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The monic irreducible polynomial x 8 + x 4 + x 3 + x 2 + 1 over GF(2) is primitive, and all 8 roots are generators of GF(2 8). All GF(2 8) have a total of 128 generators (see Number of primitive elements), and for a primitive polynomial, 8 of them are roots of the reducing polynomial
GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z. Notations Z 2 and Z 2 {\displaystyle \mathbb {Z} _{2}} may be encountered although they can be confused with the notation of 2 -adic integers .
As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64) is the prime field GF(2). The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This ...
Over GF(3) the polynomial x 2 + 1 is irreducible but not primitive because it divides x 4 − 1: its roots generate a cyclic group of order 4, while the multiplicative group of GF(3 2) is a cyclic group of order 8. The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less ...
Both the input and output are interpreted as polynomials over GF(2). First, the input is mapped to its multiplicative inverse in GF(2 8) = GF(2) [x]/(x 8 + x 4 + x 3 + x + 1), Rijndael's finite field. Zero, as the identity, is mapped to itself. This transformation is known as the Nyberg S-box after its inventor Kaisa Nyberg. [2]
[2] One use of these instructions is to improve the speed of applications doing block cipher encryption in Galois/Counter Mode , which depends on finite field GF(2 k ) multiplication. Another application is the fast calculation of CRC values , [ 3 ] including those used to implement the LZ77 sliding window DEFLATE algorithm in zlib and pngcrush .
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If q is a prime number, the elements of GF(q) can be identified with the integers modulo q. In this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive ...