### . BTC, LTC mining with Rasp PI , polynomials

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

A PAINLESS GUIDE TO CRC ERROR DETECTION ALGORITHMS ================================================== "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 ----------------- Abstract 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 Poly8-21 : -- check the original link if still available --Abstract -------- 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.,,....,.,,, ----.,,....,.,,, 0000,,....,.,,, 0000,,....,.,,, ----,,....,.,,, 0001,....,.,,, 0000,....,.,,, ----,....,.,,, 0011....,.,,, 0000....,.,,, ----....,.,,, 0110...,.,,, 0000...,.,,, ----...,.,,, 1100..,.,,, 1001..,.,,, ====..,.,,, 0110.,.,,, 0000.,.,,, ----.,.,,, 1100,.,,, 1001,.,,, ====,.,,, 0111.,,, 0000.,,, ----.,,, 1110,,, 1001,,, ====,,, 1011,, 1001,, ====,, 0101, 0000, ---- 1011 1001 ==== 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 polynomial: 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 polynomials: (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 becomes: = 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 suppressed." 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: 10011011 +11001010 -------- 01010001 -------- There are only four cases for each bit position: 0+0=0 0+1=1 1+0=1 1+1=0 (no carry) Subtraction is identical: 10011011 -11001010 -------- 01010001 -------- with 0-0=0 0-1=1 (wraparound) 1-0=1 1-1=0 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. 1101 x 1011 ---- 1101 1101. 0000.. 1101... ------- 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]). 1100001010 _______________ 10011 ) 11010110110000 10011,,.,,.... -----,,.,,.... 10011,.,,.... 10011,.,,.... -----,.,,.... 00001.,,.... 00000.,,.... -----.,,.... 00010,,.... 00000,,.... -----,,.... 00101,.... 00000,.... -----,.... 01011.... 00000.... -----.... 10110... 10011... -----... 01010.. 00000.. -----.. 10100. 10011. -----. 01110 00000 ----- 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 later. 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: 0111010110 = .......11. + ....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,,.,,.... -----,,.,,.... 10011,.,,.... 10011,.,,.... -----,.,,.... 00001.,,.... 00000.,,.... -----.,,.... 00010,,.... 00000,,.... -----,,.... 00101,.... 00000,.... -----,.... 01011.... 00000.... -----.... 10110... 10011... -----... 01010.. 00000.. -----.. 10100. 10011. -----. 01110 00000 ----- 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 checksums. 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 cleaner. 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 persist. 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]___

cgminer **CPU** mining for Raspberry PI : get cgminer 2.0.8 from http://ck.kolivas.org/apps/cgminer/2.0/

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

- 0.14MH/s with “c” SHA256 algorithm version. (the better )
- On a Nec-powermate VL6 , under Xubuntu, the same test gives 0.56MH/s ( Pentium 4 , 2.8GHz cpu )