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Date:	7 Dec 2014 16:33:54 -0500
From:	"George Spelvin" <linux@...izon.com>
To:	hannes@...essinduktion.org, linux@...izon.com
Cc:	davem@...emloft.net, dborkman@...hat.com,
	herbert@...dor.apana.org.au, linux-kernel@...r.kernel.org,
	netdev@...r.kernel.org, tgraf@...g.ch, tytso@....edu
Subject: Re: Where exactly will arch_fast_hash be used

On Sun, 2014-12-07 at 15:06 +0100, Hannes Frederic Sowa wrote:
> In case of openvswitch it shows a performance improvment. The seed
> parameter could be used as an initial biasing of the crc32 function, but
> in case of openvswitch it is only set to 0.

NACK.

This is the Fatal Error in thinking that Herbert was warning about.
The seed parameter doesn't affect CRC32 collisions *at all* if the inputs
are the same size.

For fixed-size inputs, a non-zero seed is equivalent to XORing a
constant into the output of the CRC computation.


for *different* sized inputs, a non-zero seed detects zero-padding
better than a zero one, but *which* non-zero value is also irrelevant;
all-ones is the traditional choice because it's simplest in hardware.


A CRC is inherently linear.  CRC(a^b) = CRC(a) ^ CRC(b).  This makes
them easy to analyze mathematically and gives them a number of nice
properties for detecting hardware corruption.

But that same simplicity makes it *ridiculously* easy to generate
collisions if you try.


One way of looking at a CRC is to say that each bit in the input
has a CRC.  The CRC of a message string is just the XOR of the CRCs
of the individual bits that are set in the message.

Now, a CRC polynomial is chosen so that all of the bits of a
message have different CRCs.  Obviously, there's a limit: when the
message is 2^n bits long, it's not possible for all the bits to
have different, non-zero n-bit CRCs.

But a CRC is a really efficient way of assigning different bit patterns
to different input bits up to that limit.

(Something like CRC32c is also chosen so that, for messages up to a
reasonable length, no 3-bit, 4-bit, etc. combinations have CRCs that
XOR to zero.)


But, and this might be what Herbert was trying to say and I was
misunderstanding, if you then *truncate* that CRC, the CRCs of the
message bits lose that uniqueness guarantee.  They're just pseudorandom
numbers, and a CRC loses its special collision-resistance properties.

It's just an ordinary random hash, and thanks to the birthday paradox,
you're likely to find two bits whose CRCs agree in any particular 8 bits
within roughly sqrt(2*256) or 22 bits.

Here are a few such collisions for the least significant 8 bits of CRC32c:

Msg1	CRC32c		Msg2	CRC32c		Match
1<<11	3fc5f181	1<<30	bf672381	81
1<<12	9d14c3b8	1<<31	dd45aab8	b8
1<<5	c79a971f	1<<44	6006181f	1f
1<<15	13a29877	1<<45	b2f53777	77

There's nothing special about the lsbits of the CRC.
Within 64 bits, the most significant 8 bits have it worse:

1<<5	c79a971f	1<<17	c76580d9	c7
1<<6	e13b70f7	1<<18	e144fb14	e1
1<<19	70a27d8a	1<<38	7022df58	70
1<<20	38513ec5	1<<39	38116fac	38
1<<13	4e8a61dc	1<<52	4e2dfd53	4e
1<<23	a541927e	1<<53	a5e0c5d1	a5


Now, I'd like to stress that this collision rate is no worse than any
*other* hash function.  A truncated CRC loses its special resistance to
the birthday paradox (you'd have been much smarter to use 8-bit CRC),
but it doesn't become especially bad.  A truncated SHA-1 will have
coillisions just as often.

The concern with a CRC is that, once you've found one collision, you've
found a huge number of them.  Just XOR the bit pattern of your choice
into both of the colliding messages, and you have a new collision.


For another example, if you consider the CRC32c of all possible 1-byte
messages *and then take only the low byte*, there are only 128 possible
values.

It turns out that the byte 0x5d has a CRC32c of 0xee0d9600.  This ends
in 00, so if I XOR 0x5d into anything, the low 8 bits of the CRC
don't change.

Likewise, the message "23 00" has a CRC32c of 0x00ee0d96.  So you can
XOR 0x23 into the second-last byte of anything, and the high 8 bits of
the CRC don't change.
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