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Re: Predict Random Numbers

by no_slogan (Deacon)
on Oct 20, 2002 at 05:54 UTC ( #206628=note: print w/ replies, xml ) Need Help??


in reply to Predict Random Numbers

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 ($e, ($e-1)&0x1ff, ($e-2)&0x1ff) { push @{$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) brute-force 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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