Messages in this thread Patch in this message | | | From | Theodore Ts'o <> | Subject | [PATCH, RFC 12/12] random: adjust the generator polynomials in the mixing function slightly | Date | Sun, 22 Sep 2013 16:38:58 -0400 |
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Our mixing functions were analyzed by Lacharme, Roeck, Strubel, and Videau in their paper, "The Linux Pseudorandom Number Generator Revisited" (see: http://eprint.iacr.org/2012/251.pdf).
They suggested a slight change to improve our mixing functions slightly. I also adjusted the comments to better explain what is going on, and to document why the polynomials were changed.
Signed-off-by: "Theodore Ts'o" <tytso@mit.edu> --- drivers/char/random.c | 104 ++++++++++++++++++++++++-------------------------- 1 file changed, 50 insertions(+), 54 deletions(-)
diff --git a/drivers/char/random.c b/drivers/char/random.c index 86879d1..5726ff6 100644 --- a/drivers/char/random.c +++ b/drivers/char/random.c @@ -321,23 +321,62 @@ static const int trickle_thresh = (INPUT_POOL_WORDS * 28) << ENTROPY_SHIFT; static DEFINE_PER_CPU(int, trickle_count); /* - * A pool of size .poolwords is stirred with a primitive polynomial - * of degree .poolwords over GF(2). The taps for various sizes are - * defined below. They are chosen to be evenly spaced (minimum RMS - * distance from evenly spaced; the numbers in the comments are a - * scaled squared error sum) except for the last tap, which is 1 to - * get the twisting happening as fast as possible. + * Originally, we used a primitive polynomial of degree .poolwords + * over GF(2). The taps for various sizes are defined below. They + * were chosen to be evenly spaced except for the last tap, which is 1 + * to get the twisting happening as fast as possible. + * + * For the purposes of better mixing, we use the CRC-32 polynomial as + * well to make a (modified) twisted Generalized Feedback Shift + * Reigster + * + * (See M. Matsumoto & Y. Kurita, 1992. Twisted GFSR generators. ACM + * Transactions on Modeling and Computer Simulation 2(3):179-194. + * Also see M. Matsumoto & Y. Kurita, 1994. Twisted GFSR generators + * II. ACM Transactions on Mdeling and Computer Simulation 4:254-266) + * + * Thanks to Colin Plumb for suggesting this. + * + * The mixing operation is much less sensitive than the output hash, + * where we use SHA-1. All that we want of mixing operation is that + * it be a good non-cryptographic hash; i.e. it not produce collisions + * when fed "random" data of the sort we expect to see. As long as + * the pool state differs for different inputs, we have preserved the + * input entropy and done a good job. The fact that an intelligent + * attacker can construct inputs that will produce controlled + * alterations to the pool's state is not important because we don't + * consider such inputs to contribute any randomness. The only + * property we need with respect to them is that the attacker can't + * increase his/her knowledge of the pool's state. Since all + * additions are reversible (knowing the final state and the input, + * you can reconstruct the initial state), if an attacker has any + * uncertainty about the initial state, he/she can only shuffle that + * uncertainty about, but never cause any collisions (which would + * decrease the uncertainty). + * + * Our mixing functions were analyzed by Lacharme, Roeck, Strubel, and + * Videau in their paper, "The Linux Pseudorandom Number Generator + * Revisited" (see: http://eprint.iacr.org/2012/251.pdf). In their + * paper, they point out that we are not using a true Twisted GFSR, + * since Matsumoto & Kurita used a trinomial feedback polynomial (that + * is, with only three taps, instead of the six that we are using). + * As a result, the resulting polynomial is neither primitive nor + * irreducible, and hence does not have a maximal period over + * GF(2**32). They suggest a slight change to the generator + * polynomial which improves the resulting TGFSR polynomial to be + * irreducible, which we have made here. */ - static struct poolinfo { int poolbitshift, poolwords, poolbytes, poolbits, poolfracbits; #define S(x) ilog2(x)+5, (x), (x)*4, (x)*32, (x) << (ENTROPY_SHIFT+5) int tap1, tap2, tap3, tap4, tap5; } poolinfo_table[] = { - /* x^128 + x^103 + x^76 + x^51 +x^25 + x + 1 -- 105 */ - { S(128), 103, 76, 51, 25, 1 }, - /* x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 -- 15 */ - { S(32), 26, 20, 14, 7, 1 }, + /* was: x^128 + x^103 + x^76 + x^51 +x^25 + x + 1 */ + /* x^128 + x^104 + x^76 + x^51 +x^25 + x + 1 */ + { S(128), 104, 76, 51, 25, 1 }, + /* was: x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 */ + /* x^32 + x^26 + x^19 + x^14 + x^7 + x + 1 */ + { S(32), 26, 19, 14, 7, 1 }, #if 0 /* x^2048 + x^1638 + x^1231 + x^819 + x^411 + x + 1 -- 115 */ { S(2048), 1638, 1231, 819, 411, 1 }, @@ -368,49 +407,6 @@ static struct poolinfo { }; /* - * For the purposes of better mixing, we use the CRC-32 polynomial as - * well to make a twisted Generalized Feedback Shift Reigster - * - * (See M. Matsumoto & Y. Kurita, 1992. Twisted GFSR generators. ACM - * Transactions on Modeling and Computer Simulation 2(3):179-194. - * Also see M. Matsumoto & Y. Kurita, 1994. Twisted GFSR generators - * II. ACM Transactions on Mdeling and Computer Simulation 4:254-266) - * - * Thanks to Colin Plumb for suggesting this. - * - * We have not analyzed the resultant polynomial to prove it primitive; - * in fact it almost certainly isn't. Nonetheless, the irreducible factors - * of a random large-degree polynomial over GF(2) are more than large enough - * that periodicity is not a concern. - * - * The input hash is much less sensitive than the output hash. All - * that we want of it is that it be a good non-cryptographic hash; - * i.e. it not produce collisions when fed "random" data of the sort - * we expect to see. As long as the pool state differs for different - * inputs, we have preserved the input entropy and done a good job. - * The fact that an intelligent attacker can construct inputs that - * will produce controlled alterations to the pool's state is not - * important because we don't consider such inputs to contribute any - * randomness. The only property we need with respect to them is that - * the attacker can't increase his/her knowledge of the pool's state. - * Since all additions are reversible (knowing the final state and the - * input, you can reconstruct the initial state), if an attacker has - * any uncertainty about the initial state, he/she can only shuffle - * that uncertainty about, but never cause any collisions (which would - * decrease the uncertainty). - * - * The chosen system lets the state of the pool be (essentially) the input - * modulo the generator polymnomial. Now, for random primitive polynomials, - * this is a universal class of hash functions, meaning that the chance - * of a collision is limited by the attacker's knowledge of the generator - * polynomail, so if it is chosen at random, an attacker can never force - * a collision. Here, we use a fixed polynomial, but we *can* assume that - * ###--> it is unknown to the processes generating the input entropy. <-### - * Because of this important property, this is a good, collision-resistant - * hash; hash collisions will occur no more often than chance. - */ - -/* * Static global variables */ static DECLARE_WAIT_QUEUE_HEAD(random_read_wait); -- 1.7.12.rc0.22.gcdd159b
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