## Priebel/162bit

Quadratic sieve; factoring a 162bit number. D. Priebel, Tenn. Tech Univ
Name 162bit Priebel 2252 3,606 3,597 37,118 37,118 Combinatorial Problem No 2009 D. Priebel T. Davis
Structural Rank 3,460 false 376 122 0 0% 0% no no binary
SVD Statistics
Matrix Norm 5.880771e+01
Minimum Singular Value 0
Condition Number Inf
Rank 3,460
sprank(A)-rank(A) 0
Null Space Dimension 137
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 = 3408489886335277144344023699527218196631767672957 (162-bits) passes primality test, n is composite, continuing... 1) Initial bound: 80000, pi(80000) estimate: 7086, largest found: 71549 (actual bound) 2) Number of quadratic residues estimate: 4725, actual number found: 3596 3) Modular square roots found: 7192(2x residues) 4) Constructing smooth number list [sieving] (can take a while)... Sieving for: 3606 5. Constructing a matrix of size: 3606x3597 Set a total of 37118 exponents, with 1815 negatives Matrix solution found with: 1474 combinations Divisor: 1816046478796474796528999 (probably prime) Divisor: 1876873706775468629074043 (probably prime)```