primitive polynomial sentence in Hindi
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- 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.
- 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 ).
- 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.
- 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.
- 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 ).
- 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 ).
- 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?
- :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.
- 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.
- 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 ).
- 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 ).
- 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 ).
- From Gauss'lemma it follows that is reducible in as well, and in fact can be written as the product " GH " } } of two non-constant polynomials ( in case is not primitive, one applies the lemma to the primitive polynomial ( where the integer is the content of ) to obtain a decomposition for it, and multiplies into one of the factors to obtain a decomposition for ).
- The cyclic redundancy check ( CRC ) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF ( 2 ) and dividing it by a fixed generator polynomial also over GF ( 2 ); see Mathematics of CRC . Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of for a degree " n " primitive polynomial.
- The cyclic redundancy check ( CRC ) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF ( 2 ) and dividing it by a fixed generator polynomial also over GF ( 2 ); see Mathematics of CRC . Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of for a degree " n " primitive polynomial.
- Note first that in " F " [ " X " ] \ { 0 } any class of associate elements ( whose elements are related by multiplication by nonzero elements of the field " F " ) meets the set of primitive elements in " R " [ " X " ] : starting from an arbitrary element of the class, one can first ( if necessary ) multiply by a nonzero element of " R " to enter into the subset " R " [ " X " ] ( removing denominators ), then divide by the greatest common divisor of all coefficients to obtain a primitive polynomial.
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