January 14, 2014

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Polynomials : extract from ( http://www.ross.net/crc/download/crc_v3.txt )

"Everything you wanted to know about CRC algorithms, but were afraid
to ask for fear that errors in your understanding might be detected."

Version : 3.
Date    : 19 August 1993.
Author  : Ross N. Williams.
Net     : ross@guest.adelaide.edu.au.
FTP     : ftp.adelaide.edu.au/pub/rocksoft/crc_v3.txt
Company : Rocksoft^tm Pty Ltd.
Snail   : 16 Lerwick Avenue, Hazelwood Park 5066, Australia.
Fax     : +61 8 373-4911 (c/- Internode Systems Pty Ltd).
Phone   : +61 8 379-9217 (10am to 10pm Adelaide Australia time).
Note    : "Rocksoft" is a trademark of Rocksoft Pty Ltd, Australia.
Status  : Copyright (C) Ross Williams, 1993. However, permission is
          granted to make and distribute verbatim copies of this
          document provided that this information block and copyright
          notice is included. Also, the C code modules included
          in this document are fully public domain.
Thanks  : Thanks to Jean-loup Gailly (jloup@chorus.fr) and Mark Adler
          (me@quest.jpl.nasa.gov) who both proof read this document
          and picked out lots of nits as well as some big fat bugs.

Table of Contents
 1. Introduction: Error Detection
 2. The Need For Complexity
 3. The Basic Idea Behind CRC Algorithms
 4. Polynomical Arithmetic
 5. Binary Arithmetic with No Carries
 6. A Fully Worked Example
 7. Choosing A Poly
 8-21 : -- check the original link if still available --
This document explains CRCs (Cyclic Redundancy Codes) and their
table-driven implementations in full, precise detail. Much of the
literature on CRCs, and in particular on their table-driven
implementations, is a little obscure (or at least seems so to me).
This document is an attempt to provide a clear and simple no-nonsense
explanation of CRCs and to absolutely nail down every detail of the
operation of their high-speed implementations. In addition to this,
this document presents a parameterized model CRC algorithm called the
"Rocksoft^tm Model CRC Algorithm". The model algorithm can be
parameterized to behave like most of the CRC implementations around,
and so acts as a good reference for describing particular algorithms.
A low-speed implementation of the model CRC algorithm is provided in
the C programming language. Lastly there is a section giving two forms
of high-speed table driven implementations, and providing a program
that generates CRC lookup tables.

1. Introduction: Error Detection
The aim of an error detection technique is to enable the receiver of a
message transmitted through a noisy (error-introducing) channel to
determine whether the message has been corrupted. To do this, the
transmitter constructs a value (called a checksum) that is a function
of the message, and appends it to the message. The receiver can then
use the same function to calculate the checksum of the received
message and compare it with the appended checksum to see if the
message was correctly received. For example, if we chose a checksum
function which was simply the sum of the bytes in the message mod 256
(i.e. modulo 256), then it might go something as follows. All numbers
are in decimal.

   Message                    :  6 23  4
   Message with checksum      :  6 23  4 33
   Message after transmission :  6 27  4 33

In the above, the second byte of the message was corrupted from 23 to
27 by the communications channel. However, the receiver can detect
this by comparing the transmitted checksum (33) with the computer
checksum of 37 (6 + 27 + 4). If the checksum itself is corrupted, a
correctly transmitted message might be incorrectly identified as a
corrupted one. However, this is a safe-side failure. A dangerous-side
failure occurs where the message and/or checksum is corrupted in a
manner that results in a transmission that is internally consistent.
Unfortunately, this possibility is completely unavoidable and the best
that can be done is to minimize its probability by increasing the
amount of information in the checksum (e.g. widening the checksum from
one byte to two bytes).

Other error detection techniques exist that involve performing complex
transformations on the message to inject it with redundant
information. However, this document addresses only CRC algorithms,
which fall into the class of error detection algorithms that leave the
data intact and append a checksum on the end. i.e.:

      <original intact message> <checksum>

2. The Need For Complexity
In the checksum example in the previous section, we saw how a
corrupted message was detected using a checksum algorithm that simply
sums the bytes in the message mod 256:

   Message                    :  6 23  4
   Message with checksum      :  6 23  4 33
   Message after transmission :  6 27  4 33

A problem with this algorithm is that it is too simple. If a number of
random corruptions occur, there is a 1 in 256 chance that they will
not be detected. For example:

   Message                    :  6 23  4
   Message with checksum      :  6 23  4 33
   Message after transmission :  8 20  5 33

To strengthen the checksum, we could change from an 8-bit register to
a 16-bit register (i.e. sum the bytes mod 65536 instead of mod 256) so
as to apparently reduce the probability of failure from 1/256 to
1/65536. While basically a good idea, it fails in this case because
the formula used is not sufficiently "random"; with a simple summing
formula, each incoming byte affects roughly only one byte of the
summing register no matter how wide it is. For example, in the second
example above, the summing register could be a Megabyte wide, and the
error would still go undetected. This problem can only be solved by
replacing the simple summing formula with a more sophisticated formula
that causes each incoming byte to have an effect on the entire
checksum register.

Thus, we see that at least two aspects are required to form a strong
checksum function:

   WIDTH: A register width wide enough to provide a low a-priori
          probability of failure (e.g. 32-bits gives a 1/2^32 chance
          of failure).

   CHAOS: A formula that gives each input byte the potential to change
          any number of bits in the register.

Note: The term "checksum" was presumably used to describe early
summing formulas, but has now taken on a more general meaning
encompassing more sophisticated algorithms such as the CRC ones. The
CRC algorithms to be described satisfy the second condition very well,
and can be configured to operate with a variety of checksum widths.

3. The Basic Idea Behind CRC Algorithms
Where might we go in our search for a more complex function than
summing? All sorts of schemes spring to mind. We could construct
tables using the digits of pi, or hash each incoming byte with all the
bytes in the register. We could even keep a large telephone book
on-line, and use each incoming byte combined with the register bytes
to index a new phone number which would be the next register value.
The possibilities are limitless.

However, we do not need to go so far; the next arithmetic step
suffices. While addition is clearly not strong enough to form an
effective checksum, it turns out that division is, so long as the
divisor is about as wide as the checksum register.

The basic idea of CRC algorithms is simply to treat the message as an
enormous binary number, to divide it by another fixed binary number,
and to make the remainder from this division the checksum. Upon
receipt of the message, the receiver can perform the same division and
compare the remainder with the "checksum" (transmitted remainder).

Example: Suppose the the message consisted of the two bytes (6,23) as
in the previous example. These can be considered to be the hexadecimal
number 0617 which can be considered to be the binary number
0000-0110-0001-0111. Suppose that we use a checksum register one-byte
wide and use a constant divisor of 1001, then the checksum is the
remainder after 0000-0110-0001-0111 is divided by 1001. While in this
case, this calculation could obviously be performed using common
garden variety 32-bit registers, in the general case this is messy. So
instead, we'll do the division using good-'ol long division which you
learnt in school (remember?). Except this time, it's in binary:

          ...0000010101101 = 00AD =  173 = QUOTIENT
9= 1001 ) 0000011000010111 = 0617 = 1559 = DIVIDEND
DIVISOR   0000.,,....,.,,,
                      0010 = 02 = 2 = REMAINDER

In decimal this is "1559 divided by 9 is 173 with a remainder of 2".

Although the effect of each bit of the input message on the quotient
is not all that significant, the 4-bit remainder gets kicked about
quite a lot during the calculation, and if more bytes were added to
the message (dividend) it's value could change radically again very
quickly. This is why division works where addition doesn't.

In case you're wondering, using this 4-bit checksum the transmitted
message would look like this (in hexadecimal): 06172 (where the 0617
is the message and the 2 is the checksum). The receiver would divide
0617 by 9 and see whether the remainder was 2.

4. Polynomical Arithmetic
While the division scheme described in the previous section is very
very similar to the checksumming schemes called CRC schemes, the CRC
schemes are in fact a bit weirder, and we need to delve into some
strange number systems to understand them.

The word you will hear all the time when dealing with CRC algorithms
is the word "polynomial". A given CRC algorithm will be said to be
using a particular polynomial, and CRC algorithms in general are said
to be operating using polynomial arithmetic. What does this mean?

Instead of the divisor, dividend (message), quotient, and remainder
(as described in the previous section) being viewed as positive
integers, they are viewed as polynomials with binary coefficients.
This is done by treating each number as a bit-string whose bits are
the coefficients of a polynomial. For example, the ordinary number 23
(decimal) is 17 (hex) and 10111 binary and so it corresponds to the

   1*x^4 + 0*x^3 + 1*x^2 + 1*x^1 + 1*x^0

or, more simply:

   x^4 + x^2 + x^1 + x^0

Using this technique, the message, and the divisor can be represented
as polynomials and we can do all our arithmetic just as before, except
that now it's all cluttered up with Xs. For example, suppose we wanted
to multiply 1101 by 1011. We can do this simply by multiplying the

(x^3 + x^2 + x^0)(x^3 + x^1 + x^0)
= (x^6 + x^4 + x^3
 + x^5 + x^3 + x^2
 + x^3 + x^1 + x^0) = x^6 + x^5 + x^4 + 3*x^3 + x^2 + x^1 + x^0

At this point, to get the right answer, we have to pretend that x is 2
and propagate binary carries from the 3*x^3 yielding

   x^7 + x^3 + x^2 + x^1 + x^0

It's just like ordinary arithmetic except that the base is abstracted
and brought into all the calculations explicitly instead of being
there implicitly. So what's the point?

The point is that IF we pretend that we DON'T know what x is, we CAN'T
perform the carries. We don't know that 3*x^3 is the same as x^4 + x^3
because we don't know that x is 2. In this true polynomial arithmetic
the relationship between all the coefficients is unknown and so the
coefficients of each power effectively become strongly typed;
coefficients of x^2 are effectively of a different type to
coefficients of x^3.

With the coefficients of each power nicely isolated, mathematicians
came up with all sorts of different kinds of polynomial arithmetics
simply by changing the rules about how coefficients work. Of these
schemes, one in particular is relevant here, and that is a polynomial
arithmetic where the coefficients are calculated MOD 2 and there is no
carry; all coefficients must be either 0 or 1 and no carries are
calculated. This is called "polynomial arithmetic mod 2". Thus,
returning to the earlier example:

(x^3 + x^2 + x^0)(x^3 + x^1 + x^0)
= (x^6 + x^4 + x^3
 + x^5 + x^3 + x^2
 + x^3 + x^1 + x^0)
= x^6 + x^5 + x^4 + 3*x^3 + x^2 + x^1 + x^0

Under the other arithmetic, the 3*x^3 term was propagated using the
carry mechanism using the knowledge that x=2. Under "polynomial
arithmetic mod 2", we don't know what x is, there are no carries, and
all coefficients have to be calculated mod 2. Thus, the result

= x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + x^0

As Knuth [Knuth81] says (p.400):

   "The reader should note the similarity between polynomial
   arithmetic and multiple-precision arithmetic (Section 4.3.1), where
   the radix b is substituted for x. The chief difference is that the
   coefficient u_k of x^k in polynomial arithmetic bears little or no
   relation to its neighboring coefficients x^{k-1} [and x^{k+1}], so
   the idea of "carrying" from one place to another is absent. In fact
   polynomial arithmetic modulo b is essentially identical to multiple
   precision arithmetic with radix b, except that all carries are

Thus polynomical arithmetic mod 2 is just binary arithmetic mod 2 with
no carries. While polynomials provide useful mathematical machinery in
more analytical approaches to CRC and error-correction algorithms, for
the purposes of exposition they provide no extra insight and some
encumbrance and have been discarded in the remainder of this document
in favour of direct manipulation of the arithmetical system with which
they are isomorphic: binary arithmetic with no carry.

5. Binary Arithmetic with No Carries
Having dispensed with polynomials, we can focus on the real arithmetic
issue, which is that all the arithmetic performed during CRC
calculations is performed in binary with no carries. Often this is
called polynomial arithmetic, but as I have declared the rest of this
document a polynomial free zone, we'll have to call it CRC arithmetic
instead. As this arithmetic is a key part of CRC calculations, we'd
better get used to it. Here we go:

Adding two numbers in CRC arithmetic is the same as adding numbers in
ordinary binary arithmetic except there is no carry. This means that
each pair of corresponding bits determine the corresponding output bit
without reference to any other bit positions. For example:


There are only four cases for each bit position:

   1+1=0  (no carry)

Subtraction is identical:



   0-1=1  (wraparound)

In fact, both addition and subtraction in CRC arithmetic is equivalent
to the XOR operation, and the XOR operation is its own inverse. This
effectively reduces the operations of the first level of power
(addition, subtraction) to a single operation that is its own inverse.
This is a very convenient property of the arithmetic.

By collapsing of addition and subtraction, the arithmetic discards any
notion of magnitude beyond the power of its highest one bit. While it
seems clear that 1010 is greater than 10, it is no longer the case
that 1010 can be considered to be greater than 1001. To see this, note
that you can get from 1010 to 1001 by both adding and subtracting the
same quantity:

   1010 = 1010 + 0011
   1010 = 1010 - 0011

This makes nonsense of any notion of order.

Having defined addition, we can move to multiplication and division.
Multiplication is absolutely straightforward, being the sum of the
first number, shifted in accordance with the second number.

      x 1011
     1111111  Note: The sum uses CRC addition

Division is a little messier as we need to know when "a number goes
into another number". To do this, we invoke the weak definition of
magnitude defined earlier: that X is greater than or equal to Y iff
the position of the highest 1 bit of X is the same or greater than the
position of the highest 1 bit of Y. Here's a fully worked division
(nicked from [Tanenbaum81]).

10011 ) 11010110110000
                  1110 = Remainder

That's really it. Before proceeding further, however, it's worth our
while playing with this arithmetic a bit to get used to it.

We've already played with addition and subtraction, noticing that they
are the same thing. Here, though, we should note that in this
arithmetic A+0=A and A-0=A. This obvious property is very useful

In dealing with CRC multiplication and division, it's worth getting a
feel for the concepts of MULTIPLE and DIVISIBLE.

If a number A is a multiple of B then what this means in CRC
arithmetic is that it is possible to construct A from zero by XORing
in various shifts of B. For example, if A was 0111010110 and B was 11,
we could construct A from B as follows:

                = .......11.
                + ....11....
                + ...11.....

However, if A is 0111010111, it is not possible to construct it out of
various shifts of B (can you see why? - see later) so it is said to be
not divisible by B in CRC arithmetic.

Thus we see that CRC arithmetic is primarily about XORing particular
values at various shifting offsets.

6. A Fully Worked Example
Having defined CRC arithmetic, we can now frame a CRC calculation as
simply a division, because that's all it is! This section fills in the
details and gives an example.

To perform a CRC calculation, we need to choose a divisor. In maths
marketing speak the divisor is called the "generator polynomial" or
simply the "polynomial", and is a key parameter of any CRC algorithm.
It would probably be more friendly to call the divisor something else,
but the poly talk is so deeply ingrained in the field that it would
now be confusing to avoid it. As a compromise, we will refer to the
CRC polynomial as the "poly". Just think of this number as a sort of
parrot. "Hello poly!"

You can choose any poly and come up with a CRC algorithm. However,
some polys are better than others, and so it is wise to stick with the
tried an tested ones. A later section addresses this issue.

The width (position of the highest 1 bit) of the poly is very
important as it dominates the whole calculation. Typically, widths of
16 or 32 are chosen so as to simplify implementation on modern
computers. The width of a poly is the actual bit position of the
highest bit. For example, the width of 10011 is 4, not 5. For the
purposes of example, we will chose a poly of 10011 (of width W of 4).

Having chosen a poly, we can proceed with the calculation. This is
simply a division (in CRC arithmetic) of the message by the poly. The
only trick is that W zero bits are appended to the message before the
CRC is calculated. Thus we have:

   Original message                : 1101011011
   Poly                            :      10011
   Message after appending W zeros : 11010110110000

Now we simply divide the augmented message by the poly using CRC
arithmetic. This is the same division as before:

            1100001010 = Quotient (nobody cares about the quotient)
10011 ) 11010110110000 = Augmented message (1101011011 + 0000)
=Poly  10011,,.,,....
                  1110 = Remainder = THE CHECKSUM!!!!

The division yields a quotient, which we throw away, and a remainder,
which is the calculated checksum. This ends the calculation.

Usually, the checksum is then appended to the message and the result
transmitted. In this case the transmission would be: 11010110111110.

At the other end, the receiver can do one of two things:

   a. Separate the message and checksum. Calculate the checksum for
      the message (after appending W zeros) and compare the two

   b. Checksum the whole lot (without appending zeros) and see if it
      comes out as zero!

These two options are equivalent. However, in the next section, we
will be assuming option b because it is marginally mathematically

A summary of the operation of the class of CRC algorithms:

   1. Choose a width W, and a poly G (of width W).
   2. Append W zero bits to the message. Call this M'.
   3. Divide M' by G using CRC arithmetic. The remainder is the checksum.

That's all there is to it.

7. Choosing A Poly
Choosing a poly is somewhat of a black art and the reader is referred
to [Tanenbaum81] (p.130-132) which has a very clear discussion of this
issue. This section merely aims to put the fear of death into anyone
who so much as toys with the idea of making up their own poly. If you
don't care about why one poly might be better than another and just
want to find out about high-speed implementations, choose one of the
arithmetically sound polys listed at the end of this section and skip
to the next section.

First note that the transmitted message T is a multiple of the poly.
To see this, note that 1) the last W bits of T is the remainder after
dividing the augmented (by zeros remember) message by the poly, and 2)
addition is the same as subtraction so adding the remainder pushes the
value up to the next multiple. Now note that if the transmitted
message is corrupted in transmission that we will receive T+E where E
is an error vector (and + is CRC addition (i.e. XOR)). Upon receipt of
this message, the receiver divides T+E by G. As T mod G is 0, (T+E)
mod G = E mod G. Thus, the capacity of the poly we choose to catch
particular kinds of errors will be determined by the set of multiples
of G, for any corruption E that is a multiple of G will be undetected.
Our task then is to find classes of G whose multiples look as little
like the kind of line noise (that will be creating the corruptions) as
possible. So let's examine the kinds of line noise we can expect.

SINGLE BIT ERRORS: A single bit error means E=1000...0000. We can
ensure that this class of error is always detected by making sure that
G has at least two bits set to 1. Any multiple of G will be
constructed using shifting and adding and it is impossible to
construct a value with a single bit by shifting an adding a single
value with more than one bit set, as the two end bits will always

TWO-BIT ERRORS: To detect all errors of the form 100...000100...000
(i.e. E contains two 1 bits) choose a G that does not have multiples
that are 11, 101, 1001, 10001, 100001, etc. It is not clear to me how
one goes about doing this (I don't have the pure maths background),
but Tanenbaum assures us that such G do exist, and cites G with 1 bits
(15,14,1) turned on as an example of one G that won't divide anything
less than 1...1 where ... is 32767 zeros.

ERRORS WITH AN ODD NUMBER OF BITS: We can catch all corruptions where
E has an odd number of bits by choosing a G that has an even number of
bits. To see this, note that 1) CRC multiplication is simply XORing a
constant value into a register at various offsets, 2) XORing is simply
a bit-flip operation, and 3) if you XOR a value with an even number of
bits into a register, the oddness of the number of 1 bits in the
register remains invariant. Example: Starting with E=111, attempt to
flip all three bits to zero by the repeated application of XORing in
11 at one of the two offsets (i.e. "E=E XOR 011" and "E=E XOR 110")
This is nearly isomorphic to the "glass tumblers" party puzzle where
you challenge someone to flip three tumblers by the repeated
application of the operation of flipping any two. Most of the popular
CRC polys contain an even number of 1 bits. (Note: Tanenbaum states
more specifically that all errors with an odd number of bits can be
caught by making G a multiple of 11).

BURST ERRORS: A burst error looks like E=000...000111...11110000...00.
That is, E consists of all zeros except for a run of 1s somewhere
inside. This can be recast as E=(10000...00)(1111111...111) where
there are z zeros in the LEFT part and n ones in the RIGHT part. To
catch errors of this kind, we simply set the lowest bit of G to 1.
Doing this ensures that LEFT cannot be a factor of G. Then, so long as
G is wider than RIGHT, the error will be detected. See Tanenbaum for a
clearer explanation of this; I'm a little fuzzy on this one. Note:
Tanenbaum asserts that the probability of a burst of length greater
than W getting through is (0.5)^W.

That concludes the section on the fine art of selecting polys.

Some popular polys are:
16 bits: (16,12,5,0)                                [X25 standard]
         (16,15,2,0)                                ["CRC-16"]
32 bits: (32,26,23,22,16,12,11,10,8,7,5,4,2,1,0)    [Ethernet]

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after ./configure and make install, launch ./cgminer –algo auto

  • 0.14MH/s with “c” SHA256 algorithm version. (the better )
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