## Group Priebel

Group Description ```Quadratic sieve matrices from David Priebel, Tenn. Tech. Univ. Each column in the matrix corresponds to a number in the factor base less than some bound B. Each row corresponds to a smooth number (able to be completely factored over the factor base). Each value in a row binary vector corresponds to the exponent of the factor base mod 2. For example: factor base: 2 7 23 smooth numbers: 46, 28, 322 2^1 * 23^1 = 46 2^2 * 7^1 = 28 2^1 * 7^1 * 23^1 = 322 Matrix: 101 010 111 A solution to the matrix is considered to be a set of rows which when combined in GF2 produce a null vector. Thus, if you multiply each of the smooth numbers which correspond to that particular set of rows you will get a number with only even exponents, making it a perfect square. In the above example you can see that combining the 3 vectors results in a null vector and, indeed, it is a perfect square: 644^2. Problem.A: A GF(2) matrix constructed from the exponents of the factorization of the smooth numbers over the factor base. A solution of this matrix is a kernel (nullspace). Such a solution has a 1/2 chance of being a factorization of N. Problem.aux.factor_base: The factor base used. factor_base(j) corresponds to column j of the matrix. Note that a given column may or may not have nonzero elements in the matrix. Problem.aux.smooth_number: The smooth numbers, smooth over the factor base. smooth_number(i) corresponds to row i of the matrix. Problem.aux.solution: A sample solution to the matrix. Combine, in GF(2) the rows with these indicies to produce a solution to the matrix with the additional property that it factors N (a matrix solution only has 1/2 probability of factoring N). Problem specific information: n = 787911249838484617926390474950774839909 (130-bits) Divisor: 23850290715477455017 (probably prime) Divisor: 33035708421246870877 (probably prime) n = 27393004579711727757848513391018843988362569 (145-bits) Divisor: 4762476283061573160587 (probably prime) Divisor: 5751840629031342254587 (probably prime) n = 3408489886335277144344023699527218196631767672957 (162-bits) Divisor: 1816046478796474796528999 (probably prime) Divisor: 1876873706775468629074043 (probably prime) n = 73363722971930954428433124842779099222294372095286387 (176-bits) Divisor: 236037985789994529800050193 (probably prime) Divisor: 310813205452462332837525059 (probably prime) n = 4232562527578032866150921497850842593296760823796443077101 (192-bits) Divisor: 83135929635332984850508004533 (probably prime) Divisor: 50911351399373573113182167897 (probably prime) n = 239380926372595066574100671394554319947805305453767699448870971 (208-bits) Divisor: 12216681953629826483019726942851 (probably prime) Divisor: 19594594283554225566102143686121 (probably prime)```
Displaying all 6 collection matrices
Id Name Group Rows Cols Nonzeros
2250 130bit Priebel 584 575 6,120
2251 145bit Priebel 1,002 993 11,315
2252 162bit Priebel 3,606 3,597 37,118
2253 176bit Priebel 7,441 7,431 82,270
2254 192bit Priebel 13,691 13,682 154,303
2255 208bit Priebel 24,430 24,421 299,756