## Priebel/176bit

Quadratic sieve; factoring a 176bit number. D. Priebel, Tenn. Tech Univ
Name 176bit Priebel 2253 7,441 7,431 82,270 82,270 Combinatorial Problem No 2009 D. Priebel T. Davis
Structural Rank 7,110 false 893 282 0 0% 0% no no binary
SVD Statistics
Matrix Norm 9.619242e+01
Minimum Singular Value 1.319478e-47
Condition Number 7.290185e+48
Rank 7,110
sprank(A)-rank(A) 0
Null Space Dimension 321
Full Numerical Rank? no
Download ```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 = 73363722971930954428433124842779099222294372095286387 (176-bits) passes primality test, n is composite, continuing... 1) Initial bound: 180000, pi(180000) estimate: 14875, largest found: 162359 (actual bound) 2) Number of quadratic residues estimate: 9918, actual number found: 7431 3) Modular square roots found: 14862(2x residues) 4) Constructing smooth number list [sieving] (can take a while)... Sieving for: 7441 5. Constructing a matrix of size: 7441x7432 Set a total of 82270 exponents, with 3725 negatives Matrix solution found with: 2983 combinations Divisor: 236037985789994529800050193 (probably prime) Divisor: 310813205452462332837525059 (probably prime)```