[PATCH net-next 2/2] reciprocal_divide: correction/update of the algorithm

Daniel Borkmann
2014-01-15T23:24:26Z

From: Hannes Frederic Sowa <hannes@stressinduktion.org>

Jakub Zawadzki noticed that some divisions by reciprocal_divide()
were not correct [1][2], which he could also show with BPF code
after divisions are transformed into reciprocal_value() for runtime
invariant which can be passed to reciprocal_divide() later on;
reverse in BPF dump ended up with a different, off-by-one K.

Also, reciprocal_value() and reciprocal_divide() always return 0
for divisions by 1. This is a bit worrisome as those functions
also get used in mm/slab.c and lib/flex_array.c, apparently for
index calculation to access array slots. Bonding already seems to
check for the 1-divisor case and handles that correctly. We don't
know about other problems, besides possibly flex array, yet.

In order to fix that, we propose an extension from the original
implementation from commit 6a2d7a955d8d resp. [3][4], by using
the algorithm proposed in "Division by Invariant Integers Using
Multiplication" [5], Torbjörn Granlund and Peter L. Montgomery,
that is, pseudocode for q = n/d where q,n,d is in u32 universe:

1) Initialization:

int l = ceil(log_2 d)
uword m' = floor((1<<32)*((1<<l)-d)/d)+1
int sh_1 = min(l,1)
int sh_2 = max(l-1,0)

2) For q = n/d, all uword:

uword t = (n*m')>>32
q = (t+((n-t)>>sh_1))>>sh_2

The assembler implementation from Agner Fog [6] also helped a lot
while implementing. We have tested the implementation on x86_64,
ppc64, i686, s390x; on x86_64/haswell we're still half the latency
compared to normal divide.

Joint work with Daniel Borkmann.

[1] http://www.wireshark.org/~darkjames/reciprocal-buggy.c
[2] http://www.wireshark.org/~darkjames/set-and-dump-filter-k-bug.c
[3] https://gmplib.org/~tege/division-paper.pdf
[4] http://homepage.cs.uiowa.edu/~jones/bcd/divide.html
[5] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1.2556
[6] http://www.agner.org/optimize/asmlib.zip

Fixes: 6a2d7a955d8d ("SLAB: use a multiply instead of a divide in obj_to_index()")
Fixes: 704f15ddb5fc2a ("flex_array: avoid divisions when accessing elements")
Reported-by: Jakub Zawadzki <darkjames-ws@darkjames.pl>
Cc: Eric Dumazet <eric.dumazet@gmail.com>
Cc: Austin S Hemmelgarn <ahferroin7@gmail.com>
Cc: linux-kernel@vger.kernel.org
Cc: Jesse Gross <jesse@nicira.com>
Cc: Jamal Hadi Salim <jhs@mojatatu.com>
Cc: Stephen Hemminger <stephen@networkplumber.org>
Cc: Matt Mackall <mpm@selenic.com>
Cc: Pekka Enberg <penberg@kernel.org>
Cc: Christoph Lameter <cl@linux-foundation.org>
Cc: Andy Gospodarek <andy@greyhouse.net>
Cc: Veaceslav Falico <vfalico@redhat.com>
Cc: Jay Vosburgh <fubar@us.ibm.com>
Cc: Jakub Zawadzki <darkjames-ws@darkjames.pl>
Signed-off-by: Daniel Borkmann <dborkman@redhat.com>
Signed-off-by: Hannes Frederic Sowa <hannes@stressinduktion.org>
---
drivers/net/bonding/bond_main.c | 16 ++++++++-------
drivers/net/bonding/bond_netlink.c | 4 ----
drivers/net/bonding/bond_options.c | 9 +++-----
drivers/net/bonding/bond_sysfs.c | 5 -----
drivers/net/bonding/bonding.h | 3 +++
include/linux/flex_array.h | 3 ++-
include/linux/reciprocal_div.h | 42 +++++++++++++++++++++++---------------
include/linux/slab_def.h | 4 +++-
include/net/red.h | 3 ++-
lib/flex_array.c | 3 ++-
lib/reciprocal_div.c | 28 +++++++++++++++++++++----
net/sched/sch_netem.c | 4 +++-
12 files changed, 76 insertions(+), 48 deletions(-)

diff --git a/drivers/net/bonding/bond_main.c b/drivers/net/bonding/bond_main.c
index f2fe6cb..8cf03eb 100644
--- a/drivers/net/bonding/bond_main.c
+++ b/drivers/net/bonding/bond_main.c
@@ -79,7 +79,6 @@
#include <net/pkt_sched.h>
#include <linux/rculist.h>
#include <net/flow_keys.h>
-#include <linux/reciprocal_div.h>
#include "bonding.h"
#include "bond_3ad.h"
#include "bond_alb.h"
@@ -3551,8 +3550,9 @@ static void bond_xmit_slave_id(struct bonding *bond, struct sk_buff *skb, int sl
*/
static u32 bond_rr_gen_slave_id(struct bonding *bond)
{
- int packets_per_slave = bond->params.packets_per_slave;
u32 slave_id;
+ struct reciprocal_value reciprocal_packets_per_slave;
+ int packets_per_slave = bond->params.packets_per_slave;

switch (packets_per_slave) {
case 0:
@@ -3562,8 +3562,10 @@ static u32 bond_rr_gen_slave_id(struct bonding *bond)
slave_id = bond->rr_tx_counter;
break;
default:
+ reciprocal_packets_per_slave =
+ bond->params.reciprocal_packets_per_slave;
slave_id = reciprocal_divide(bond->rr_tx_counter,
- packets_per_slave);
+ reciprocal_packets_per_slave);
break;
}
bond->rr_tx_counter++;
@@ -4297,10 +4299,10 @@ static int bond_check_params(struct bond_params *params)
params->resend_igmp = resend_igmp;
params->min_links = min_links;
params->lp_interval = lp_interval;
- if (packets_per_slave > 1)
- params->packets_per_slave = reciprocal_value(packets_per_slave);
- else
- params->packets_per_slave = packets_per_slave;
+ params->packets_per_slave = packets_per_slave;
+ params->reciprocal_packets_per_slave =
+ reciprocal_value(packets_per_slave);
+
if (primary) {
strncpy(params->primary, primary, IFNAMSIZ);
params->primary[IFNAMSIZ - 1] = 0;
diff --git a/drivers/net/bonding/bond_netlink.c b/drivers/net/bonding/bond_netlink.c
index 555c783..9b13791 100644
--- a/drivers/net/bonding/bond_netlink.c
+++ b/drivers/net/bonding/bond_netlink.c
@@ -19,7 +19,6 @@
#include <linux/if_ether.h>
#include <net/netlink.h>
#include <net/rtnetlink.h>
-#include <linux/reciprocal_div.h>
#include "bonding.h"

static const struct nla_policy bond_policy[IFLA_BOND_MAX + 1] = {
@@ -416,9 +415,6 @@ static int bond_fill_info(struct sk_buff *skb,
goto nla_put_failure;

packets_per_slave = bond->params.packets_per_slave;
- if (packets_per_slave > 1)
- packets_per_slave = reciprocal_value(packets_per_slave);
-
if (nla_put_u32(skb, IFLA_BOND_PACKETS_PER_SLAVE,
packets_per_slave))
goto nla_put_failure;
diff --git a/drivers/net/bonding/bond_options.c b/drivers/net/bonding/bond_options.c
index 945a666..bacfd53 100644
--- a/drivers/net/bonding/bond_options.c
+++ b/drivers/net/bonding/bond_options.c
@@ -16,7 +16,6 @@
#include <linux/netdevice.h>
#include <linux/rwlock.h>
#include <linux/rcupdate.h>
-#include <linux/reciprocal_div.h>
#include "bonding.h"

int bond_option_mode_set(struct bonding *bond, int mode)
@@ -671,11 +670,9 @@ int bond_option_packets_per_slave_set(struct bonding *bond,
pr_warn("%s: Warning: packets_per_slave has effect only in balance-rr mode\n",
bond->dev->name);

- if (packets_per_slave > 1)
- bond->params.packets_per_slave =
- reciprocal_value(packets_per_slave);
- else
- bond->params.packets_per_slave = packets_per_slave;
+ bond->params.packets_per_slave = packets_per_slave;
+ bond->params.reciprocal_packets_per_slave =
+ reciprocal_value(packets_per_slave);

return 0;
}
diff --git a/drivers/net/bonding/bond_sysfs.c b/drivers/net/bonding/bond_sysfs.c
index 011f163..c083e9a 100644
--- a/drivers/net/bonding/bond_sysfs.c
+++ b/drivers/net/bonding/bond_sysfs.c
@@ -39,7 +39,6 @@
#include <net/net_namespace.h>
#include <net/netns/generic.h>
#include <linux/nsproxy.h>
-#include <linux/reciprocal_div.h>

#include "bonding.h"

@@ -1374,10 +1373,6 @@ static ssize_t bonding_show_packets_per_slave(struct device *d,
{
struct bonding *bond = to_bond(d);
unsigned int packets_per_slave = bond->params.packets_per_slave;
-
- if (packets_per_slave > 1)
- packets_per_slave = reciprocal_value(packets_per_slave);
-
return sprintf(buf, "%u\n", packets_per_slave);
}

diff --git a/drivers/net/bonding/bonding.h b/drivers/net/bonding/bonding.h
index 955dc48..502dda8 100644
--- a/drivers/net/bonding/bonding.h
+++ b/drivers/net/bonding/bonding.h
@@ -23,6 +23,8 @@
#include <linux/netpoll.h>
#include <linux/inetdevice.h>
#include <linux/etherdevice.h>
+#include <linux/reciprocal_div.h>
+
#include "bond_3ad.h"
#include "bond_alb.h"

@@ -171,6 +173,7 @@ struct bond_params {
int resend_igmp;
int lp_interval;
int packets_per_slave;
+ struct reciprocal_value reciprocal_packets_per_slave;
};

struct bond_parm_tbl {
diff --git a/include/linux/flex_array.h b/include/linux/flex_array.h
index 6843cf1..b6efb0c 100644
--- a/include/linux/flex_array.h
+++ b/include/linux/flex_array.h
@@ -2,6 +2,7 @@
#define _FLEX_ARRAY_H

#include <linux/types.h>
+#include <linux/reciprocal_div.h>
#include <asm/page.h>

#define FLEX_ARRAY_PART_SIZE PAGE_SIZE
@@ -22,7 +23,7 @@ struct flex_array {
int element_size;
int total_nr_elements;
int elems_per_part;
- u32 reciprocal_elems;
+ struct reciprocal_value reciprocal_elems;
struct flex_array_part *parts[];
};
/*
diff --git a/include/linux/reciprocal_div.h b/include/linux/reciprocal_div.h
index f9c90b3..be66c3a 100644
--- a/include/linux/reciprocal_div.h
+++ b/include/linux/reciprocal_div.h
@@ -4,29 +4,37 @@
#include <linux/types.h>

/*
- * This file describes reciprocical division.
+ * This algorithm is based on the paper "Division by Invariant
+ * Integers Using Multiplication" by Torbjörn Granlund and Peter
+ * L. Montgomery.
*
- * This optimizes the (A/B) problem, when A and B are two u32
- * and B is a known value (but not known at compile time)
+ * The assembler implementation from Agner Fog, which this code is
+ * based on, can be found here:
+ * http://www.agner.org/optimize/asmlib.zip
*
- * The math principle used is :
- * Let RECIPROCAL_VALUE(B) be (((1LL << 32) + (B - 1))/ B)
- * Then A / B = (u32)(((u64)(A) * (R)) >> 32)
+ * This optimization for A/B is helpful if the divisor B is mostly
+ * constant. The reciprocal of B is calculated in the slow-path with
+ * reciprocal_value(). The fast-path can then just use a much faster
+ * multiplication operation with a variable dividend A to calculate
+ * the division A/B.
*
- * This replaces a divide by a multiply (and a shift), and
- * is generally less expensive in CPU cycles.
+ * RECIPROCAL_VALUE_TO_ZERO can be used to express an element, which
+ * used as the argument to reciprocal_divide always yields zero.
*/

-/*
- * Computes the reciprocal value (R) for the value B of the divisor.
- * Should not be called before each reciprocal_divide(),
- * or else the performance is slower than a normal divide.
- */
-extern u32 reciprocal_value(u32 B);
+struct reciprocal_value {
+ u32 m;
+ u8 sh1, sh2;
+};

+#define RECIPROCAL_VALUE_RESULT_TO_ZERO ((struct reciprocal_value){.sh1 = 32})

-static inline u32 reciprocal_divide(u32 A, u32 R)
+struct reciprocal_value reciprocal_value(u32 d);
+
+static inline u32 reciprocal_divide(u32 a, struct reciprocal_value R)
{
- return (u32)(((u64)A * R) >> 32);
+ u32 t = (u32)(((u64)a * R.m) >> 32);
+ return (t + ((a - t) >> R.sh1)) >> R.sh2;
}
-#endif
+
+#endif /* _LINUX_RECIPROCAL_DIV_H */
diff --git a/include/linux/slab_def.h b/include/linux/slab_def.h
index 09bfffb..96e8aba 100644
--- a/include/linux/slab_def.h
+++ b/include/linux/slab_def.h
@@ -1,6 +1,8 @@
#ifndef _LINUX_SLAB_DEF_H
#define _LINUX_SLAB_DEF_H

+#include <linux/reciprocal_div.h>
+
/*
* Definitions unique to the original Linux SLAB allocator.
*/
@@ -12,7 +14,7 @@ struct kmem_cache {
unsigned int shared;

unsigned int size;
- u32 reciprocal_buffer_size;
+ struct reciprocal_value reciprocal_buffer_size;
/* 2) touched by every alloc & free from the backend */

unsigned int flags; /* constant flags */
diff --git a/include/net/red.h b/include/net/red.h
index 168bb2f..76e0b5f 100644
--- a/include/net/red.h
+++ b/include/net/red.h
@@ -130,7 +130,8 @@ struct red_parms {
u32 qth_max; /* Max avg length threshold: Wlog scaled */
u32 Scell_max;
u32 max_P; /* probability, [0 .. 1.0] 32 scaled */
- u32 max_P_reciprocal; /* reciprocal_value(max_P / qth_delta) */
+ /* reciprocal_value(max_P / qth_delta) */
+ struct reciprocal_value max_P_reciprocal;
u32 qth_delta; /* max_th - min_th */
u32 target_min; /* min_th + 0.4*(max_th - min_th) */
u32 target_max; /* min_th + 0.6*(max_th - min_th) */
diff --git a/lib/flex_array.c b/lib/flex_array.c
index 6948a66..704efa1 100644
--- a/lib/flex_array.c
+++ b/lib/flex_array.c
@@ -90,8 +90,9 @@ struct flex_array *flex_array_alloc(int element_size, unsigned int total,
{
struct flex_array *ret;
int elems_per_part = 0;
- int reciprocal_elems = 0;
int max_size = 0;
+ struct reciprocal_value reciprocal_elems =
+ RECIPROCAL_VALUE_RESULT_TO_ZERO;

if (element_size) {
elems_per_part = FLEX_ARRAY_ELEMENTS_PER_PART(element_size);
diff --git a/lib/reciprocal_div.c b/lib/reciprocal_div.c
index 75510e9..f49742d 100644
--- a/lib/reciprocal_div.c
+++ b/lib/reciprocal_div.c
@@ -1,11 +1,31 @@
+#include <linux/kernel.h>
+#include <linux/bug.h>
#include <asm/div64.h>
#include <linux/reciprocal_div.h>
#include <linux/export.h>

-u32 reciprocal_value(u32 k)
+/*
+ * For a description of the algorithm please have a look at
+ * include/linux/reciprocal_div.h
+ */
+
+struct reciprocal_value reciprocal_value(u32 d)
{
- u64 val = (1LL << 32) + (k - 1);
- do_div(val, k);
- return (u32)val;
+ struct reciprocal_value R;
+ u64 m;
+ int l;
+
+ BUILD_BUG_ON(reciprocal_divide(0U, RECIPROCAL_VALUE_RESULT_TO_ZERO));
+ BUILD_BUG_ON(reciprocal_divide(1U, RECIPROCAL_VALUE_RESULT_TO_ZERO));
+ BUILD_BUG_ON(reciprocal_divide(~(0U), RECIPROCAL_VALUE_RESULT_TO_ZERO));
+
+ l = fls(d - 1);
+ m = ((1ULL << 32) * ((1ULL << l) - d));
+ do_div(m, d);
+ ++m;
+ R.m = (u32)m;
+ R.sh1 = min(l, 1);
+ R.sh2 = max(l-1, 0);
+ return R;
}
EXPORT_SYMBOL(reciprocal_value);
diff --git a/net/sched/sch_netem.c b/net/sched/sch_netem.c
index 3019c10..3bf6495 100644
--- a/net/sched/sch_netem.c
+++ b/net/sched/sch_netem.c
@@ -91,7 +91,7 @@ struct netem_sched_data {
u64 rate;
s32 packet_overhead;
u32 cell_size;
- u32 cell_size_reciprocal;
+ struct reciprocal_value cell_size_reciprocal;
s32 cell_overhead;

struct crndstate {
@@ -718,6 +718,8 @@ static void get_rate(struct Qdisc *sch, const struct nlattr *attr)
q->cell_size = r->cell_size;
if (q->cell_size)
q->cell_size_reciprocal = reciprocal_value(q->cell_size);
+ else
+ q->cell_size_reciprocal = RECIPROCAL_VALUE_RESULT_TO_ZERO;
q->cell_overhead = r->cell_overhead;
}

--
1.8.3.1