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Longer titles found: Finite field arithmetic (view), Factorization of polynomials over finite fields (view), Primitive element (finite field) (view), Quasi-finite field (view), Pseudo-finite field (view), Hyper-finite field (view), Conway polynomial (finite fields) (view)

searching for finite field 123 found (705 total)

alternate case: Finite field

Diffie–Hellman key exchange (4,969 words) [view diff] case mismatch in snippet view article find links to article

supercomputers. The simplest and the original implementation, later formalized as Finite Field Diffie-Hellman in RFC 7919, of the protocol uses the multiplicative group

x)^{2}}}.} When the integers are replaced by the polynomial ring F[u] for a finite field F, one can ask how often a finite set of polynomials fi(x) in F[u][x]

group of a finite field is trivial. In fact, this characterization immediately yields a proof of the theorem as follows: let K be a finite field. Since the

projective special linear group of the 2-dimensional vector space over the finite field of 19 elements, L2(19). It has Schläfli type {5,3,5} with 5 hemi-dodecahedral

projective special linear group of the 2-dimensional vector space over the finite field with 11 elements L2(11). It was discovered in 1977 by Branko Grünbaum

finite field with q elements, then ζ X ( s ) = 1 1 − q − s . {\displaystyle \zeta _{X}(s)={\frac {1}{1-q^{-s}}}.} For a variety X over a finite field

Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical

absolute Galois group of the finite field. The dual group G* is then the reductive algebraic group over the finite field associated to the dual root datum

In algebraic geometry, given a smooth projective curve X over a finite field F q {\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over

In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field

have only an equivalence class of norm: the sup norm pg. 58-59. Given a finite field extension K / F {\displaystyle K/F} over a locally compact field F {\displaystyle

underlying set of the finite field of order pq of the form x→axσ+b where a has norm 1 and σ is an automorphism of the finite field, where p and q are distinct

endomorphisms of the Jacobian J0(N)Fp of the modular curve X0N over the finite field Fp. The Eichler–Shimura congruence relation and its generalizations to

In mathematics, a nilcurve is a pointed stable curve over a finite field with an indigenous bundle whose p-curvature is square nilpotent. Nilcurves were

is an odd prime. Here Fp{\displaystyle \mathbf {F} _{p}} denotes the finite field with p{\displaystyle p} elements; {0,1,…,p−1}{\displaystyle \{0,1,\dots

In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Eisenstein

generate numbers in a finite field (for example G F ( 2 n + 1 ) {\displaystyle \mathrm {GF} (2^{n}+1)} ). A root of unity under a finite field GF(r), is an element

the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm

for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant of

corresponds to the Frobenius automorphism in the Galois group of the finite field extension Fj /F. In the unramified case the order of DPj is f and IPj

also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of the base field is a conjugacy-closed

where D is a central division algebra over k. Special fields: Over a finite field, d = 1; over the reals, d = 1 or 2; over a p-adic field or a number field

Characterization of the Three-Dimensional Projective Unitary Group over a Finite Field. He was a professor at Rutgers University. In 1976 he found strong evidence

it is a necessary condition for representability of a matroid over a finite field. Let M be a matroid and let ρ be its rank function, Ingleton's inequality

version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922. Theorem 1 (Schwartz

K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen. The map sending

integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers. Local

of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions O(X) on X is a finitely

ISBN 0691123306. Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms. Annals of Mathematical Studies, Princeton 2012. With

generated as an associative algebra over another field k, then K is a finite field extension of k (that is, it is also finitely generated as a vector space)

number of strands in the braid, q {\displaystyle q} , the size of the finite field F q {\displaystyle \mathbb {F} _{q}} , M ∗ {\displaystyle M_{*}} , the

matrices under multiplication is the subgroup of scalar matrices. Fix a finite field F q {\displaystyle \mathbb {F} _{q}} , let P ( n ) {\displaystyle P(n)}

(1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states that the torus thus built is a variety

of the curve lies in a quadratic extension of the prime field of K, a finite field of order p2. Suppose E is in Legendre form, defined by the equation y

a completely integrable system whose twisted generalization over a finite field was used by Ngô Bảo Châu in his proof of the fundamental lemma in the

encryption J. H. Ellis, January 1970. Non-Secret Encryption Using a Finite Field MJ Williamson, January 21, 1974. Thoughts on Cheaper Non-Secret Encryption

This fact was essentially known to Leonard Dickson (1901), even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field

The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the m {\displaystyle m} "most

would be an element of the finite field. The operator ⋅ represents ordinary multiplication (repeated addition in the finite field) which is the same as the

Wrońskian with differentiation replaced by the Frobenius endomorphism over a finite field. Alternant matrix Vandermonde matrix Peano published his example twice

the Finite Field Kakeya Conjecture using the polynomial method. Finite Field Kakeya Conjecture: Let F q {\displaystyle \mathbb {F} _{q}} be a finite field

creating ever-larger arrays to hold growing patterns. The Game of Life on a finite field is sometimes explicitly studied; some implementations, such as Golly

the second residue field, and the pattern continues until we reach a finite field. Two-dimensional local fields are divided into the following classes:

groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as

discrete valuation ring O {\displaystyle {\mathcal {O}}} , then there is a finite field extension L / K {\displaystyle L/K} such that A ( L ) = A ⊗ K L {\displaystyle

quadratic system S=(Q1, ..., Qm) of m=kn equations in n unknowns over a finite field GF(q). The keystream generation process simply consists in iterating

quadratic system S=(Q1, ..., Qm) of m=kn equations in n unknowns over a finite field GF(q). The keystream generation process simply consists in iterating

= 0 {\displaystyle GH^{\top }=P-P=0} . Negation is performed in the finite field Fq. Note that if the characteristic of the underlying field is 2 (i.e

finitely generated as an associative algebra over a field k, then it is a finite field extension of k (that is, it is also finitely generated as a vector space)

evaluate each gate. The function is now defined as a "circuit" over a finite field, as opposed to the binary circuits used for Yao. Such a circuit is called

of given prime exponent (essentially the same as vector spaces over a finite field) and divisible torsion-free abelian groups (essentially the same as vector

{\displaystyle g\geq 1} . The general formula of Hyperelliptic curve over a finite field K {\displaystyle K} is given by C : y 2 + h ( x ) y = f ( x ) ∈ K [ x

S256 (SHA-2 256-bit) Key Agreement: DH3k (Finite Field Diffie-Hellman with 3072-bit Prime) DH2k (Finite Field Diffie-Hellman with 2048-bit Prime) Prsh

by Ezra Brown and Richard K. Guy, 2020, ISBN 978-1-4704-5279-7 The Finite Field Distance Problem, by David J. Covert, 2021, ISBN 978-1-4704-6031-0 Carus

the degree of the polynomial alone. When considering varieties over a finite field, or a field of characteristic p {\displaystyle p} , more powerful tools

of the polynomial ring F[x]. An element y = f(x) of F(x) determines a finite field extension [F(x) : F(y)]. This extension is generally not Galois but has

Weil that G {\displaystyle G} -bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group G {\displaystyle

Linear PCP is a PCP in which the proof is a vector of elements of a finite field π ∈ F n {\displaystyle \pi \in \mathbb {F} ^{n}} , and such that the

solve the following problem: given a list of invertible matrices over a finite field, determine the composition series of the group. Implementations of algorithms

n!} , and the number of elements of the general linear group over a finite field is (up to some factor) the q-factorial [ n ] q ! {\displaystyle [n]_{q}

miniaturization is achieved by reducing the value of the Planck constant within a finite field, which it claims is the only conceivable way to do it. However, in reducing

→ F n {\displaystyle F^{n}\to F^{n}} where F {\displaystyle F} is a finite field or the algebraic closure of such a field. A second application of the

finite field Fq, or more generally over any field, is a sequence of elements of that field satisfying the EDS recursion. An EDS over a finite field is

 331–343. doi:10.1007/978-1-4684-5386-7_18 – via Springer Link. The finite-field model of the photon is both a particle and a wave, and hence we refer

and there are infinitely many smooth hyperplane sections in X. Over a finite field, the above open subset may not contain rational points and in general

non-zero characteristic p has p-cohomological dimension at most 1. Every finite field has absolute Galois group isomorphic to Z ^ {\displaystyle {\hat {\mathbb

constant abelian varieties over function fields in one variable over a finite field. He also gave the first examples of nonzero abelian varieties with finite

are also André planes. Let F = G F ( q ) {\displaystyle F=GF(q)} be a finite field, and let K = G F ( q n ) {\displaystyle K=GF(q^{n})} be a degree n {\displaystyle

There is a quadratic residue code of length p {\displaystyle p} over the finite field G F ( l ) {\displaystyle GF(l)} whenever p {\displaystyle p} and l {\displaystyle

abelian group scheme. If G is the discrete group with n elements over the finite field Fp of order p, then the Frobenius homomorphism F is the identity homomorphism

system of linear equations admits infinitively many solutions, if K is a finite field, the number of solutions is finite, namely | K | n − r a n k ( A ) {\displaystyle

from a smooth projective surface onto a smooth projective curve over a finite field. Suppose that the generic fiber F of f, which is a curve over the function

modulo p should also have a full set of algebraic solutions, over the finite field with p elements. Grothendieck's conjecture is that these necessary conditions

discussion below considers the case where the matrix entries are drawn from a finite field Fp for some prime p, and where all arithmetic is performed in that field

with bona fide Artin representations. These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic

are not linear; indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero

replaced by the Dickson invariant. The projective orthogonal group over a finite field is used in the construction of a family of finite simple groups of Lie

described in other terms as group UT(3,p) of unitriangular matrices over finite field with p elements, also called the Heisenberg group mod p. For p = 2, both

spanned by (001111), (111100), and (0101ωω), where the elements of the finite field F4 are 0, 1, ω, ω. The group PGL3(F4) acts on the 2-dimensional projective

the Galois group, π is the Frobenius map minus the identity, and C the finite field of order p. Taking A to be a ring of truncated Witt vectors gives Witt's

largest RSA key publicly known to be cracked is RSA-250 with 829 bits. The Finite Field Diffie-Hellman algorithm has roughly the same key strength as RSA for

comprising a vector space of dimension n {\displaystyle n} over the finite field of p {\displaystyle p} elements F p {\displaystyle \mathbb {F} _{p}}

1007/3-540-49162-7_12 "Reducing elliptic curve logarithms to logarithms in a finite field" (with T. Okamoto and S. Vanstone), IEEE Transactions on Information

algorithm for efficiently computing the roots of a polynomial over a finite field. In 1977, Robert M. Solovay and Volker Strassen discovered a polynomial-time

Comparable Algorithm Strengths Security Bits Symmetric Key Finite Field/Discrete Logarithm (DSA, DH, MQV) Integer Factorization (RSA) Elliptic Curve (ECDSA

the Galois group, π is the Frobenius map minus the identity, and C the finite field of order p. Taking A to be a ring of truncated Witt vectors gives Witt's

field has Tsen rank zero if and only if it is algebraically closed. A finite field has Tsen rank 1: this is the Chevalley–Warning theorem. If F is algebraically

Nationality Australian, UK Known for Electron transfer, molecular electronics, finite-field response Awards Fellow of the Royal Society (FRS) (1988) Centenary Medal

x n ) {\displaystyle f(x_{1},\dots ,x_{n})} is a polynomial over a finite field of size q with q > deg(ƒ) + 1. Then ƒ is randomly testable of order deg(ƒ)

of the problem, the n points lie in a projective plane defined over a finite field.(Padmanabhan & Shukla 2020). The Handbook of Combinatorics, edited by

fixed-point theorem. Let X {\displaystyle X} be a variety defined over the finite field k {\displaystyle k} with q {\displaystyle q} elements and let X ¯ {\displaystyle

which ensures forward secrecy by leaving ephemeral Diffie–Hellman (finite field and elliptic curve variants) as the only remaining key exchange mechanism

angles of a triangle is at least π. This theorem states that if E/F is a finite field extension, and E is Pythagorean, then so is F. As a consequence, no algebraic

S3 which may also be considered as the general linear group over the finite field with two elements, S3 ≅GL(2,2)). When one quotients Spin(8) by one central

Gundy & Silverstein 1971). In this example, Ω = [0, 1] and Σn is the finite field generated by the dyadic partition of [0, 1] into 2n intervals of length

Decentralized Power Supply Digital Audio Broadcasting Digital Signal Processing Finite Field Calculations Fuzzy Technology Gerontotechnology Integrated Supply Chain

is n!, and the number of elements of the general linear group over a finite field is related to the q-factorial [ n ] q ! {\displaystyle [n]_{q}!} ; thus

occur in the context of multiplication by p on an elliptic curve over a finite field of characteristic p. If the characteristic of a field F is a (non-zero)

construct a Cartan subalgebra is by means of a regular element. Over a finite field, the question of the existence is still open.[citation needed] For a

Huang, Ming-Deh (1992). Primality testing and Abelian varieties over finite field. Lecture notes in mathematics. Vol. 1512. Springer-Verlag. ISBN 3-540-55308-8

"Substitution automata, functional equations and "functions algebraic over a finite field"". Papers in algebra, analysis and statistics (Hobart, 1981). Contemporary

multi-precision integers; prime number generation and verification; finite field arithmetic, including GF(p) and GF(2n); elliptical curves; and polynomial

quotient of a principal ring, is itself a principal ring. 6. Let k be a finite field and put A = k [ x , y ] {\displaystyle A=k[x,y]} , m = ⟨ x , y ⟩ {\displaystyle

is generated over K by a single element, t, as a field. If E/F is a finite field extension then it follows from the definitions that E is a finitely generated

Williamson's January 1974 internal GCHQ note "Non-Secret Encryption Using a Finite Field" (A couple of typos in this pdf: Extended Euclidean Algorithm modulus

be NP-hard). More formally: Let F {\displaystyle \mathbf {F} } be a finite field, E {\displaystyle E} be an elliptic curve with Weierstrass equation having

JSTARS-2014-00702.R1, in press. Simulating images of passive sensors with finite field of view by coupling 3-D radiative transfer model and sensor perspective

to break public-key cryptography schemes, such as The RSA scheme The Finite Field Diffie-Hellman key exchange The Elliptic Curve Diffie-Hellman key exchange

q)} be the projective space of dimension n {\displaystyle n} over the finite field F q {\displaystyle \mathbb {F} _{q}} and let f {\displaystyle f} be a

above procedure, the derivative of the KS matrix is obtained using the finite field, and then the density matrix derivative is obtained by a single-step

the rationality of the zeta function of an algebraic variety over a finite field and the theory of p-adic differential equations) and contains numerous

\end{aligned}}} where the first equality uses the Binomial Theorem in a finite field, which is ( x + y ) M p ≡ x M p + y M p ( mod M p ) {\displaystyle (x+y)^{M_{p}}\equiv

{\displaystyle \mathbb {R} ,} Q p , {\displaystyle \mathbb {Q} _{p},} and their finite field extensions, including C , {\displaystyle \mathbb {C} ,} are called local

sets of Fq-rational points of an affine algebraic group, where Fq is a finite field. If an author requires the base field of an affine variety to be algebraically

PGL(2, 7) acts on the eight points in the projective line over the finite field GF(7), while PGL(2, 4), which is isomorphic to the alternating group

Geometries, Oxford University Press 1991 Projective geometry over a finite field and Generalized Polygons in F. Buekenhout Handbook of incidence geometry

M. (2012), Convolution and Equidistribution : Sato-Tate Theorems for Finite Field Mellin Transformations, Princeton University Press, p. 146, ISBN 9780691153315

statistical tests consist of all multivariate linear functions over some finite field F {\displaystyle \mathbb {F} } , one speaks of epsilon-biased generators

example of non-classical curves. These are projective curves defined over finite field G F ( q 2 ) {\displaystyle GF(q^{2})} by equation y q + y = x q + 1 {\displaystyle

George Lusztig (The mod-2 cohomology of the orthogonal groups over a finite field). In 1975 he was a lecturer (Professore Incaricato) at the University

is a perfect field – for example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable

on the representation theory of the group of infinite matrices over a finite field". Journal of Mathematical Sciences. 147 (6): 7129–7144. arXiv:0705.3605

Steinberg representation of G(Fq) of dimension qN. (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation