If you liked that, here's another way to discover the internal state of
a linear congruential pseudorandom number generator. It's only effective
when the internal state of the PRNG is small enough. As long as that's true,
it's fairly fast, since it doesn't need to use any BigInts.
This code is set up to use the same constants as the rand() function in
ActiveState Perl for Windows. It can be modified for other generators, but
there are some magic numbers that need to be tweaked. It takes four output
bytes from the PRNG, and comes up with a small number of possible seeds
(often just one). If that bothers you, give it a fifth byte to work with,
and that will narrow it down to one almost every time.
It works by looking at the first two output bytes from the PRNG, and
checking each of the roughly 2**15 seed values that would produce them.
This is one of the steps in the
cryptanalysis of the PKZIP cipher
by Biham and Kocher, which is implemented (though somewhat differently) in
pkcrack.
use strict;
use warnings;
use integer;
# PRNG parameters
my $m = 0x343fd;
my $b = 0x269ec3;
my $m1 = 0x39b33155; # multiplicative inverse of $m mod 2**31
# build the lookup tables
my (@diff_tbl, @prod_tbl);
for my $x (0 .. 0x2bc) { # 0x2bc is a magic number... see below
my $p = ($x * $m1) & 0x7fffffff;
my $e = ($p >> 22) & 0x1ff;
for ($e, ($e1)&0x1ff, ($e2)&0x1ff) {
push @{$diff_tbl[$_]}, $x;
push @{$prod_tbl[$_]}, $p;
}
}
# "Unknown" values: internal states of the PRNG
my $X0 = int(rand 2**31);
my $X1 = ($X0 * $m + $b) & 0x7fffffff;
my $X2 = ($X1 * $m + $b) & 0x7fffffff;
my $X3 = ($X2 * $m + $b) & 0x7fffffff;
# "Known" values: a few bytes of output from the PRNG
my $x0 = $X0 & 0x7f800000;
my $x1 = $X1 & 0x7f800000;
my $x2 = $X2 & 0x7f800000;
my $x3 = $X3 & 0x7f800000;
# find the lowest nonnegative $d such that
# $x0 == ((($x1 + $d  $b) * $m1) & 0x7f800000)
my $prev = (($x1  $b) * $m1) & 0x7fffffff;
my $ent = (($x0  $prev) >> 22) & 0x1ff;
my ($d, $p);
for (0 .. $#{$prod_tbl[$ent]}) {
$p = $prev + $prod_tbl[$ent][$_];
if (($p & 0x7f800000) == $x0) { $d = $diff_tbl[$ent][$_]; last }
}
die "can't happen\n" unless defined $d;
$p &= 0x7fffff;
my $q = ($x1 + $d) * $m + $b;
# modest (about 2**15 steps) bruteforce search for the right $d
while ($d < 0x800000) {
# invariant: $p == ((($x1 + $d  $b) * $m1) & 0x7fffff)
# invariant: $q == ($x1 + $d) * $m + $b
# is this a solution?
if (($q & 0x7f800000) == $x2) {
my $r = $q * $m + $b;
if (($r & 0x7f800000) == $x3) {
printf "guessed x0 = %08x\n", (($x1 + $d  $b) * $m1) & 0x7fffff
+ff;
}
}
# find the next admissible value of $d
# there is much magic here... see below for a hint
if ($p < 0x67ceeb) { $d += 0x0c1; $p += 0x183115; $q += 0x27641b
+d }
elsif ($p >= 0x6a1b54) { $d += 0x1fc; $p = 0x6a1b54; $q += 0x67aea0
+c }
else { $d += 0x2bd; $p = 0x51ea3f; $q += 0x8f12bc
+9 }
}
printf "actual X0 = %08x\n", $X0;
# calculate the magic numbers
my $low = 0;
my $high = 0x800000;
my $x = 0;
while ($low < $high) {
$x++;
my $y = ($x * $m1) & 0x7fffffff;
my $yhi = ($y >> 23) & 0xff;
my $yneg = ($y) & 0x7fffff;
if ($yhi == 0 && $yneg > $low) {
printf "if y < 0x%x then x += 0x%x, y += 0x%x\n", $yneg, $x, $y;
$low = $yneg;
}
elsif ($yhi == 0xff && $yneg < $high) {
printf "if y >= 0x%x then x += 0x%x, y = 0x%x\n", $yneg, $x, $yne
+g;
$high = $yneg;
}
}
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