I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows.
How does one go about finding such a linearly dependent set? Why is it the case that such a set will have entries that sum to an even number in each column?
More detail: We pick a factor base of $h$ primes. Then, we find numbers that can be factored into multiples of the factor base, and write them as vectors in $\mathbb{F}_2^h$. Finally, we find a collection of these vectors that is linearly dependent over $\mathbb{F}_2$. It is the last part that I am confused about.
1) Do row reduction.
2) The linearly dependent set is $ \sum \alpha_i v_i = 0 $, where $ \alpha_i = 0, 1$ and not all 0. Notice that each coordinate sums to 0. What does this imply?