding to
two irreducible polynomials. This formula is the analogue of the formulas
for Dirichlet
$L$
-functions over
$\mathbb{Q}$
obtained by Conrey [‘The mean-square of Dirichlet
$L$
-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law
for the twisted second moment of Dirichlet
$L$
-functions’, Forum Math.
23(6) (2011), 1323–1337].
We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.
A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.
We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.
For a finite field of odd cardinality
$q$
, we show that the sequence of iterates of
$aX^{2}+c$
, starting at
$0$
, always recurs after
$O(q/\text{log}\log q)$
steps. For
$X^{2}+1$
, the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of
$X^{3}+c$
, when
$q$
is 2 mod 3.
For a
$t$
-nomial
$f(x)=\sum _{i=1}^{t}c_{i}x^{a_{i}}\in \mathbb{F}_{q}[x]$
, we show that the number of distinct, nonzero roots of
$f$
is bounded above by
$2(q-1)^{1-\unicode[STIX]{x1D700}}C^{\unicode[STIX]{x1D700}}$
, where
$\unicode[STIX]{x1D700}=1/(t-1)$
and
$C$
is the size of the largest coset in
$\mathbb{F}_{q}^{\ast }$
on which
$f$
vanishes completely. Additionally, we describe a number-theoretic parameter depending only on
$q$
and the exponents
$a_{i}$
which provides a general and easily computable upper bound for
$C$
. We thus obtain a strict improvement over an earlier bound of Canetti et al. which is related to the uniformity of the Diffie–Hellman distribution. Finally, we conjecture that
$t$
-nomials over prime fields have only
$O(t\log p)$
roots in
$\mathbb{F}_{p}^{\ast }$
when
$C=1$
.
Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.
The problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.
We determine several variants of the classical interpolation formula for finite fields which produce polynomials that induce a desirable mapping on the nonspecified elements, and without increasing the number of terms in the formula. As a corollary, we classify those permutation polynomials over a finite field which are their own compositional inverse, extending work of C. Wells.
In this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form
${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, where
${L}_{i} (x)$
and
$B(x)$
are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms
$xh({\lambda }_{j} (x))$
and
$xh({\mu }_{j} (x))$
, where
${\lambda }_{j} (x)$
is the
$j$
th elementary symmetric polynomial of
$x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $
and
${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$
. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form
${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, which extends a result of Kyureghyan.
Let
${ \mathbb{F} }_{q} $
be the finite field of characteristic
$p$
containing
$q= {p}^{r} $
elements and
$f(x)= a{x}^{n} + {x}^{m} $
, a binomial with coefficients in this field. If some conditions on the greatest common divisor of
$n- m$
and
$q- 1$
are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if
$f(x)= a{x}^{n} + {x}^{m} $
permutes
${ \mathbb{F} }_{p} $
, where
$n\gt m\gt 0$
and
$a\in { \mathbb{F} }_{p}^{\ast } $
, then
$p- 1\leq (d- 1)d$
, where
$d= \gcd (n- m, p- 1)$
, and that this bound of
$p$
, in terms of
$d$
only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of
${ \mathbb{F} }_{q} $
from a permutation binomial over
${ \mathbb{F} }_{q} $
.
Given
$f(x,y)\in \mathbb Z[x,y]$
with no common components with
$x^a-y^b$
and
$x^ay^b-1$
, we prove that for
$p$
sufficiently large, with
$C(f)$
exceptions, the solutions
$(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$
of
$f(x,y)=0$
satisfy
$ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$
where
$c$
is a constant and
${\rm ord}(r)$
is the order of
$r$
in the multiplicative group
$\overline {\mathbb F}_p^*$
. Moreover, for most
$p\lt N$
,
$N$
being a large number, we prove that, with
$C(f)$
exceptions,
${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$
where
$\epsilon (p)$
is an arbitrary function tending to
$0$
when
$p$
goes to
$\infty $
.
We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.
We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(ℤpn,ℤpn), and a basis for the (pn−1−n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the -dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form.
We introduce a class of polynomials which induce a permutation on the set of polynomials in one variable of degree less than m over a finite field. We call then Am-permutation polynomials. We also give three criteria to characterize such polynomials.