For some very specific values, such as
FF FF FF FF 80 20 83 B8 ED
the CRC32 (using polynomial 0x04C11DB7 and pre and post-inversion) is 0xFFFFFFFF (crccalc.com).
Adding any number of trailing zeroes does not change the result (since that just multiplies the message polynomial).
My doubt is that, according to Wikipedia, post inversion was supposed to prevent just that:
A similar solution can be applied at the end of the message, inverting the CRC register before it is appended to the message. Again, any non-zero change will do; inverting all the bits (XORing with an all-ones pattern) is simply the most common.
That doesn't seem to be the case. Also, this answer by Mark Adler suggests that the post-inversion is just so the CRC of an empty message is 0x00000000.
Is the Wikipedia article incorrect or did I misunderstand something?
For any n-bit CRC and any current state of the CRC, there will exist a sequence of n bits in the message that will bring the internal CRC register to all zero bits. And many sequences of more than n bits that will do the same. From there on, any application of zero bits will leave the register all zeros.
That n-bit sequence is easily found. It is the internal CRC register bits themselves at that point. For example, the standard CRC-32 that you reference, when applied to the nine-byte message "123456789" is 0xcbf43926. Since the final exclusive-or is 0xffffffff, the internal CRC register at the end is the complement of that, 0x340bc6d9. This is a reflected CRC, so you need to feed that value starting from its least significant bit. Then you find that the CRC-32 of "123456789\xd9\xc6\x0b\x34" is 0xffffffff. Now I can follow that message with any number of zeros, and still get 0xffffffff. E.g. "123456789\xd9\xc6\x0b\x34\x00\x00\x00".
However that is the only such sequence of four bytes that will do that. In general, the probability of bringing the internal CRC register to all zeros given any appended sequence of n or more random bits will be 2-n. So unless you're being deliberate about it, this insensitivity to a subsequent sequence of zero bits will happen very infrequently.
Initializing the internal CRC register to a non-zero value, as many CRC definitions do, avoids this behavior at the very start of the process. It may not be unusual for the start of a message to be a sequence of zeros, so you would like for the CRC to be sensitive to the length of that sequence.
The final exclusive-or doesn't change the behavior. All it does is change the final CRC value that you would be stuck at once you arrive at the state of the internal CRC register being zero.
As you noted, the final exclusive-or is often set equal to the initial CRC register value, so that you get what some might consider to be a "nice" behavior that the CRC of an empty message is zero.
If the the CRC of the message before post-inversion is 0, such as the question's example, adding trailing zeroes won't change the CRC. The post-inversion detection of trailing zeroes only works when the CRC before post inversion is not 0.
Related
Can anyone with good knowledge of CRC calculation verify that this code
https://github.com/psvanstrom/esphome-p1reader/blob/main/p1reader.h#L120
is actually calculating crc according to this description?
CRC is a CRC16 value calculated over the preceding characters in the data message (from
“/” to “!” using the polynomial: x16+x15+x2
+1). CRC16 uses no XOR in, no XOR out and is
computed with least significant bit first. The value is represented as 4 hexadecimal characters (MSB first).
There's nothing in the linked code about where it starts and ends, and how the result is eventually represented, but yes, that code implements that specification.
Are there CRC algorithms in which the remainder is checked for zero during the generation, and changed when that happens?
When calculating a CRC, if you start with an initial value (the CRC value, aka the remainder value) of 0, leading zeroes in the data being CRCd will not have an effect. So the CRC for "\0\0\0Hello" will be the same as "Hello".
https://xcore.github.io/doc_tips_and_tricks/crc.html#the-initial-value
This means that if, as the CRC is being calculated, the value becomes zero at some point, any zeroes immediately following that point will have no effect on the CRC. If one or more of the zeroes is lost, the CRC will not be changed.
I want to generate a CRC for some data, so that I can determine when a byte is lost. Is this an esoteric application of the CRC? Because when looking for examples of calculating a CRC, I have not found any examples where the CRC value is checked for 0 during the computation, and some non-zero bits fed in a that point. I would think that this would be the only way to be sure to detect a loss of one or more bits at a point in the data, if the CRC up to that point happened to be 0.
To detect and/or correct dropped bits, something like Marker Code or Watermark Code is needed.
https://link.springer.com/article/10.1007/BF03219806
https://ieeexplore.ieee.org/document/866775
Marker codes and Watermark codes are usually implemented in hardware, such as the read / write logic in a hard drive, and operate at the bit level and the media interface. Additional bits are intermixed with data bits to allow detection of lost synchronization or signal, which could otherwise lead to dropped or inserted bits. Here is a better link that describes this. The turbopaper.pdf file mentions LDPC, low density parity codes, which Wiki has an article for. Note that when receiving or reading data, the Marker | Watermark code is handled first at the hardware interface, and the LDPC (or CRC or Reed Solomon) is done after the hardware has detected no dropped or inserted bits.
http://www.inference.org.uk/ear23/turbopaper.pdf
https://en.wikipedia.org/wiki/Low-density_parity-check_code
If you did that, then it wouldn't be a CRC anymore. So, no. By definition there are no such CRC algorithms.
I've seen 8-bit, 16-bit, and 32-bit CRCs.
At what point do I need to jump to a wider CRC?
My gut reaction is that it is based on the data length:
1-100 bytes: 8-bit CRC
101 - 1000 bytes: 16-bit CRC
1001 - ??? bytes: 32-bit CRC
EDIT:
Looking at the Wikipedia page about CRC and Lott's answer, here' what we have:
<64 bytes: 8-bit CRC
<16K bytes: 16-bit CRC
<512M bytes: 32-bit CRC
It's not a research topic. It's really well understood: http://en.wikipedia.org/wiki/Cyclic_redundancy_check
The math is pretty simple. An 8-bit CRC boils all messages down to one of 256 values. If your message is more than a few bytes long, the possibility of multiple messages having the same hash value goes up higher and higher.
A 16-bit CRC, similarly, gives you one of the 65,536 available hash values. What are the odds of any two messages having one of these values?
A 32-bit CRC gives you about 4 billion available hash values.
From the wikipedia article: "maximal total blocklength is equal to 2**r − 1". That's in bits. You don't need to do much research to see that 2**9 - 1 is 511 bits. Using CRC-8, multiple messages longer than 64 bytes will have the same CRC checksum value.
The effectiveness of a CRC is dependent on multiple factors. You not only need to select the SIZE of the CRC but also the GENERATING POLYNOMIAL to use. There are complicated and non-intuitive trade-offs depending on:
The expected bit error rate of the channel.
Whether the errors tend to occur in bursts or tend to be spread out (burst is common)
The length of the data to be protected - maximum length, minimum length and distribution.
The paper Cyclic Redundancy Code Polynominal Selection For Embedded Networks, by Philip Koopman and Tridib Chakravarty, publised in the proceedings of the 2004 International Conference on Dependable Systems and Networks gives a very good overview and makes several recomendations. It also provides a bibliography for further understanding.
http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf
The choice of CRC length versus file size is mainly relevant in cases where one is more likely to have an input which differs from the "correct" input by three or fewer bits than to have a one which is massively different. Given two inputs which are massively different, the possibility of a false match will be about 1/256 with most forms of 8-bit check value (including CRC), 1/65536 with most forms of 16-bit check value (including CRC), etc. The advantage of CRC comes from its treatment of inputs which are very similar.
With an 8-bit CRC whose polynomial generates two periods of length 128, the fraction of single, double, or triple bit errors in a packet shorter than that which go undetected won't be 1/256--it will be zero. Likewise with a 16-bit CRC of period 32768, using packets of 32768 bits or less.
If packets are longer than the CRC period, however, then a double-bit error will go undetected if the distance between the erroneous bits is a multiple of the CRC period. While that might not seem like a terribly likely scenario, a CRC8 will be somewhat worse at catching double-bit errors in long packets than at catching "packet is totally scrambled" errors. If double-bit errors are the second most common failure mode (after single-bit errors), that would be bad. If anything that corrupts some data is likely to corrupt a lot of it, however, the inferior behavior of CRCs with double-bit errors may be a non-issue.
I think the size of the CRC has more to do with how unique of a CRC you need instead of of the size of the input data. This is related to the particular usage and number of items on which you're calculating a CRC.
The CRC should be chosen specifically for the length of the messages, it is not just a question of the size of the CRC: http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf
Here is a nice "real world" evaluation of CRC-N
http://www.backplane.com/matt/crc64.html
I use CRC-32 and file-size comparison and have NEVER, in the billions of files checked, run into a matching CRC-32 and File-Size collision. But I know a few exist, when not purposely forced to exist. (Hacked tricks/exploits)
When doing comparison, you should ALSO be checking "data-sizes". You will rarely have a collision of the same data-size, with a matching CRC, within the correct sizes.
Purposely manipulated data, to fake a match, is usually done by adding extra-data until the CRC matches a target. However, that results in a data-size that no-longer matches. Attempting to brute-force, or cycle through random, or sequential data, of the same exact size, would leave a real narrow collision-rate.
You can also have collisions within the data-size, just by the generic limits of the formulas used, and constraints of using bits/bytes and base-ten systems, which depends on floating-point values, which get truncated and clipped.
The point you would want to think about going larger, is when you start to see many collisions which can not be "confirmed" as "originals". (When they both have the same data-size, and (when tested backwards, they have a matching CRC. Reverse/byte or reverse/bits, or bit-offsets)
In any event, it should NEVER be used as the ONLY form of comparison, just for a quick form of comparison, for indexing.
You can use a CRC-8 to index the whole internet, and divide everything into one of N-catagories. You WANT those collisions. Now, with those pre-sorted, you only have to check one of N-directories, looking for "file-size", or "reverse-CRC", or whatever other comparison you can do to that smaller data-set, fast...
Doing a CRC-32 forwards and backwards on the same blob of data is more reliable than using CRC-64 in just one direction. (Or an MD5, for that matter.)
You can detect a single bit error with a CRC in any size packet. Detecting double bit errors or correction of single bit errors is limited to the number of distinct values the CRC can take, so for 8 bits, that would 256; for 16 bits, 65535; etc. 2^n; In practice, though, CRCs actually take on fewer distinct values for single bit errors. For example what I call the 'Y5' polynomial, the 0x5935 polynomial only takes on up to 256 different values before they repeat going back farther, but on the other hand it is able to correct double bit errors that distance, which is 30 bytes plus 2 bytes for errors in the CRC itself.
The number of bits you can correct with forward error correction is also limited by the Hamming Distance of the polynomial. For example, if the Hamming distance is three, you have to flip three bits to change from a set of bits that represents one valid message with matching CRC to another valid message with its own matching CRC. If that is the case, you can correct one bit with confidence. If the Hamming distance were 5, you could correct two bits. But when correcting multiple bits, you are effectively indexing multiple positions, so you need twice as many bits to represent the indexes of two corrected bits rather than one.
With forward error correction, you calculate the CRC on a packet and CRC together, and get a residual value. A good message with zero errors will always have the expected residual value (zero unless there's a nonzero initial value for the CRC register), and each bit position of error has a unique residual value, so use it to identify the position. If you ever get a CRC result with that residual, you know which bit (or bits) to flip to correct the error.
I know that CRC is a linear function which means CRC(x xor y) = CRC(x) xor CRC(y), but I don't know how to prove this property for CRC.
Does anyone have any idea?
Thanks a lot!
That is not generally true. It is only true for CRCs that have the property that a CRC of a string of zeros is always zero. (That property is easily derived from your equation.) Most CRCs have pre and post processing, for which one of the purposes of the pre-processing is to assure that that is not the case. You wouldn't want a check algorithm to not be able to distinguish how many zeros there are in a string of zeros. Similarly, for such a check algorithm you could prepend any number of zeros to a message with no change in the check value.
A "pure" CRC without pre or post processing does have the linearity property you define. This can be seen by looking at what CRC register implementation does with a single bit and how that changes if you invert the bit. The one bit rolled off of one end of the register, which is determined by the bit you fed into the other end, determines if the register is exclusive-ored with the polynomial word. If that bit is inverted, that reverses that decision. So the exclusive-or of those two CRCs is the polynomial word. If you feed a single one bit out to that end of the register initialized as zero (this is where the no pre-processing is important), you get the polynomial word. So the CRC of the exclusive-or of the messages is equal to the exclusive-or of the CRCs. This is then extended to all bits by applying this finding one bit at a time.
I've seen 8-bit, 16-bit, and 32-bit CRCs.
At what point do I need to jump to a wider CRC?
My gut reaction is that it is based on the data length:
1-100 bytes: 8-bit CRC
101 - 1000 bytes: 16-bit CRC
1001 - ??? bytes: 32-bit CRC
EDIT:
Looking at the Wikipedia page about CRC and Lott's answer, here' what we have:
<64 bytes: 8-bit CRC
<16K bytes: 16-bit CRC
<512M bytes: 32-bit CRC
It's not a research topic. It's really well understood: http://en.wikipedia.org/wiki/Cyclic_redundancy_check
The math is pretty simple. An 8-bit CRC boils all messages down to one of 256 values. If your message is more than a few bytes long, the possibility of multiple messages having the same hash value goes up higher and higher.
A 16-bit CRC, similarly, gives you one of the 65,536 available hash values. What are the odds of any two messages having one of these values?
A 32-bit CRC gives you about 4 billion available hash values.
From the wikipedia article: "maximal total blocklength is equal to 2**r − 1". That's in bits. You don't need to do much research to see that 2**9 - 1 is 511 bits. Using CRC-8, multiple messages longer than 64 bytes will have the same CRC checksum value.
The effectiveness of a CRC is dependent on multiple factors. You not only need to select the SIZE of the CRC but also the GENERATING POLYNOMIAL to use. There are complicated and non-intuitive trade-offs depending on:
The expected bit error rate of the channel.
Whether the errors tend to occur in bursts or tend to be spread out (burst is common)
The length of the data to be protected - maximum length, minimum length and distribution.
The paper Cyclic Redundancy Code Polynominal Selection For Embedded Networks, by Philip Koopman and Tridib Chakravarty, publised in the proceedings of the 2004 International Conference on Dependable Systems and Networks gives a very good overview and makes several recomendations. It also provides a bibliography for further understanding.
http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf
The choice of CRC length versus file size is mainly relevant in cases where one is more likely to have an input which differs from the "correct" input by three or fewer bits than to have a one which is massively different. Given two inputs which are massively different, the possibility of a false match will be about 1/256 with most forms of 8-bit check value (including CRC), 1/65536 with most forms of 16-bit check value (including CRC), etc. The advantage of CRC comes from its treatment of inputs which are very similar.
With an 8-bit CRC whose polynomial generates two periods of length 128, the fraction of single, double, or triple bit errors in a packet shorter than that which go undetected won't be 1/256--it will be zero. Likewise with a 16-bit CRC of period 32768, using packets of 32768 bits or less.
If packets are longer than the CRC period, however, then a double-bit error will go undetected if the distance between the erroneous bits is a multiple of the CRC period. While that might not seem like a terribly likely scenario, a CRC8 will be somewhat worse at catching double-bit errors in long packets than at catching "packet is totally scrambled" errors. If double-bit errors are the second most common failure mode (after single-bit errors), that would be bad. If anything that corrupts some data is likely to corrupt a lot of it, however, the inferior behavior of CRCs with double-bit errors may be a non-issue.
I think the size of the CRC has more to do with how unique of a CRC you need instead of of the size of the input data. This is related to the particular usage and number of items on which you're calculating a CRC.
The CRC should be chosen specifically for the length of the messages, it is not just a question of the size of the CRC: http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf
Here is a nice "real world" evaluation of CRC-N
http://www.backplane.com/matt/crc64.html
I use CRC-32 and file-size comparison and have NEVER, in the billions of files checked, run into a matching CRC-32 and File-Size collision. But I know a few exist, when not purposely forced to exist. (Hacked tricks/exploits)
When doing comparison, you should ALSO be checking "data-sizes". You will rarely have a collision of the same data-size, with a matching CRC, within the correct sizes.
Purposely manipulated data, to fake a match, is usually done by adding extra-data until the CRC matches a target. However, that results in a data-size that no-longer matches. Attempting to brute-force, or cycle through random, or sequential data, of the same exact size, would leave a real narrow collision-rate.
You can also have collisions within the data-size, just by the generic limits of the formulas used, and constraints of using bits/bytes and base-ten systems, which depends on floating-point values, which get truncated and clipped.
The point you would want to think about going larger, is when you start to see many collisions which can not be "confirmed" as "originals". (When they both have the same data-size, and (when tested backwards, they have a matching CRC. Reverse/byte or reverse/bits, or bit-offsets)
In any event, it should NEVER be used as the ONLY form of comparison, just for a quick form of comparison, for indexing.
You can use a CRC-8 to index the whole internet, and divide everything into one of N-catagories. You WANT those collisions. Now, with those pre-sorted, you only have to check one of N-directories, looking for "file-size", or "reverse-CRC", or whatever other comparison you can do to that smaller data-set, fast...
Doing a CRC-32 forwards and backwards on the same blob of data is more reliable than using CRC-64 in just one direction. (Or an MD5, for that matter.)
You can detect a single bit error with a CRC in any size packet. Detecting double bit errors or correction of single bit errors is limited to the number of distinct values the CRC can take, so for 8 bits, that would 256; for 16 bits, 65535; etc. 2^n; In practice, though, CRCs actually take on fewer distinct values for single bit errors. For example what I call the 'Y5' polynomial, the 0x5935 polynomial only takes on up to 256 different values before they repeat going back farther, but on the other hand it is able to correct double bit errors that distance, which is 30 bytes plus 2 bytes for errors in the CRC itself.
The number of bits you can correct with forward error correction is also limited by the Hamming Distance of the polynomial. For example, if the Hamming distance is three, you have to flip three bits to change from a set of bits that represents one valid message with matching CRC to another valid message with its own matching CRC. If that is the case, you can correct one bit with confidence. If the Hamming distance were 5, you could correct two bits. But when correcting multiple bits, you are effectively indexing multiple positions, so you need twice as many bits to represent the indexes of two corrected bits rather than one.
With forward error correction, you calculate the CRC on a packet and CRC together, and get a residual value. A good message with zero errors will always have the expected residual value (zero unless there's a nonzero initial value for the CRC register), and each bit position of error has a unique residual value, so use it to identify the position. If you ever get a CRC result with that residual, you know which bit (or bits) to flip to correct the error.