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About 10,000 papers for “finite fields”

Finite Fields

Series on university mathematics · 1996

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Finite fields with prime order are the most directly accessible fields using modular arithmetic. This chapter discusses order, finiteness, generators, and properties of an important group helping students in understanding the associated concepts of finite fields, polynomials, and their execution in computer science, including important theorems, methods, and algorithms. Abstract algebra emerged around the start of the 20th century under the name modern algebra. The formal definition of certain algebraic structures began to emerge in the 19th century, such as results on permutation group, algebra of structure, and classification of various mathematical objects. The formal definition of certain algebraic structures began to emerge in the 19th century, such as results on permutation group, algebra of structure, and classification of various mathematical objects. Finite fields are the field with finite elements; such cardinality is referred to as the order of the field.

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Finite fields

Robert J. McEliece · 2002

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Finite Fields

Allen C. Hibbard, Kenneth M. Levasseur · 1999

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Finite Fields

Undergraduate texts in mathematics · 2006

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Finite Fields

J. W. P. Hirschfeld · 1998

Abstract 1.1 Definitions and existence (i) A field is a set K closed under two operations+, × such that (a) (K, +) is an abelian group with identity O; (b) (Ko, ×) is an abelian group with identity 1, where Ko= K\{O); (c) x(y + z) = xy + xz, (x + y)z = xz + yz, for all .r,y, z in K. (ii) A finite field is a field with only a finite number of elements. (iii) The characteristic of a finite field K is the smallest positive integer (and hence a prime) p such that px = 0 for all x in K.

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Finite Fields

Scott A. Vanstone, Paul C. van Oorschot · 1989

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Finite Fields

Christoph Schwarzweller · Formalized Mathematics · 2024

Summary We continue the formalization of field theory in Mizar. Here we prove existence and uniqueness of finite fields by constructing the splitting field of the polynomial X (p n ) − X over the prime field of a field with characteristic p . We also define the Frobenius morphism and show that the automorphisms of a field with p n elements are exactly the powers 0, . . ., n − 1 of the Frobenius morphism, that is the automorphism group is generated by the Frobenius morphism.

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Finite Fields

1990

Cited by 1

Finite Fields

Marco Hien · Mathematics study resources · 2024

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Finite Fields

Joseph Rotman · Universitext · 1990

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Finite fields

2016

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Finite fields

I. Isaacs · Graduate studies in mathematics · 2009

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Finite Fields

Louis Rowen, Uzi Vishne · Algebra · 2025

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Finite Fields

2025

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Finite Fields

Lindsay N. Childs · Undergraduate texts in mathematics · 1979

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Finite Fields

Solomon W. Golomb, Guang Gong · Cambridge University Press eBooks · 2005

Finite fields are used in most of the known constructions of pseudorandom sequences and analysis of periods, correlations, and linear spans of linear feedback shift register (LFSR) sequences and nonlinear generated sequences. They are also important in many cryptographic primitive algorithms, such as the Diffie-Hellman key exchange, the Digital Signature Standard (DSS), the El Gamal public-key encryption, elliptic curve public-key cryptography, and LFSR (or Torus) based public-key cryptography. Finite fields and shift register sequences are also used in algebraic error-correcting codes, in code-division multiple-access (CDMA) communications, and in many other applications beyond the scope of this book. This chapter gives a description of these fields and some properties that are frequently used in sequence design and cryptography. Section 3.1 introduces definitions of algebraic structures of groups, rings and fields, and polynomials. Section 3.2 shows the construction of the finite field GF(pn). Section 3.3 presents the basic theory of finite fields. Section 3.4 discusses minimal polynomials. Section 3.5 introduces subfields, trace functions, bases, and computation of the minimal polynomials over intermediate subfields. Computation of a power of a trace function is shown in Section 3.6. And, the last section presents some counting numbers related to finite fields.

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Finite fields

Rudolf Lide, Harald Niederreiter · 1996

Cited by 2,216