IEN 45
Section 2.4.4.5
TCP Checksum Function Design
William W. Plummer
Bolt Beranek and Newman, Inc.
50 Moulton Street
Cambridge MA 02138
5 June 1978
Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
1. Introduction
Checksums are included in packets in order that errors
encountered during transmission may be detected. For Internet
protocols such as TCP [1,9] this is especially important because
packets may have to cross wireless networks such as the Packet
Radio Network [2] and Atlantic Satellite Network [3] where
packets may be corrupted. Internet protocols (e.g., those for
real time speech transmission) can tolerate a certain level of
transmission errors and forward error correction techniques or
possibly no checksum at all might be better. The focus in this
paper is on checksum functions for protocols such as TCP where
the required reliable delivery is achieved by retransmission.
Even if the checksum appears good on a message which has been
received, the message may still contain an undetected error. The
probability of this is bounded by 2**(-C) where C is the number
of checksum bits. Errors can arise from hardware (and software)
malfunctions as well as transmission errors. Hardware induced
errors are usually manifested in certain well known ways and it
is desirable to account for this in the design of the checksum
function. Ideally no error of the "common hardware failure" type
would go undetected.
An example of a failure that the current checksum function
handles successfully is picking up a bit in the network interface
(or I/O buss, memory channel, etc.). This will always render the
checksum bad. For an example of how the current function is
inadequate, assume that a control signal stops functioning in the
network interface and the interface stores zeros in place of the
real data. These "all zero" messages appear to have valid
checksums. Noise on the "There's Your Bit" line of the ARPANET
Interface [4] may go undetected because the extra bits input may
cause the checksum to be perturbed (i.e., shifted) in the same
way as the data was.
Although messages containing undetected errors will occasionally
be passed to higher levels of protocol, it is likely that they
will not make sense at that level. In the case of TCP most such
messages will be ignored, but some could cause a connection to be
aborted. Garbled data could be viewed as a problem for a layer
of protocol above TCP which itself may have a checksuming scheme.
This paper is the first step in design of a new checksum function
for TCP and some other Internet protocols. Several useful
properties of the current function are identified. If possible
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Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
these should be retained in any new function. A number of
plausible checksum schemes are investigated. Of these only the
"product code" seems to be simple enough for consideration.
2. The Current TCP Checksum Function
The current function is oriented towards sixteen-bit machines
such as the PDP-11 but can be computed easily on other machines
(e.g., PDP-10). A packet is thought of as a string of 16-bit
bytes and the checksum function is the one's complement sum (add
with end-around carry) of those bytes. It is the one's
complement of this sum which is stored in the checksum field of
the TCP header. Before computing the checksum value, the sender
places a zero in the checksum field of the packet. If the
checksum value computed by a receiver of the packet is zero, the
packet is assumed to be valid. This is a consequence of the
"negative" number in the checksum field exactly cancelling the
contribution of the rest of the packet.
Ignoring the difficulty of actually evaluating the checksum
function for a given packet, the way of using the checksum
described above is quite simple, but it assumes some properties
of the checksum operator (one's complement addition, "+" in what
follows):
(P1) + is commutative. Thus, the order in which
the 16-bit bytes are "added" together is
unimportant.
(P2) + has at least one identity element (The
current function has two: +0 and -0). This
allows the sender to compute the checksum
function by placing a zero in the packet checksum
field before computing the value.
(P3) + has an inverse. Thus, the receiver may
evaluate the checksum function and expect a zero.
(P4) + is associative, allowing the checksum field
to be anywhere in the packet and the 16-bit bytes
to be scanned sequentially.
Mathematically, these properties of the binary operation "+" over
the set of 16-bit numbers forms an Abelian group [5]. Of course,
there are many Abelian groups but not all would be satisfactory
for use as checksum operators. (Another operator readily
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TCP Checksum Function Design William W. Plummer
available in the PDP-11 instruction set that has all of these
properties is exclusive-OR, but XOR is unsatisfactory for other
reasons.)
Albeit imprecise, another property which must be preserved in any
future checksum scheme is:
(P5) + is fast to compute on a variety of machines
with limited storage requirements.
The current function is quite good in this respect. On the
PDP-11 the inner loop looks like:
LOOP: ADD (R1)+,R0 ; Add the next 16-bit byte
ADC R0 ; Make carry be end-around
SOB R2,LOOP ; Loop over entire packet.
( 4 memory cycles per 16-bit byte )
On the PDP-10 properties P1-4 are exploited further and two
16-bit bytes per loop are processed:
LOOP: ILDB THIS,PTR ; Get 2 16-bit bytes
ADD SUM,THIS ; Add into current sum
JUMPGE SUM,CHKSU2 ; Jump if fewer than 8 carries
LDB THIS,[POINT 20,SUM,19] ; Get left 16 and carries
ANDI SUM,177777 ; Save just low 16 here
ADD SUM,THIS ; Fold in carries
CHKSU2: SOJG COUNT,LOOP ; Loop over entire packet
( 3.1 memory cycles per 16-bit byte )
The "extra" instruction in the loops above are required to
convert the two's complement ADD instruction(s) into a one's
complement add by making the carries be end-around. One's
complement arithmetic is better than two's complement because it
is equally sensitive to errors in all bit positions. If two's
complement addition were used, an even number of 1's could be
dropped (or picked up) in the most significant bit channel
without affecting the value of the checksum. It is just this
property that makes some sort of addition preferable to a simple
exclusive-OR which is frequently used but permits an even number
of drops (pick ups) in any bit channel. RIM10B paper tape format
used on PDP-10s [10] uses two's complement add because space for
the loader program is extremely limited.
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Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
Another property of the current checksum scheme is:
(P6) Adding the checksum to a packet does not change
the information bytes. Peterson [6] calls this a
"systematic" code.
This property allows intermediate computers such as gateway
machines to act on fields (i.e., the Internet Destination
Address) without having to first decode the packet. Cyclical
Redundancy Checks used for error correction are not systematic
either. However, most applications of CRCs tend to emphasize
error detection rather than correction and consequently can send
the message unchanged, with the CRC check bits being appended to
the end. The 24-bit CRC used by ARPANET IMPs and Very Distant
Host Interfaces [4] and the ANSI standards for 800 and 6250 bits
per inch magnetic tapes (described in [11]) use this mode.
Note that the operation of higher level protocols are not (by
design) affected by anything that may be done by a gateway acting
on possibly invalid packets. It is permissible for gateways to
validate the checksum on incoming packets, but in general
gateways will not know how to do this if the checksum is a
protocol-specific feature.
A final property of the current checksum scheme which is actually
a consequence of P1 and P4 is:
(P7) The checksum may be incrementally modified.
This property permits an intermediate gateway to add information
to a packet, for instance a timestamp, and "add" an appropriate
change to the checksum field of the packet. Note that the
checksum will still be end-to-end since it was not fully
recomputed.
3. Product Codes
Certain "product codes" are potentially useful for checksuming
purposes. The following is a brief description of product codes
in the context of TCP. More general treatment can be found in
Avizienis [7] and probably other more recent works.
The basic concept of this coding is that the message (packet) to
be sent is formed by transforming the original source message and
adding some "check" bits. By reading this and applying a
(possibly different) transformation, a receiver can reconstruct
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TCP Checksum Function Design William W. Plummer
the original message and determine if it has been corrupted
during transmission.
Mo Ms Mr
----- ----- -----
| A | code | 7 | decode | A |
| B | ==> | 1 | ==> | B |
| C | | 4 | | C |
----- |...| -----
| 2 | check plus "valid" flag
----- info
Original Sent Reconstructed
With product codes the transformation is Ms = K * Mo . That is,
the message sent is simply the product of the original message
Mo and some well known constant K . To decode, the received
Ms is divided by K which will yield Mr as the quotient and
0 as the remainder if Mr is to be considered the same as Mo .
The first problem is selecting a "good" value for K, the "check
factor". K must be relatively prime to the base chosen to
express the message. (Example: Binary messages with K
incorrectly chosen to be 8. This means that Ms looks exactly
like Mo except that three zeros have been appended. The only
way the message could look bad to a receiver dividing by 8 is if
the error occurred in one of those three bits.)
For TCP the base R will be chosen to be 2**16. That is, every
16-bit byte (word on the PDP-11) will be considered as a digit of
a big number and that number is the message. Thus,
Mo = SIGMA [ Bi * (R**i)] , Bi is i-th byte
i=0 to N
Ms = K * Mo
Corrupting a single digit of Ms will yield Ms' = Ms +or-
C*(R**j) for some radix position j . The receiver will compute
Ms'/K = Mo +or- C(R**j)/K. Since R and K are relatively prime,
C*(R**j) cannot be any exact multiple of K. Therefore, the
division will result in a non-zero remainder which indicates that
Ms' is a corrupted version of Ms. As will be seen, a good
choice for K is (R**b - 1), for some b which is the "check
length" which controls the degree of detection to be had for
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Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
burst errors which affect a string of digits (i.e., 16-bit bytes)
in the message. In fact b will be chosen to be 1, so K will
be 2**16 - 1 so that arithmetic operations will be simple. This
means that all bursts of 15 or fewer bits will be detected.
According to [7] this choice for b results in the following
expression for the fraction of undetected weight 2 errors:
f = 16(k-1)/[32(16k-3) + (6/k)] where k is the message length.
For large messages f approaches 3.125 per cent as k goes to
infinity.
Multiple precision multiplication and division are normally quite
complex operations, especially on small machines which typically
lack even single precision multiply and divide operations. The
exception to this is exactly the case being dealt with here --
the factor is 2**16 - 1 on machines with a word length of 16
bits. The reason for this is due to the following identity:
Q*(R**j) = Q, mod (R-1) 0 <= Q < R
That is, any digit Q in the selected radix (0, 1, ... R-1)
multiplied by any power of the radix will have a remainder of Q
when divided by the radix minus 1.
Example: In decimal R = 10. Pick Q = 6.
6 = 0 * 9 + 6 = 6, mod 9
60 = 6 * 9 + 6 = 6, mod 9
600 = 66 * 9 + 6 = 6, mod 9 etc.
More to the point,
rem(31415/9) = rem((30000+1000+400+10+5)/9)
= (3 mod 9) + (1 mod 9) + (4 mod 9) + (1 mod 9) + (5 mod 9)
= (3+1+4+1+5) mod 9
= 14 mod 9
= 5
So, the remainder of a number divided by the radix minus one can
be found by simply summing the digits of the number. Since the
radix in the TCP case has been chosen to be 2**16 and the check
factor is 2**16 - 1, a message can quickly be checked by summing
all of the 16-bit words (on a PDP-11), with carries being
end-around. If zero is the result, the message can be considered
valid. Thus, checking a product coded message is exactly the
same complexity as with the current TCP checksum!
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Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
In order to form Ms, the sender must multiply the multiple
precision "number" Mo by 2**16 - 1. Or, Ms = (2**16)Mo - Mo.
This is performed by shifting Mo one whole word's worth of
precision and subtracting Mo. Since carries must propagate
between digits, but it is only the current digit which is of
interest, one's complement arithmetic is used.
(2**16)Mo = Mo0 + Mo1 + Mo2 + ... + MoX + 0
- Mo = - ( Mo0 + Mo1 + ......... + MoX)
--------- ----------------------------------
Ms = Ms0 + Ms1 + ... - MoX
A loop which implements this function on a PDP-11 might look
like:
LOOP: MOV -2(R2),R0 ; Next byte of (2**16)Mo
SBC R0 ; Propagate carries from last SUB
SUB (R2)+,R0 ; Subtract byte of Mo
MOV R0,(R3)+ ; Store in Ms
SOB R1,LOOP ; Loop over entire message
( 8 memory cycles per 16-bit byte)
Note that the coding procedure is not done in-place since it is
not systematic. In general the original copy, Mo, will have to
be retained by the sender for retransmission purposes and
therefore must remain readable. Thus the MOV R0,(R3)+ is
required which accounts for 2 of the 8 memory cycles per loop.
The coding procedure will add exactly one 16-bit word to the
message since Ms < (2**16)Mo . This additional 16 bits will be
at the tail of the message, but may be moved into the defined
location in the TCP header immediately before transmission. The
receiver will have to undo this to put Ms back into standard
format before decoding the message.
The code in the receiver for fully decoding the message may be
inferred by observing that any word in Ms contains the
difference between two successive words of Mo minus the carries
from the previous word, and the low order word contains minus the
low word of Mo. So the low order (i.e., rightmost) word of Mr is
just the negative of the low order byte of Ms. The next word of
Mr is the next word of Ms plus the just computed word of Mr
plus the carry from that previous computation.
A slight refinement of the procedure is required in order to
protect against an all-zero message passing to the destination.
This will appear to have a valid checksum because Ms'/K = 0/K
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TCP Checksum Function Design William W. Plummer
= 0 with 0 remainder. The refinement is to make the coding be
Ms = K*Mo + C where C is some arbitrary, well-known constant.
Adding this constant requires a second pass over the message, but
this will typically be very short since it can stop as soon as
carries stop propagating. Chosing C = 1 is sufficient in most
cases.
The product code checksum must be evaluated in terms of the
desired properties P1 - P7. It has been shown that a factor of
two more machine cycles are consumed in computing or verifying a
product code checksum (P5 satisfied?).
Although the code is not systematic, the checksum can be verified
quickly without decoding the message. If the Internet
Destination Address is located at the least significant end of
the packet (where the product code computation begins) then it is
possible for a gateway to decode only enough of the message to
see this field without having to decode the entire message.
Thus, P6 is at least partially satisfied. The algebraic
properties P1 through P4 are not satisfied, but only a small
amount of computation is needed to account for this -- the
message needs to be reformatted as previously mentioned.
P7 is satisfied since the product code checksum can be
incrementally updated to account for an added word, although the
procedure is somewhat involved. Imagine that the original
message has two halves, H1 and H2. Thus, Mo = H1*(R**j) + H2.
The timestamp word is to be inserted between these halves to form
a modified Mo' = H1*(R**(j+1)) + T*(R**j) + H2. Since K has
been chosen to be R-1, the transmitted message Ms' = Mo'(R-1).
Then,
Ms' = Ms*R + T(R-1)(R**j) + P2((R-1)**2)
= Ms*R + T*(R**(j+1)) + T*(R**j) + P2*(R**2) - 2*P2*R - P2
Recalling that R is 2**16, the word size on the PDP-11,
multiplying by R means copying down one word in memory. So,
the first term of Ms' is simply the unmodified message copied
down one word. The next term is the new data T added into the
Ms' being formed beginning at the (j+1)th word. The addition is
fairly easy here since after adding in T all that is left is
propagating the carry, and that can stop as soon as no carry is
produced. The other terms can be handle similarly.
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Internet Experiment Note 45 5 June 1978 TCP Checksum Function Design William W. Plummer 4. More Complicated Codes There exists a wealth of theory on error detecting and correcting codes. Peterson [6] is an excellent reference. Most of these "CRC" schemes are designed to be implemented using a shift register with a feedback network composed of exclusive-ORs. Simulating such a logic circuit with a program would be too slow to be useful unless some programming trick is discovered. One such trick has been proposed by Kirstein [8]. Basically, a few bits (four or eight) of the current shift register state are combined with bits from the input stream (from Mo) and the result is used as an index to a table which yields the new shift register state and, if the code is not systematic, bits for the output stream (Ms). A trial coding of an especially "good" CRC function using four-bit bytes showed showed this technique to be about four times as slow as the current checksum function. This was true for both the PDP-10 and PDP-11 machines. Of the desirable properties listed above, CRC schemes satisfy only P3 (It has an inverse.), and P6 (It is systematic.). Placement of the checksum field in the packet is critical and the CRC cannot be incrementally modified. Although the bulk of coding theory deals with binary codes, most of the theory works if the alphabet contains q symbols, where q is a power of a prime number. For instance q taken as 2**16 should make a great deal of the theory useful on a word-by-word basis. 5. Outboard Processing When a function such as computing an involved checksum requires extensive processing, one solution is to put that processing into an outboard processor. In this way "encode message" and "decode message" become single instructions which do not tax the main host processor. The Digital Equipment Corporation VAX/780 computer is equipped with special hardware for generating and checking CRCs [13]. In general this is not a very good solution since such a processor must be constructed for every different host machine which uses TCP messages. It is conceivable that the gateway functions for a large host may be performed entirely in an "Internet Frontend Machine". This machine would be responsible for forwarding packets received - 9 -
Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
either from the network(s) or from the Internet protocol modules
in the connected host, and for reassembling Internet fragments
into segments and passing these to the host. Another capability
of this machine would be to check the checksum so that the
segments given to the host are known to be valid at the time they
leave the frontend. Since computer cycles are assumed to be both
inexpensive and available in the frontend, this seems reasonable.
The problem with attempting to validate checksums in the frontend
is that it destroys the end-to-end character of the checksum. If
anything, this is the most powerful feature of the TCP checksum!
There is a way to make the host-to-frontend link be covered by
the end-to-end checksum. A separate, small protocol must be
developed to cover this link. After having validated an incoming
packet from the network, the frontend would pass it to the host
saying "here is an Internet segment for you. Call it #123". The
host would save this segment, and send a copy back to the
frontend saying, "Here is what you gave me as #123. Is it OK?".
The frontend would then do a word-by-word comparison with the
first transmission, and tell the host either "Here is #123
again", or "You did indeed receive #123 properly. Release it to
the appropriate module for further processing."
The headers on the messages crossing the host-frontend link would
most likely be covered by a fairly strong checksum so that
information like which function is being performed and the
message reference numbers are reliable. These headers would be
quite short, maybe only sixteen bits, so the checksum could be
quite strong. The bulk of the message would not be checksumed of
course.
The reason this scheme reduces the computing burden on the host
is that all that is required in order to validate the message
using the end-to-end checksum is to send it back to the frontend
machine. In the case of the PDP-10, this requires only 0.5
memory cycles per 16-bit byte of Internet message, and only a few
processor cycles to setup the required transfers.
6. Conclusions
There is an ordering of checksum functions: first and simplest is
none at all which provides no error detection or correction.
Second, is sending a constant which is checked by the receiver.
This also is extremely weak. Third, the exclusive-OR of the data
may be sent. XOR takes the minimal amount of computer time to
generate and check, but is not a good checksum. A two's
complement sum of the data is somewhat better and takes no more
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TCP Checksum Function Design William W. Plummer
computer time to compute. Fifth, is the one's complement sum
which is what is currently used by TCP. It is slightly more
expensive in terms of computer time. The next step is a product
code. The product code is strongly related to one's complement
sum, takes still more computer time to use, provides a bit more
protection against common hardware failures, but has some
objectionable properties. Next is a genuine CRC polynomial code,
used for checking purposes only. This is very expensive for a
program to implement. Finally, a full CRC error correcting and
detecting scheme may be used.
For TCP and Internet applications the product code scheme is
viable. It suffers mainly in that messages must be (at least
partially) decoded by intermediate gateways in order that they
can be forwarded. Should product codes not be chosen as an
improved checksum, some slight modification to the existing
scheme might be possible. For instance the "add and rotate"
function used for paper tape by the PDP-6/10 group at the
Artificial Intelligence Laboratory at M.I.T. Project MAC [12]
could be useful if it can be proved that it is better than the
current scheme and that it can be computed efficiently on a
variety of machines.
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Internet Experiment Note 45 5 June 1978
TCP Checksum Function Design William W. Plummer
References
[1] Cerf, V.G. and Kahn, Robert E., "A Protocol for Packet
Network Communications," IEEE Transactions on Com-
munications, vol. COM-22, No. 5, May 1974.
[2] Kahn, Robert E., "The Organization of Computer Resources
into a Packet Radio Network", IEEE Transactions on
Communications, vol. COM-25, no. 1, pp. 169-178, January 1977.
[3] Jacobs, Irwin, et al., "CPODA - A Demand Assignment
Protocol for SatNet", Fifth Data Communications
Symposium, September 27-9, 1977, Snowbird, Utah
[4] Bolt Beranek and Newman, Inc. "Specifications for the
Interconnection of a Host and an IMP", Report 1822, January
1976 edition.
[5] Dean, Richard A., "Elements of Abstract Algebra",
John Wyley and Sons, Inc., 1966
[6] Peterson, W. Wesley, "Error Correcting Codes", M.I.T. Press
Cambridge MA, 4th edition, 1968.
[7] Avizienis, Algirdas, "A Study of the Effectiveness of
Fault-Detecting Codes for Binary Arithmetic", Jet
Propulsion Laboratory Technical Report No. 32-711,
September 1, 1965.
[8] Kirstein, Peter, private communication
[9] Cerf, V. G. and Postel, Jonathan B., "Specification
of Internetwork Transmission Control Program Version 3",
University of Southern California Information Sciences
Institute, January 1978.
[10] Digital Equipment Corporation, "PDP-10 Reference Handbook",
1970, pp. 114-5.
[11] Swanson, Robert, "Understanding Cyclic Redundancy Codes",
Computer Design, November, 1975, pp. 93-99.
[12] Clements, Robert C., private communication.
[13] Conklin, Peter F., and Rodgers, David P., "Advanced Minicomputer
Designed by Team Evaluation of Hardware/Software Tradeoffs",
Computer Design, April 1978, pp. 136-7.
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