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Algebraic structure
a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field
Finite_field
Arithmetic in a field with a finite number of elements
mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an
Finite_field_arithmetic
Method of exchanging cryptographic keys
supercomputers. The simplest and the original implementation, later formalized as Finite Field Diffie–Hellman in RFC 7919, of the protocol uses the multiplicative group
Diffie–Hellman_key_exchange
Algebraic structure with addition, multiplication, and division
of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied in mathematics
Field_(mathematics)
quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is
Quasi-finite_field
pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect
Pseudo-finite_field
Cryptographic algorithm created by Adi Shamir
{\displaystyle S} can be represented as an element a 0 {\displaystyle a_{0}} of a finite field G F ( q ) {\displaystyle \mathrm {GF} (q)} (where q {\displaystyle q}
Shamir's_secret_sharing
Shape containing unit line segments in all directions
conjecture could be carried over to the Euclidean case. Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ Fn be a Kakeya set, i.e. for each vector
Kakeya_set
Dimension of the extension field viewed as a vector space over the base field
be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements)
Degree_of_a_field_extension
Algebraic curve in mathematics
a finite field Fp is, in some sense, a generating function assembling the information of the number of points of E with values in the finite field extensions
Elliptic_curve
mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable
Hyper-finite_field
Construction of a larger algebraic field by "adding elements" to a smaller field
s ) / K {\displaystyle K(s)/K} is not finite, the field K ( s ) {\displaystyle K(s)} is isomorphic to the field of rational fractions in s {\displaystyle
Field_extension
Injective polynomial functions are bijective
algebraic closures. That is, for any field F {\displaystyle F} that is itself finite or that is the closure of a finite field, if a polynomial P {\displaystyle
Ax–Grothendieck_theorem
Geometric system with a finite number of points
higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine
Finite_geometry
Roots of multiple multivariate polynomials
given on fields k in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields. Searching
System of polynomial equations
System_of_polynomial_equations
Finitely generated extension field of positive transcendence degree
function field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension
Algebraic_function_field
Mathematical concept
kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible
Global_field
Polynomial without nontrivial factorization
over the integers, the rational numbers, finite fields and finitely generated field extension of these fields. All these algorithms use the algorithms
Irreducible_polynomial
Geometric concept of a 2D space with "points at infinity" adjoined
common case, they are also known as field planes, and if the field is a finite field they can be called finite field planes or Galois planes. A subplane
Projective_plane
Mathematical group
all finite Galois extensions E / F {\displaystyle E/F} for a fixed field. The inverse limit is denoted Gal ( F ¯ / F ) := lim ← E / F finite separable
Galois_group
Mathematical function
the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K. Let K be a field and
Field_trace
Best results achieved to date
p} , the multiplicative group of a finite field, and the group of points on an elliptic curve over a finite field.[citation needed] The current[needs
Discrete_logarithm_records
Mathematical group based upon a finite number of elements
of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite
Finite_group
Result in algebra
theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for finite rings, there is no distinction
Wedderburn's_little_theorem
for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them. All factorization
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Generator of the multiplicative group of a finite field
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called
Primitive element (finite field)
Primitive_element_(finite_field)
irreducible representation of the finite group Sp4(Fq), where Fq is a finite field with q elements. In this finite-field case, for q odd, θ10 has dimension
Θ10
Polynomial that permutes a ring
ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function
Permutation_polynomial
Branch of number theory
algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring
Algebraic_number_theory
Specific algebraic group
spaces and buildings. In most places we suppose that the base field is perfect (for example finite or characteristic zero). This hypothesis is required to have
Algebraic_torus
Uniform coding for primitive elements of all finite fields
In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define
Conway polynomial (finite fields)
Conway_polynomial_(finite_fields)
Sporadic simple group
the Huang–Jackson–Lee–Poonen–Pries–Zhang realization. Let F211 be the finite field with 211 elements. Its group of units has order 211 − 1 = 2047 = 23 ·
Mathieu_group_M23
with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension
Local_class_field_theory
K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen. The map sending a finite-dimensional
K-groups_of_a_field
Algebraic structure where all polynomials have roots
all finite fields of a fixed characteristic p (p prime) is an algebraically closed field, which is, in fact, the algebraic closure of the field F p {\displaystyle
Algebraically_closed_field
Smallest integer n for which n equals 0 in a ring
the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive
Characteristic_(algebra)
Mathematical group
refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The important
Group_of_Lie_type
Theoretical object in mathematics
the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could
Field_with_one_element
Projective variety that is also an algebraic group
defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain
Abelian_variety
Approach to public-key cryptography
over finite fields. ECC allows smaller keys to provide equivalent security, compared to cryptosystems based on modular exponentiation in finite fields, such
Elliptic-curve_cryptography
Family of polynomials
{\displaystyle \mathbb {F} _{q}} , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian G r ( k , F q n ) {\displaystyle
Gaussian_binomial_coefficient
Line with a point at infinity added
a finite field Fq of q elements has q + 1 points. In all other respects it is no different from projective lines defined over other types of fields. In
Projective_line
Change of basis applied in quantum computing
permutation. The discrete Fourier transform can also be formulated over a finite field F q {\displaystyle F_{q}} , and a quantum version can be defined. Consider
Quantum_Fourier_transform
Linear representation in mathematics
linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional
Steinberg_representation
Type of zeta function
all points whose residue field is finite. The cardinality of this field is denoted N(x). If X is the spectrum of a finite field with q elements, then ζ
Arithmetic_zeta_function
Certain polynomial equations in enough variables over a finite field have solutions
that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly
Chevalley–Warning_theorem
Given a finite field extension K / F {\displaystyle K/F} over a locally compact field F {\displaystyle F} , there is at most one unique field norm | ⋅
Locally_compact_field
Natural number
space. The integers modulo 2 form the finite field F 2 {\displaystyle \mathbb {F} _{2}} , the smallest finite field. It has two elements, usually denoted
2
Locally compact infinite topological field
fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation whose residue field is finite.
Local_field
Expresses the number of points of a variety over a finite field
Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology
Grothendieck_trace_formula
Conjecture in number theory
polynomial ring F [ u ] {\displaystyle F[u]} for a finite field F {\displaystyle F} , one can ask how often a finite set of polynomials f i ( x ) {\displaystyle
Bateman–Horn_conjecture
Minimal polynomial of a primitive element in a finite field
In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(pm). This means
Primitive polynomial (field theory)
Primitive_polynomial_(field_theory)
Concept in field theory mathematics
(field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Let K be a field and L a finite extension
Field_norm
Field theory is the branch of algebra that studies fields
example, Complex conjugate. Finite field A field with finitely many elements, a.k.a. Galois field. Ordered field A field with a total order compatible
Glossary_of_field_theory
a finite field, using only multiplication of the matrix by long, thin matrices. Such matrices are considered as vectors of tuples of finite-field entries
Block_Lanczos_algorithm
Discrete group type in group theory
over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field. In two dimensions, the finite reflection
Reflection_group
Finite extension of the rationals
\mathbb {Q} } such that the field extension K / Q {\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K {\displaystyle
Algebraic_number_field
conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion: K i ( X ) ⊗ Q =
Parshin's_conjecture
Algebraic structure
{C} } ; every finite field F q {\displaystyle \mathbb {F} _{q}} ; every algebraically closed field; the union of a set of perfect fields totally ordered
Perfect_field
Group without normal subgroups other than the trivial group and itself
of Lie type over the field with one element, which unites this family with the next, and thus all families of non-abelian finite simple groups may be
Simple_group
Group of unitary matrices
defined over fields other than the complex numbers. The hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. Since
Unitary_group
Theorem in arithmetic combinatorics
arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set A {\displaystyle A} of integers, at least one of the sets A + A {\displaystyle
Erdős–Szemerédi_theorem
Map raising elements to the pth power, in characteristic p
rings with prime characteristic p, an important class that includes finite fields. The endomorphism maps every element to its pth power. In certain contexts
Frobenius_endomorphism
projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of Fq. Making the
Local_zeta_function
Generalisation of Fourier transform to any ring
does not make sense in an arbitrary field. If F = G F ( q ) {\displaystyle F=\mathrm {GF} (q)} is a finite field, where q is a prime power, then the existence
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Abstract ring with finite number of elements
finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring
Finite_ring
Generalization of vector spaces from fields to rings
an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and
Module_(mathematics)
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of
List_of_finite_simple_groups
Result concerning ideals of commutative rings
algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces. The following
Prime_avoidance_lemma
Mathematical group that can be generated as the set of powers of a single element
the Frobenius mapping. Conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G. All
Cyclic_group
Characterization of how many integers are prime
that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical
Prime_number_theorem
Algorithm for computing kernel vectors
Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalization by Don Coppersmith of an algorithm due to Doug Wiedemann
Block_Wiedemann_algorithm
Irreducible polynomial whose roots are nth roots of unity
p ≤ q ≤ r {\displaystyle p\leq q\leq r} are three odd primes. Over a finite field with a prime number p of elements, for any integer n that is not a multiple
Cyclotomic_polynomial
Completion of the usual space with "points at infinity"
a finite field, GF(q), whose order (that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field has
Projective_space
Special fields Over a finite field, d = 1; over the reals, d = 1 or 2; over a p-adic field or a number field, or any local or global function field, d is
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Number with an integer power equal to 1
roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly
Root_of_unity
Quantum field theory at non-zero temperatures
theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate
Thermal_quantum_field_theory
Type of algebraic field extension
Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable. It follows that
Separable_extension
On generating functions from counting points on algebraic varieties over finite fields
over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as
Weil_conjectures
Fingerprinting algorithm
finite field. It was proposed by Michael O. Rabin. Given an n-bit message m0,...,mn-1, we view it as a polynomial of degree n-1 over the finite field
Rabin_fingerprint
German mathematician, specialist in combinatorics
optimisation, and operations research. Jungnickel wrote about finite fields in 1993: Finite fields, Structure and Arithmetics. A reviewer notes that "The author
Dieter_Jungnickel
Syllabus in college and university mathematics
Finite Mathematics, Academic Press Business mathematics § Undergraduate Discrete mathematics Finite geometry Finite group, Finite ring, Finite field Finite
Finite_mathematics
Mathematical theorem used in cryptography
specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a
Normal_basis
Computational method
number fields, a fundamental step is a factorization of a polynomial over a finite field. Polynomial rings over the integers or over a field are unique
Factorization_of_polynomials
Theorem in number theory
that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies
Serre's_modularity_conjecture
leading to the Iwahori–Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel
Hecke_algebra_of_a_pair
Commutative ring with no zero divisors other than zero
field. In particular, all finite integral domains are finite fields (more generally, by Wedderburn's little theorem, finite domains are finite fields)
Integral_domain
Network coding concept
cryptography over a finite field is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Let F q {\displaystyle
Homomorphic signatures for network coding
Homomorphic_signatures_for_network_coding
In algebraic geometry, a point with rational coordinates
X(k) is finite.) In the opposite direction, a variety X over a number field k is said to have potentially dense rational points if there is a finite extension
Rational_point
Substitution box used in the Rijndael cipher
multiplicative inverse in GF(28) = GF(2) [x]/(x8 + x4 + x3 + x + 1), Rijndael's finite field. Zero, as the identity, is mapped to itself. This transformation is known
Rijndael_S-box
Construction in group theory
special linear groups PSL(n, Fq) for a finite field Fq are often written as PSL(n, q) or Ln(q). They are finite simple groups whenever n is at least 2
Projective_linear_group
Mathematical problem
are known to exist when n is a prime number or power of a prime (see Finite field construction below). However, the number of MOLS that may exist for a
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Problem of inverting exponentiation in groups
such that Q = k P {\displaystyle Q=kP} . When the underlying field F is a finite field, this problem has cryptographic applications. Powers obey the
Discrete_logarithm
Number in {..., –2, –1, 0, 1, 2, ...}
is also a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, Z {\displaystyle
Integer
Turkish cryptographic engineer
research and work in cryptographic engineering, secure hardware design, finite field arithmetic, and side‑channel security. He has retired from Computer Science
Çetin_Kaya_Koç
Conjecture on zeros of the zeta function
which have been proved, such as the Riemann hypothesis for curves over finite fields, which was proved by André Weil. The Riemann zeta function ζ {\displaystyle
Riemann_hypothesis
Concept in quantum information theory
r {\displaystyle d=p^{r}} is a power of a prime, we make use of the finite field F d {\displaystyle \mathbb {F} _{d}} to construct a maximal set of d + 1
Mutually_unbiased_bases
Area of combinatorics
higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine
Algebraic_combinatorics
Mathematical curves that are isomorphic over algebraic closures
, but over the field extension K [ X ] / ( X 2 + X + d ) {\displaystyle K[X]/(X^{2}+X+d)} . If K {\displaystyle K} is a finite field with q {\displaystyle
Twists_of_elliptic_curves
Algebraic structure
modules. Suppose K is finitely generated over its prime field (e.g. a finite field, an algebraic number field, a global function field), of characteristic
Tate_module
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