Cleaned up and using a test harness (i.e. some internal test data and print to stdout) you code looks like this:

#!/usr/bin/perl
use strict;
use warnings;

my $file1 = <<FILE;
chirac prime paris
chirac prime jacques
chirac prime president 
chirac paris france
chirac paris french
FILE
my $file2 = <<FILE;
chirac presidential migration 
chirac presidential paris 
chirac prime president
chirac presidential 007
chirac paris migration 
chirac paris french
FILE

#open my $inA, '<', $ARGV[0] or die "Can;t open $ARGV[0]: $!\n";
#open my $inB, '<', $ARGV[1] or die "Can;t open $ARGV[0]: $!\n";
open my $inA, '<', \$file1;
open my $inB, '<', \$file2;

#print "bonjour\n";
#print "choose the output file name\n";
#
#chomp(my $fic2 = <STDIN>);
#open my $outFile, '>', $fic2 or die "Can't create $fic2: $!\n";

my @aLines;

while (my $ligne = <$inA>) {
    chomp $ligne;
    push @aLines, lc($ligne);
}

while (my $che = <$inB>) {
    chomp $che;

    my @bWords = split(/\s/, $che);

    foreach my $kh (@aLines) {
        my @aWords = split(/\s/, $kh);
        my $total = 0;
        
        for my $bWord (@bWords) {
            my $matched;
            
            for my $aWord (@aWords) {
                $matched = $bWord eq $aWord;
                last if $matched;
            }

            $total++ if $matched;
        }

        #print the retrieved line
        #print $outFile "$.: $kh\n";
        print "$.: $kh\n";
    }
}
[download]

Prints:

1: chirac prime paris
1: chirac prime jacques
1: chirac prime president 
1: chirac paris france
1: chirac paris french
2: chirac prime paris
2: chirac prime jacques
2: chirac prime president 
2: chirac paris france
2: chirac paris french
3: chirac prime paris
3: chirac prime jacques
3: chirac prime president 
3: chirac paris france
3: chirac paris french
4: chirac prime paris
4: chirac prime jacques
4: chirac prime president 
4: chirac paris france
4: chirac paris french
5: chirac prime paris
5: chirac prime jacques
5: chirac prime president 
5: chirac paris france
5: chirac paris french
6: chirac prime paris
6: chirac prime jacques
6: chirac prime president 
6: chirac paris france
6: chirac paris french
[download]

which doesn't work as advertised. Making this code generate the wrong answer more quickly is possible, but probably not what you actually want to do! Maybe you should more fully describe what it is you want to achieve? Are you looking for matching lines (in which case the word matching stuff and nested loops is bogus), or do you want to match lines that have some minimum number of matching words, or something else? We can't tell unless you tell us. Tell us why you are doing this and we may be able to make better guesses!