RE: I-D ACTION:draft-cavanna-iscsi-crc-vs-cksum-00.txt

["THALER,PAT (A-Roseville,ex1)" <pat_thaler@agilent.com>]

Julian,

It is relatively easy to investigate the Hamming distance and burst
detection of Adler32 and Fletcher32 using mathematics. In the argument
below, I will use "value" to identify the quantities that get summed since
Adler sums bytes and Fletcher32 sums 16-bits.

Both Adler and Fletcher calculate for each value:

 S1 = (S1 + value) mod m
 S2 = (S2 + S1) mod m

where m is 65521 for Adler and 65535 for Fletcher.

Which means that for an n value string of values V1 to Vn:
 S1 = (V1 + V2 + ... + Vn (+ 1 for Adler)) mod m
 S2 = (n * V1 + (n-1) * V2 + ... + Vn (+ n for Adler)) mod m

A two bit error can only change two values. For a two bit error to be
undetectable the two changed values must produce no change to S1 and S2. 

Let 
 x be the change to the first errored value
 y be the change to the second errored value
 k be the separation between the first and second errored values.

Then for an undetectable error:

 delta S1 = (x + y) mod m = 0
 delta S2 = ((k + 1) * x + y) mod m = 0
          = (k * x + x + y) mod m =0
          = (k * x) mod m = 0          because (x + y) mod m = 0

Therefore, for an undetectable 2 bit error, k * x must be a multiple of m. 

For Adler, m is prime and the maximum value of x is 255 so the only way to
satisfy the condition is for k to equal 65521. Then x can be any value so it
could be a one bit error such as changing the lsb of the first value from 0
to 1. y would then need to be the inverse changing the lsb of the second
value from 1 to 0. 

For Fletcher, m is factorable to 3, 5, 17, 257. For x to provide one or more
of those factors, it would have to have more than a 1 bit change and y would
have to have the inverse of that change which would mean an error of at
least 4 bits. The only way to get a two bit undetectable error is for k to
equal 65535.

So, Fletcher and Adler do have 2-bit undetectable errors if the messages are
longer than the modulus.

What about burst protection? For Fletcher, we know that a change of a value
from all 0's to all 1's produces an an undetectable error so we know that
burst protection is less than 16. k = 1 for a burst spread across two
consecutive values. Therefore, the only way for a burst spread across two
consecutive values to produce an undetectable error is for x to equal 65535.
A burst shorter than 16 can't do it.

For Adler, corrupting two consecutive bytes can't produce an undetectable
error because that would be the case above where k equals 1. There is no way
for a burst across two consectutive values to be undetectable in Adler since
x is less than the modulus. 

For a burst across three values:

let x equal the error in the first value
    y equal the error in the second value
    z equal the error in the third value.

For the burst to be undetectable

 Delta S1 = (x + y + z) mod m = 0
 Delta S2 = (3x + 2y + z) mod m = 0

Since the maximum values of x, y, and z are 255 and m is more than 6 times
that, the sums above will always be less than m and we can drop the mod m.

 Delta S2 = (3x + 2y + z) = 2x + y + (x + y + z) = 0
          = 2x + y = 0                    because x + y + z = 0

        y = - 2x

 Delta S1 = x + y + z = 0
          = x - 2x + z = 0
          = -x + z
        z = x

So an undetectable error is created when in three consecutive values the
errors are
    x
  -2x
    x

This could for instance be errors of 1, -2, and 1. Since the error to the
first and third values are the same, the error must span at least 2 * the
length of the value + 1. So for Adler, it must be a 17 bit burst at least.

Also, note that this can be created by a 3 bit error and is equally
applicable to Adler and Fletcher. 

Therefore, for messages shorter than the modulus + 1, the Hamming distance
of Adler32 and Fletcher 32 is 3 and, for messages longer than that, the
Hamming distance is 2.

Regards, 
Pat Thaler


 
 

-----Original Message-----
From: julian_satran@il.ibm.com [mailto:julian_satran@il.ibm.com]

Pat,

I did not run a check. That is very expensive.
With CRCs there are methods that enable you to run a check on the
complement code
(that has only 2**32 different patterns for any block length) and derive
from there the distances (Fujiwara has done this in 1989 for the IEEE
CRC-32 and about some more recent experiments I'll get back to the list).

And that is the trouble with this whole line of argument.
There are no numbers to prove Adler32 or Fletcher32 and there are plenty
for CRCs.

The big question is then is there anybody out there that wants to build a
modern bridge based only on its beauty?

Regards,
Julo


pat_thaler@agilent.com on 05/03/2001 23:27:49

Please respond to pat_thaler@agilent.com

To:   Julian Satran/Haifa/IBM@IBMIL
cc:
Subject:  RE: I-D ACTION:draft-cavanna-iscsi-crc-vs-cksum-00.txt




Julian,

I know that Hamming distance gets down to 2 for errors that are separated
by
the modulus (or a multiple of it). Is there another case?

Pat

> - Adler and Fletcher are weak and there is no theory behind your
> distribution statements, nor any simulation results as far as I know.  We
> found that on very simple sequences the Hamming distance gets
> down to 2 (or
> lower) and the burst protection is probably not better than 16 bit.
There
> is even a simple formula for what sequences will get you false codes (see
> bellow for a reference)
>