random32: upgrade taus88 generator to taus113 from errata paper

Since we use prandom*() functions quite often in networking code
i.e. in UDP port selection, netfilter code, etc, upgrade the PRNG
from Pierre L'Ecuyer's original paper "Maximally Equidistributed
Combined Tausworthe Generators", Mathematics of Computation, 65,
213 (1996), 203--213 to the version published in his errata paper [1].

The Tausworthe generator is a maximally-equidistributed generator,
that is fast and has good statistical properties [1].

The version presented there upgrades the 3 state LFSR to a 4 state
LFSR with increased periodicity from about 2^88 to 2^113. The
algorithm is presented in [1] by the very same author who also
designed the original algorithm in [2].

Also, by increasing the state, we make it a bit harder for attackers
to "guess" the PRNGs internal state. See also discussion in [3].

Now, as we use this sort of weak initialization discussed in [3]
only between core_initcall() until late_initcall() time [*] for
prandom32*() users, namely in prandom_init(), it is less relevant
from late_initcall() onwards as we overwrite seeds through
prandom_reseed() anyways with a seed source of higher entropy, that
is, get_random_bytes(). In other words, a exhaustive keysearch of
96 bit would be needed. Now, with the help of this patch, this
state-search increases further to 128 bit. Initialization needs
to make sure that s1 > 1, s2 > 7, s3 > 15, s4 > 127.

taus88 and taus113 algorithm is also part of GSL. I added a test
case in the next patch to verify internal behaviour of this patch
with GSL and ran tests with the dieharder 3.31.1 RNG test suite:

$ dieharder -g 052 -a -m 10 -s 1 -S 4137730333 #taus88
$ dieharder -g 054 -a -m 10 -s 1 -S 4137730333 #taus113

With this seed configuration, in order to compare both, we get
the following differences:

algorithm                 taus88           taus113
rands/second [**]         1.61e+08         1.37e+08
sts_serial(4, 1st run)    WEAK             PASSED
sts_serial(9, 2nd run)    WEAK             PASSED
rgb_lagged_sum(31)        WEAK             PASSED

We took out diehard_sums test as according to the authors it is
considered broken and unusable [4]. Despite that and the slight
decrease in performance (which is acceptable), taus113 here passes
all 113 tests (only rgb_minimum_distance_5 in WEAK, the rest PASSED).
In general, taus/taus113 is considered "very good" by the authors
of dieharder [5].

The papers [1][2] states a single warm-up step is sufficient by
running quicktaus once on each state to ensure proper initialization
of ~s_{0}:

Our selection of (s) according to Table 1 of [1] row 1 holds the
condition L - k <= r - s, that is,

  (32 32 32 32) - (31 29 28 25) <= (25 27 15 22) - (18 2 7 13)

with r = k - q and q = (6 2 13 3) as also stated by the paper.
So according to [2] we are safe with one round of quicktaus for
initialization. However we decided to include the warm-up phase
of the PRNG as done in GSL in every case as a safety net. We also
use the warm up phase to make the output of the RNG easier to
verify by the GSL output.

In prandom_init(), we also mix random_get_entropy() into it, just
like drivers/char/random.c does it, jiffies ^ random_get_entropy().
random-get_entropy() is get_cycles(). xor is entropy preserving so
it is fine if it is not implemented by some architectures.

Note, this PRNG is *not* used for cryptography in the kernel, but
rather as a fast PRNG for various randomizations i.e. in the
networking code, or elsewhere for debugging purposes, for example.

[*]: In order to generate some "sort of pseduo-randomness", since
get_random_bytes() is not yet available for us, we use jiffies and
initialize states s1 - s3 with a simple linear congruential generator
(LCG), that is x <- x * 69069; and derive s2, s3, from the 32bit
initialization from s1. So the above quote from [3] accounts only
for the time from core to late initcall, not afterwards.
[**] Single threaded run on MacBook Air w/ Intel Core i5-3317U

 [1] http://www.iro.umontreal.ca/~lecuyer/myftp/papers/tausme2.ps
 [2] http://www.iro.umontreal.ca/~lecuyer/myftp/papers/tausme.ps
 [3] http://thread.gmane.org/gmane.comp.encryption.general/12103/
 [4] http://code.google.com/p/dieharder/source/browse/trunk/libdieharder/diehard_sums.c?spec=svn490&r=490#20
 [5] http://www.phy.duke.edu/~rgb/General/dieharder.php

Joint work with Hannes Frederic Sowa.

Cc: Florian Weimer <fweimer@redhat.com>
Cc: Theodore Ts'o <tytso@mit.edu>
Signed-off-by: Daniel Borkmann <dborkman@redhat.com>
Signed-off-by: Hannes Frederic Sowa <hannes@stressinduktion.org>
Signed-off-by: David S. Miller <davem@davemloft.net>

include/linux/random.h

@@ -32,10 +32,10 @@ void prandom_seed(u32 seed);
 void prandom_reseed_late(void);
 
 struct rnd_state {
-	__u32 s1, s2, s3;
+	__u32 s1, s2, s3, s4;
 };
 
-u32 prandom_u32_state(struct rnd_state *);
+u32 prandom_u32_state(struct rnd_state *state);
 void prandom_bytes_state(struct rnd_state *state, void *buf, int nbytes);
 
 /*
@@ -55,9 +55,10 @@ static inline void prandom_seed_state(struct rnd_state *state, u64 seed)
 {
 	u32 i = (seed >> 32) ^ (seed << 10) ^ seed;
 
-	state->s1 = __seed(i, 2);
-	state->s2 = __seed(i, 8);
-	state->s3 = __seed(i, 16);
+	state->s1 = __seed(i,   2U);
+	state->s2 = __seed(i,   8U);
+	state->s3 = __seed(i,  16U);
+	state->s4 = __seed(i, 128U);
 }
 
 #ifdef CONFIG_ARCH_RANDOM

lib/random32.c

@@ -2,19 +2,19 @@
   This is a maximally equidistributed combined Tausworthe generator
   based on code from GNU Scientific Library 1.5 (30 Jun 2004)
 
-   x_n = (s1_n ^ s2_n ^ s3_n)
+  lfsr113 version:
 
-   s1_{n+1} = (((s1_n & 4294967294) <<12) ^ (((s1_n <<13) ^ s1_n) >>19))
-   s2_{n+1} = (((s2_n & 4294967288) << 4) ^ (((s2_n << 2) ^ s2_n) >>25))
-   s3_{n+1} = (((s3_n & 4294967280) <<17) ^ (((s3_n << 3) ^ s3_n) >>11))
+   x_n = (s1_n ^ s2_n ^ s3_n ^ s4_n)
 
-   The period of this generator is about 2^88.
+   s1_{n+1} = (((s1_n & 4294967294) << 18) ^ (((s1_n <<  6) ^ s1_n) >> 13))
+   s2_{n+1} = (((s2_n & 4294967288) <<  2) ^ (((s2_n <<  2) ^ s2_n) >> 27))
+   s3_{n+1} = (((s3_n & 4294967280) <<  7) ^ (((s3_n << 13) ^ s3_n) >> 21))
+   s4_{n+1} = (((s4_n & 4294967168) << 13) ^ (((s4_n <<  3) ^ s4_n) >> 12))
 
-   From: P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
-   Generators", Mathematics of Computation, 65, 213 (1996), 203--213.
-
-   This is available on the net from L'Ecuyer's home page,
+   The period of this generator is about 2^113 (see erratum paper).
 
+   From: P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
+   Generators", Mathematics of Computation, 65, 213 (1996), 203--213:
    http://www.iro.umontreal.ca/~lecuyer/myftp/papers/tausme.ps
    ftp://ftp.iro.umontreal.ca/pub/simulation/lecuyer/papers/tausme.ps
 
@@ -29,7 +29,7 @@
         that paper.)
 
    This affects the seeding procedure by imposing the requirement
-   s1 > 1, s2 > 7, s3 > 15.
+   s1 > 1, s2 > 7, s3 > 15, s4 > 127.
 
 */
 
@@ -52,11 +52,12 @@ u32 prandom_u32_state(struct rnd_state *state)
 {
 #define TAUSWORTHE(s,a,b,c,d) ((s&c)<<d) ^ (((s <<a) ^ s)>>b)
 
-	state->s1 = TAUSWORTHE(state->s1, 13, 19, 4294967294UL, 12);
-	state->s2 = TAUSWORTHE(state->s2, 2, 25, 4294967288UL, 4);
-	state->s3 = TAUSWORTHE(state->s3, 3, 11, 4294967280UL, 17);
+	state->s1 = TAUSWORTHE(state->s1,  6U, 13U, 4294967294U, 18U);
+	state->s2 = TAUSWORTHE(state->s2,  2U, 27U, 4294967288U,  2U);
+	state->s3 = TAUSWORTHE(state->s3, 13U, 21U, 4294967280U,  7U);
+	state->s4 = TAUSWORTHE(state->s4,  3U, 12U, 4294967168U, 13U);
 
-	return (state->s1 ^ state->s2 ^ state->s3);
+	return (state->s1 ^ state->s2 ^ state->s3 ^ state->s4);
 }
 EXPORT_SYMBOL(prandom_u32_state);
 
@@ -126,6 +127,21 @@ void prandom_bytes(void *buf, int bytes)
 }
 EXPORT_SYMBOL(prandom_bytes);
 
+static void prandom_warmup(struct rnd_state *state)
+{
+	/* Calling RNG ten times to satify recurrence condition */
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+	prandom_u32_state(state);
+}
+
 /**
  *	prandom_seed - add entropy to pseudo random number generator
  *	@seed: seed value
@@ -141,8 +157,9 @@ void prandom_seed(u32 entropy)
 	 */
 	for_each_possible_cpu (i) {
 		struct rnd_state *state = &per_cpu(net_rand_state, i);
-		state->s1 = __seed(state->s1 ^ entropy, 2);
-		prandom_u32_state(state);
+
+		state->s1 = __seed(state->s1 ^ entropy, 2U);
+		prandom_warmup(state);
 	}
 }
 EXPORT_SYMBOL(prandom_seed);
@@ -158,18 +175,13 @@ static int __init prandom_init(void)
 	for_each_possible_cpu(i) {
 		struct rnd_state *state = &per_cpu(net_rand_state,i);
 
-#define LCG(x)	((x) * 69069)	/* super-duper LCG */
-		state->s1 = __seed(LCG(i + jiffies), 2);
-		state->s2 = __seed(LCG(state->s1), 8);
-		state->s3 = __seed(LCG(state->s2), 16);
-
-		/* "warm it up" */
-		prandom_u32_state(state);
-		prandom_u32_state(state);
-		prandom_u32_state(state);
-		prandom_u32_state(state);
-		prandom_u32_state(state);
-		prandom_u32_state(state);
+#define LCG(x)	((x) * 69069U)	/* super-duper LCG */
+		state->s1 = __seed(LCG((i + jiffies) ^ random_get_entropy()), 2U);
+		state->s2 = __seed(LCG(state->s1),   8U);
+		state->s3 = __seed(LCG(state->s2),  16U);
+		state->s4 = __seed(LCG(state->s3), 128U);
+
+		prandom_warmup(state);
 	}
 	return 0;
 }
@@ -215,15 +227,15 @@ static void __prandom_reseed(bool late)
 
 	for_each_possible_cpu(i) {
 		struct rnd_state *state = &per_cpu(net_rand_state,i);
-		u32 seeds[3];
+		u32 seeds[4];
 
 		get_random_bytes(&seeds, sizeof(seeds));
-		state->s1 = __seed(seeds[0], 2);
-		state->s2 = __seed(seeds[1], 8);
-		state->s3 = __seed(seeds[2], 16);
+		state->s1 = __seed(seeds[0],   2U);
+		state->s2 = __seed(seeds[1],   8U);
+		state->s3 = __seed(seeds[2],  16U);
+		state->s4 = __seed(seeds[3], 128U);
 
-		/* mix it in */
-		prandom_u32_state(state);
+		prandom_warmup(state);
 	}
 out:
 	spin_unlock_irqrestore(&lock, flags);