Arithmetic in a field with a finite number of elements
In 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 infinite number of elements, like the field of rational numbers.
There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field.
Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael (AES) encryption algorithm, in tournament scheduling, and in the design of experiments.
and 23 Related for: Finite field arithmetic information
mathematics, finitefieldarithmetic is arithmetic in a finitefield (a field containing a finite number of elements) contrary to arithmetic in a field with an...
rules of arithmetic known as the field axioms. The number of elements of a finitefield is called its order or, sometimes, its size. A finitefield of order...
in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finitefields, p-adic fields, or...
and economic modeling. Arithmetic portal Mathematics portal Algorism Finitefieldarithmetic Outline of arithmetic Plant arithmetic Other symbols for the...
cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finitefields which underlie elliptic...
by the extended GCD algorithm (see Arithmetic of algebraic extensions). It follows that, to compute in a finitefield of non prime order, one needs to generate...
standard. The standard defines: arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros...
the early development of the study of arithmetic invariant of number fields such as the discriminant. Arithmetic groups can be thought of as a vast generalisation...
polynomial which is then evaluated at a key-dependent point H, using finitefieldarithmetic. The result is then encrypted, producing an authentication tag...
number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups. Let K be a field and let G = Gal(K)...
global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory...
V. Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers. Arithmetic geometry...
adds cryptography instructions supporting AES, SHA-1/SHA-256 and finitefieldarithmetic. AArch64 was introduced in Armv8-A and its subsequent revision...
the latter (conjecturally finite) being difficult to study. The theory of heights plays a prominent role in the arithmetic of abelian varieties. For instance...
be simply finite if it is a finite extension; this should not be confused with the fields themselves being finitefields (fields with finitely many elements)...
instruction set can be checked by testing one of the CPU feature bits. Finitefieldarithmetic AES instruction set FMA3 instruction set FMA4 instruction set AVX...
rings with prime characteristic p, an important class that includes finitefields. The endomorphism maps every element to its p-th power. In certain contexts...
a finitefield Fp is, in some sense, a generating function assembling the information of the number of points of E with values in the finitefield extensions...
valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology...
mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes...
Global Information