UPDATE: Try my original approach, but just shuffle at the very last step!

my %data;

<DATA>;
while (<DATA>) {
    my ($row, @values) = split;
    $data{$row} = sum @values;
}

say $data{$_} for shuffle keys %data;
[download]

---

Maybe try something like the following. I start by making a random array of numbers equal to the number of data rows you have. Next I read the file and build a hash. The keys are shifted from the random array and the values are the row sums. Then iterating over the hash (sorted by keys) will let you do whatever you want with the permuted row sums.

#!/usr/bin/env perl
use strict;
use warnings;
use feature 'say';
use List::Util qw(sum shuffle);

my %data;
my @row_nums = shuffle 1..5;

<DATA>;
while (<DATA>) {
    my ($old_row, @values) = split;
    my $new_row = shift @row_nums;
    $data{$new_row} = sum @values;
}

say $data{$_} for sort { $a <=> $b } keys %data;

__DATA__
Head1    Head2    Head3    Head4    Head5    Head6    Head7    Head8  
+  Head9    Head10    Head11
1   0   1   1   0   0   0   0   0   0   1
2   0   0   0   0   0   0   0   1   0   0
3   1   0   0   0   1   0   0   0   1   0
4   0   1   1   1   1   0   0   0   0   1
5   0   0   0   0   0   0   0   1   0   0
[download]

Quick question: Are you really doing this for 1 million different tables or are you doing 1 million permutations of the same table? If the latter, just read in and sum the records once in the original order. This will result in a 110,000 element array of row sums that you can just shuffle a million times.

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