Explain what does while(k) mean

The while is attached to the do at the top of the block, and it's one of the three types of loops that C supports: for, while, and do ... while
The do ... while statement is very similar to the while loop, except that the loop body is always executed once, because the loop / don't loop condition is only checked after the body, not before.

The condition here is just k and because C doesn't have boolean operators, a value of zero is always taken as "false" and any non-zero value is always taken as "true".
so your code:

do
    {
    ...
    } while (k);

is the same as:

do
    {
    ...
    } while (k != 0);