- A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 2 of degree 2 x ^ 2 + x + 1, 2 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 2 of degree 17 x ^ 17 + x ^ 3 + 1, 2 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 2 of degree 43 x ^ 43 + x ^ 6 + x ^ 4 + x ^ 3 + 1, 2 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 2 of degree 62 x ^ 62 + x ^ 6 + x ^ 5 + x ^ 3 + 1, 2 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 3 of degree 30 x ^ 30 + x + 2, 3 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 5 of degree 22 x ^ 22 + x ^ 2 + x + 3, 5 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 7 of degree 16 x ^ 16 + 2 x + 3, 7 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 17 of degree 14 x ^ 14 + x ^ 2 + 2 x + 6, 17 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 137 of degree 7 x ^ 7 + x + 3, 137 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 223 of degree 8 x ^ 8 + x + 51, 223 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 557 of degree 6 x ^ 6 + x + 5, 557 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 2 of degree 4 x ^ 4 + x + 1, 2 Confirming polynomial is primitive with a brute force check. WARNING: You may wait an impossibly long time! To abort, hit control-C in your console window to stop this program. x ^ 4 + x + 1, 2 confirmed primitive! +--------- OperationCount -------------------------------- | | Integer factorization: Table lookup + Trial division + Pollard Rho | | Number of trial divisions : 0 | Number of gcd's computed : 0 | Number of primality tests : 0 | Number of squarings: 0 | | Polynomial Testing | | Total num. degree 4 poly mod 2 : 16 | Number of possible primitive poly: 2 | Polynomials tested : 4 | Const. coeff. was primitive root : 2 | Free of linear factors : 1 | Irreducible to power >=1 : 1 | Had order r (x^r = integer) : 1 | Passed const. coeff. test : 1 | Had order m (x^m != integer) : 1 | +----------------------------------------------------- - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... Primitive polynomial modulo 3 of degree 5 x ^ 5 + 2 x + 1, 3 Confirming polynomial is primitive with a brute force check. WARNING: You may wait an impossibly long time! To abort, hit control-C in your console window to stop this program. x ^ 5 + 2 x + 1, 3 confirmed primitive! +--------- OperationCount -------------------------------- | | Integer factorization: Table lookup + Trial division + Pollard Rho | | Number of trial divisions : 0 | Number of gcd's computed : 0 | Number of primality tests : 0 | Number of squarings: 0 | | Polynomial Testing | | Total num. degree 5 poly mod 3 : 243 | Number of possible primitive poly: 22 | Polynomials tested : 8 | Const. coeff. was primitive root : 3 | Free of linear factors : 1 | Irreducible to power >=1 : 1 | Had order r (x^r = integer) : 1 | Passed const. coeff. test : 1 | Had order m (x^m != integer) : 1 | +----------------------------------------------------- - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... There are 6 primitive polynomials modulo 2 of degree 5 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 2 + 1, 2 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 3 + 1, 2 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 3 + x ^ 2 + x + 1, 2 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 4 + x ^ 2 + x + 1, 2 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 4 + x ^ 3 + x + 1, 2 Primitive polynomial modulo 2 of degree 5 x ^ 5 + x ^ 4 + x ^ 3 + x ^ 2 + 1, 2 - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Error. Polynomial has bad syntax or coefficients are out of range. [ Error in parser converting polynomial from string x 1 for p = 2 Expecting to see x^ or x or x ^ integer in sentence x 1 ] This program generates primitive polynomials of degree n modulo p. Usage: Generate a single random polynomial of degree n modulo p where p is a prime >= 2 and n is an integer >= 2 Primpoly p n Example: Primpoly 2 4 Self-check passes... Primitive polynomial modulo 2 of degree 4 x ^ 4 + x + 1, 2 Usage: Test whether a polynomial is primitive modulo p. Primpoly -t , p If you leave off the modulus p we default to p = 2 Examples: Primpoly -t x^4 + x + 1, 2 Self-check passes... x ^ 4 + x + 1, 2 is primitive! Primpoly -t x^4 + x + 1 Self-check passes... x ^ 4 + x + 1, 2 is primitive! Usage: Generate all primitive polynomial of degree n modulo p. Primpoly -a p n Example: Primpoly -a 2 4 Self-check passes... Primitive polynomial modulo 2 of degree 4 x ^ 4 + x + 1, 2 Primitive polynomial modulo 2 of degree 4 x ^ 4 + x ^ 3 + 1, 2 Usage: Same but show computation statistics. Primpoly -s p n Examples: Primpoly.exe -s 13 19 Self-check passes... Primitive polynomial modulo 13 of degree 19 x ^ 19 + 9 x + 2, 13 +--------- OperationCount -------------------------------- | | Integer factorization: Table lookup + Trial division + Pollard Rho | | Number of trial divisions : 0 | Number of gcd's computed : 9027 | Number of primality tests : 2 | Number of squarings: 9026 | | Polynomial Testing | | Total num. degree 19 poly mod 13 : 1461920290375446110677 | Number of possible primitive poly: 6411930599771980992 | Polynomials tested : 120 | Const. coeff. was primitive root : 46 | Free of linear factors : 11 | Irreducible to power >=1 : 1 | Had order r (x^r = integer) : 1 | Passed const. coeff. test : 1 | Had order m (x^m != integer) : 1 | +----------------------------------------------------- Usage: Print help message. Primpoly -h Primitive polynomials find many uses in mathematics and communications engineering: * Generation of pseudonoise (PN) sequences for spread spectrum communications and chip fault testing. * Generating Sobol sequences for high dimensional numerical integration. * Generation of CRC and Hamming codes. * Generation of Galois (finite) fields for use in decoding Reed-Solomon and BCH error correcting codes. For detailed technical information, see my web page http://seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/overview.html - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... x ^ 4 + x + 1, 2 is primitive! - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... x ^ 3 + x + 2, 3 is NOT primitive! - A Program for Computing Primitive Polynomials. Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the GNU General Public License. This is free software, and you are welcome to redistribute it under certain conditions; see the GNU General Public License for details. Self-check passes... x ^ 2 + x + 10, 32749 is primitive!