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Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2 , P = ( x + 1) 4 . On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have
It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54 / 6 irreducible monic polynomials of degree 6. This may be verified by factoring X 64 − X over GF(2). The elements of GF(64) are primitive n th roots of unity for some n dividing 63.
Given an n-bit message m 0,...,m n-1, we view it as a polynomial of degree n-1 over the finite field GF(2). = + + … +We then pick a random irreducible polynomial of degree k over GF(2), and we define the fingerprint of the message m to be the remainder () after division of () by () over GF(2) which can be viewed as a polynomial of degree k − 1 or as a k-bit number.
In general terms, ideal lattices are lattices corresponding to ideals in rings of the form [] / for some irreducible polynomial of degree . [1] All of the definitions of ideal lattices from prior work are instances of the following general notion: let be a ring whose additive group is isomorphic to (i.e., it is a free -module of rank), and let be an additive isomorphism mapping to some lattice ...
(A polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients. [note 2]) A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive ...
Irreducibility (mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
The LFSR is maximal-length if and only if the corresponding feedback polynomial is primitive over the Galois field GF(2). [3] [4] This means that the following conditions are necessary (but not sufficient): The number of taps is even. The set of taps is setwise co-prime; i.e., there must be no divisor other than 1 common to all taps.