I'm interested in the Massey-Omura cryptosystem at the moment and I need a primitive polynomial for GF ( 2 ^ { 256 } ) because I want to use a 256-bit long key.
22.
For example, given the primitive polynomial, we start with a user-specified 10-bit seed occupying bit positions 1 through 10, starting from the least significant bit . ( The seed need not randomly be chosen, but it can be ).
23.
In other words, integer GCD computation allows to reduce the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and to reduce the factorization over the integers to the factorization of an integer and a primitive polynomial.
24.
In other words, integer GCD computation allows to reduce the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and to reduce the factorization over the integers to the factorization of an integer and a primitive polynomial.
25.
Over GF ( 2 ), 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 ( it has " 1 " as a root ).
26.
Over GF ( 2 ), 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 ( it has " 1 " as a root ).
27.
How do you prove that x ( and any element in GF ( p ) [ x ] / f ( x ) other than the additive identity ) generates the multiplicative group of GF ( p ) [ x ] / f ( x ) if f ( x ) is a primitive polynomial?
28.
:Well, x generates the field because it is a root of a primitive polynomial . . . However, for example, if the multiplicative group of the field has size 8 and z is a generator, then z 4 cannot possibly be a generator, because z 4 will have order 2.
29.
Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma; this does not alter the set of rational roots and only strengthens the divisibility conditions.
30.
The combination of these factors means that good CRC polynomials are often primitive polynomials ( which have the best 2-bit error detection ) or primitive polynomials of degree n-1, multiplied by x + 1 ( which detects all odd numbers of bit errors, and has half the two-bit error detection ability of a primitive polynomial of degree n ).
