RE: effect of initializing CRC reg to 1's depends on implementati on? iSCSI

[Luben Tuikov <ltuikov@yahoo.com>]

--- Paul Koning <ni1d@arrl.net> wrote:
> Excerpt of message (sent 14 December 2001) by Luben
> > This is actually equivalent to XORing the first
> > 32 bits of the message with the magic constant as Vince
> > and I have shown.
> 
> Are you saying that you believe the intent of the current
> spec is NOT
> the same as the Ethernet CRC, i.e., complement the first
> 32 bits,
> multiply by x^32, then divide by G(x)?

In our profession we cannot talk about beliefs and
intentions, more so for documents, rfc, etc.

If you show the current description of the CRC of the draft
to a mathematician, they will object to the same thing I've
objected.

The reason is that they don't have the preconception about
what circuit is being used, and any such higher level
notions.

All they would have and all I had was long division in Z_2.
(Actually Z_2[x] since we deal with polynomials.) 

Then I started from first principles of CRC, polydivision,
etc, etc. -- similarly to how Williams proceeds in his
paper, but my treatment was purely algebraic.

The same applies to anyone looking at a description of an
algorithm. (An ``algorithm'' has a precise definition in
Computer Science.)

In its current form the description of the algorithm to
compute the CRC in the iSCSI draft is not consistent, just
because there is an unspoken assumption that a SMD circuit
will be used. We cannot afford to do this in a definition
document, unless we refer explicitly to it. We cannot even
assume that a circuit will be used...

Please also note that the Ethernet spec CRC algorithm, is
an optimized version of a basic
prefix-postfix-initialize-the-
CRC-register-to-some-constant algoritm. BUT as SMD it
operates on NON-AUGMENTED messages. FOR THAT REASON you
cannot say that we have to multiply M(x) by x^32!!!

I.e. SMD algoritms like the one in the Ethernet spec,
operate on non-augmented messages, while simple Division
(D) algorithms operate on augmented messages.

Any D can be optimized to SMD, but the constant has to be
changed. Also, prefixing a message is equivalent to loading
the CRC register with that prefix (WLG), but this is not so
for SMD. (elaborated below)

Further more for any SMD there is an equivalent D,
including for the Ethernet spec SMD; and for any D there is
an equivalent SMD.

> If that is your interpretation, then we absolutely MUST
> fix the spec,
> I'm quite sure that the intent of the spec is as I wrote
> it, i.e.,
> Ethernet except for the generator.  

Probably it is. But lets not hasten here.
We can further generalize the explanation of the algorithm.
 
> > If this is changed, then the message in its form:
> >   M'(x) = x^32 M(x) + x^(32+n) I(x)
> > 
> > yields to better optimizations as Vince and I have
> shown.
> > (see his circuits and worksheet4.pdf, which I posted
> > yesterday)
> > 
> > I.e. the message is augmented (mult by x^32, aka
> postfixed)
> > and prefixed by 32 1's (aka CRC=32 1's).
> 
> But that is NOT what the spec currently says.  What you
> describe may
> be a fine CRC but it is not the one that's in the spec. 
> Initializing
> the CRC register to all 1 is not the same as prefixing 32
> 1 bits onto
> the message.

It is, in a simple division, e.g. in D. This is clearly and
simply explained in the Williams paper. It is an equivalent
to a long division.

Now, in D, if I initialize the CRC to all 1's and then do
division of M(x) by G(x), so that I catch 0's at the
beginning of the message,

(which is equivalent to prefixing the message by 1's of the
number of the width of the CRC, and then do just simple
division,(CRC=0), since after so many clicks the CRC will
be loaded with the prefixing bits, and also 0/G(x) = 0)

then I can find a constant which I can XOR to the first 32
bits of the message and then perform the division, still in
D, with the same effect. In this particular case that
constant is the magic constant. (We are one step closer to
SMD...)

(Now we jump to SMD, by means of worksheet4.pdf, Prefixing
part.)

Now, in the Ethernet CRC spec, that constant is 32 1's, and
XORing 32 1's with the 32 MSb of the message is equivalent
to complementing them.

BUT the Ethernet spec uses SMD, not D, so I claim that
there is an equivalent D, which for a suitable initial CRC
(also equivalent to prefixing the message with it) value
will yield IDENTICAL results as Ethernet SMD.

Now using a bit of algebra similar to worksheet4.pdf which
I posted a couple of days ago and numerical methods
(solving a linear system of 33 unknowns) I have found such
a constant:
0x2a26f826 which in simple division, D, will yield results
identical to SMD of Ethernet.

So, here is an abstractization of the Ethernet CRC SMD,
explained in a simple, simple, simple division only way:

TX:
1. Mutliply the message, M(x), by x^32.
2. Prefix the result of 1. with 0x2a26f826.
3. Find the remainder of the result of 2.
4. Complement that remainder and add it to the 
   result of 1.
5. Send the result of 4. to the recipient.

RX: (same as steps 1-3 in TX)
1. Mutliply the message by x^32.
2. Prefix the result of 1. with 0x2a26f826.
3. Find the remainder of the result of 2.

The result of 3 should be 0x1c2d19ed (the magic constant).

Of course I've ommitted the mirroring of bytes for clarity
and brevity. It can be mentioned on the side, e.g. How to
build the message: mirror the bytes == to swapping the bits
inside the bytes, ....

Also note that step 4 in TX, adding means exactly that,
since we already mult. by x^32, so the remainder will
nicely fit in the last 32 0 bits.

Also note that step 2, can be made clearer:
let C(x), be the polynomial representation of 0x2a26f826.
Let n = deg G(x), k-1 = deg M(x) (there are k bits), then
step 2 is:

2. Add x^n x^k C(x) to the message M(x).

Or we can swap step 1 and 2:

1. Add x^k C(x) to M(x).
2. Multiply the result of 1. by x^n.

I hope this makes things a bit clearer.

So from this D specification above, I can derive SMD
exaclty as it is in the Ethernet spec. I'll include this
derivation in the paper -- it will be pure math, but in
english it goes like this: First we notice that after 32
clicks the CRC register will contain only 32 1's XOR-ed
with the message, then we see that we can just load the crc
with ones and then kick one byte off the left xor it with
the next message bit...

The above equivalence I've tested empirically.

> What does "better optimizations" mean?

See above.
 
> > In an SMD, one would have to set the CRC to the magic
> > constant and then proceed as the algorithm indicates,
> > (this is identical to what you'd find in Ethernet, ...
> 
> No it isn't.  The Ethernet spec appendix C, and, more
> importantly, the
> formal definition of the CRC in the body of the spec,
> makes it quite
> clear that it isn't.  I don't understand how you arrive
> at any other
> conclusion.

I didn't mean identical as in the constant. I meant
identical in a way of equivalence of algorithms, that one
can be derived from the other, etc. See above. Or simpler
answer: Math.

-l





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