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Public-Key Parameters
Published in Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, 2018
Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone
The arithmetic in finite fields can usually be implemented more efficiently if the irreducible polynomial chosen has few non-zero terms. Irreducible trinomials, i.e., irreducible polynomials having exactly three non-zero terms, are considered in §4.5.2. Primitive polynomials, i.e., irreducible polynomials f (x) of degree m in ℤp[x] for which x is a generator of F*pm, the multiplicative group of the finite field F2m = ℤp[x]/f(x)) (Definition 2.228), are the topic of§4.5.3. Primitive polynomials are also used in the generation of linear feed-back shift register sequences having the maximum possible period (Fact 6.12).
General linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
In general it is not always true that all field elements are powers of X. If it is the case, one speaks of a primitive polynomial and calls the corresponding field elements primitive elements. Every finite field can be described by a primitive polynomial.
A Component-Position Model, Analysis and Design for Order-of-Addition Experiments
Published in Technometrics, 2021
Jian-Feng Yang, Fasheng Sun, Hongquan Xu
We need the concept of Galois fields in order to describe our next construction method for component orthogonal arrays. A Galois field (or finite field) is a field that contains a finite number of elements, on which the operations of multiplication, addition, subtraction and division are defined and satisfy the rules of arithmetic. The number of elements, called the order of a Galois field, must be a prime power. For any prime p and positive integer u, there is a unique Galois field of order pu up to isomorphism. Let be a Galois field. For a prime number p, is simply the ring of integers modulo p. For u > 1, . In other words, the elements of are polynomials with degree less than u and coefficients from GF(p). The addition of is ordinary polynomial addition with coefficients modulo p, and the multiplication is ordinary polynomial multiplication and then modulo a given irreducible polynomial of degree u. An irreducible polynomial (or primitive polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials. For a selected primitive polynomial over GF(p), see Table A.19 of Hedayat et al. (1999).