Messages in this thread | | | Date | 7 Dec 2014 16:33:54 -0500 | From | "George Spelvin" <> | 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.
| |