Network Working Group R. Braden
Request for Comments: 1071 ISI
D. Borman
Cray Research
C. Partridge
BBN Laboratories
September 1988
Computing the Internet Checksum
Status of This Memo
This memo summarizes techniques and algorithms for efficiently
computing the Internet checksum. It is not a standard, but a set of
useful implementation techniques. Distribution of this memo is
unlimited.
1. IntrodUCtion
This memo discusses methods for efficiently computing the Internet
checksum that is used by the standard Internet protocols IP, UDP, and
TCP.
An efficient checksum implementation is critical to good performance.
As advances in implementation techniques streamline the rest of the
protocol processing, the checksum computation becomes one of the
limiting factors on TCP performance, for example. It is usually
appropriate to carefully hand-craft the checksum routine, eXPloiting
every machine-dependent trick possible; a fraction of a microsecond
per TCP data byte can add up to a significant CPU time savings
overall.
In outline, the Internet checksum algorithm is very simple:
(1) Adjacent octets to be checksummed are paired to form 16-bit
integers, and the 1's complement sum of these 16-bit integers is
formed.
(2) To generate a checksum, the checksum field itself is cleared,
the 16-bit 1's complement sum is computed over the octets
concerned, and the 1's complement of this sum is placed in the
checksum field.
(3) To check a checksum, the 1's complement sum is computed over the
same set of octets, including the checksum field. If the result
is all 1 bits (-0 in 1's complement arithmetic), the check
succeeds.
Suppose a checksum is to be computed over the sequence of octets
A, B, C, D, ... , Y, Z. Using the notation [a,b] for the 16-bit
integer a*256+b, where a and b are bytes, then the 16-bit 1's
complement sum of these bytes is given by one of the following:
[A,B] +' [C,D] +' ... +' [Y,Z] [1]
[A,B] +' [C,D] +' ... +' [Z,0] [2]
where +' indicates 1's complement addition. These cases
correspond to an even or odd count of bytes, respectively.
On a 2's complement machine, the 1's complement sum must be
computed by means of an "end around carry", i.e., any overflows
from the most significant bits are added into the least
significant bits. See the examples below.
Section 2 explores the properties of this checksum that may be
exploited to speed its calculation. Section 3 contains some
numerical examples of the most important implementation
techniques. Finally, Section 4 includes examples of specific
algorithms for a variety of common CPU types. We are grateful
to Van Jacobson and Charley Kline for their contribution of
algorithms to this section.
The properties of the Internet checksum were originally
discussed by Bill Plummer in IEN-45, entitled "Checksum Function
Design". Since IEN-45 has not been widely available, we include
it as an extended appendix to this RFC.
2. Calculating the Checksum
This simple checksum has a number of wonderful mathematical
properties that may be exploited to speed its calculation, as we
will now discuss.
(A) Commutative and Associative
As long as the even/odd assignment of bytes is respected, the
sum can be done in any order, and it can be arbitrarily split
into groups.
For example, the sum [1] could be split into:
( [A,B] +' [C,D] +' ... +' [J,0] )
+' ( [0,K] +' ... +' [Y,Z] ) [3]
(B) Byte Order Independence
The sum of 16-bit integers can be computed in either byte order.
Thus, if we calculate the swapped sum:
[B,A] +' [D,C] +' ... +' [Z,Y] [4]
the result is the same as [1], except the bytes are swapped in
the sum! To see why this is so, observe that in both orders the
carries are the same: from bit 15 to bit 0 and from bit 7 to bit
8. In other Words, consistently swapping bytes simply rotates
the bits within the sum, but does not affect their internal
ordering.
Therefore, the sum may be calculated in exactly the same way
regardless of the byte order ("big-endian" or "little-endian")
of the underlaying hardware. For example, assume a "little-
endian" machine summing data that is stored in memory in network
("big-endian") order. Fetching each 16-bit word will swap
bytes, resulting in the sum [4]; however, storing the result
back into memory will swap the sum back into network byte order.
Byte swapping may also be used explicitly to handle boundary
alignment problems. For example, the second group in [3] can be
calculated without concern to its odd/even origin, as:
[K,L] +' ... +' [Z,0]
if this sum is byte-swapped before it is added to the first
group. See the example below.
(C) Parallel Summation
On machines that have word-sizes that are multiples of 16 bits,
it is possible to develop even more efficient implementations.
Because addition is associative, we do not have to sum the
integers in the order they appear in the message. Instead we
can add them in "parallel" by exploiting the larger word size.
To compute the checksum in parallel, simply do a 1's complement
addition of the message using the native word size of the
machine. For example, on a 32-bit machine we can add 4 bytes at
a time: [A,B,C,D]+'... When the sum has been computed, we "fold"
the long sum into 16 bits by adding the 16-bit segments. Each
16-bit addition may produce new end-around carries that must be
added.
Furthermore, again the byte order does not matter; we could
instead sum 32-bit words: [D,C,B,A]+'... or [B,A,D,C]+'... and
then swap the bytes of the final 16-bit sum as necessary. See
the examples below. Any permutation is allowed that collects
all the even-numbered data bytes into one sum byte and the odd-
numbered data bytes into the other sum byte.
There are further coding techniques that can be exploited to speed up
the checksum calculation.
(1) Deferred Carries
Depending upon the machine, it may be more efficient to defer
adding end-around carries until the main summation loop is
finished.
One approach is to sum 16-bit words in a 32-bit accumulator, so
the overflows build up in the high-order 16 bits. This approach
typically avoids a carry-sensing instruction but requires twice
as many additions as would adding 32-bit segments; which is
faster depends upon the detailed hardware architecture.
(2) Unwinding Loops
To reduce the loop overhead, it is often useful to "unwind" the
inner sum loop, replicating a series of addition commands within
one loop traversal. This technique often provides significant
savings, although it may complicate the logic of the program
considerably.
(3) Combine with Data Copying
Like checksumming, copying data from one memory location to
another involves per-byte overhead. In both cases, the
bottleneck is essentially the memory bus, i.e., how fast the
data can be fetched. On some machines (especially relatively
slow and simple micro-computers), overhead can be significantly
reduced by combining memory-to-memory copy and the checksumming,
fetching the data only once for both.
(4) Incremental Update
Finally, one can sometimes avoid recomputing the entire checksum
when one header field is updated. The best-known example is a
gateway changing the TTL field in the IP header, but there are
other examples (for example, when updating a source route). In
these cases it is possible to update the checksum without
scanning the message or datagram.
To update the checksum, simply add the differences of the
sixteen bit integers that have been changed. To see why this
works, observe that every 16-bit integer has an additive inverse
and that addition is associative. From this it follows that
given the original value m, the new value m', and the old
checksum C, the new checksum C' is:
C' = C + (-m) + m' = C + (m' - m)
3. Numerical Examples
We now present explicit examples of calculating a simple 1's
complement sum on a 2's complement machine. The examples show the
same sum calculated byte by bye, by 16-bits words in normal and
swapped order, and 32 bits at a time in 3 different orders. All
numbers are in hex.
Byte-by-byte "Normal" Swapped
Order Order
Byte 0/1: 00 01 0001 0100
Byte 2/3: f2 03 f203 03f2
Byte 4/5: f4 f5 f4f5 f5f4
Byte 6/7: f6 f7 f6f7 f7f6
--- --- ----- -----
Sum1: 2dc 1f0 2ddf0 1f2dc
dc f0 ddf0 f2dc
Carrys: 1 2 2 1
-- -- ---- ----
Sum2: dd f2 ddf2 f2dd
Final Swap: dd f2 ddf2 ddf2
Byte 0/1/2/3: 0001f203 010003f2 03f20100
Byte 4/5/6/7: f4f5f6f7 f5f4f7f6 f7f6f5f4
-------- -------- --------
Sum1: 0f4f7e8fa 0f6f4fbe8 0fbe8f6f4
Carries: 0 0 0
Top half: f4f7 f6f4 fbe8
Bottom half: e8fa fbe8 f6f4
----- ----- -----
Sum2: 1ddf1 1f2dc 1f2dc
ddf1 f2dc f2dc
Carrys: 1 1 1
---- ---- ----
Sum3: ddf2 f2dd f2dd
Final Swap: ddf2 ddf2 ddf2
Finally, here an example of breaking the sum into two groups, with
the second group starting on a odd boundary:
Byte-by-byte Normal
Order
Byte 0/1: 00 01 0001
Byte 2/ : f2 (00) f200
--- --- -----
Sum1: f2 01 f201
Byte 4/5: 03 f4 03f4
Byte 6/7: f5 f6 f5f6
Byte 8/: f7 (00) f700
--- --- -----
Sum2: 1f0ea
Sum2: f0ea
Carry: 1
-----
Sum3: f0eb
Sum1: f201
Sum3 byte swapped: ebf0
-----
Sum4: 1ddf1
Sum4: ddf1
Carry: 1
-----
Sum5: ddf2
4. Implementation Examples
In this section we show examples of Internet checksum implementation
algorithms that have been found to be efficient on a variety of
CPU's. In each case, we show the core of the algorithm, without
including environmental code (e.g., subroutine linkages) or special-
case code.
4.1 "C"
The following "C" code algorithm computes the checksum with an inner
loop that sums 16-bits at a time in a 32-bit accumulator.
in 6
{
/* Compute Internet Checksum for "count" bytes
* beginning at location "addr".
*/
register long sum = 0;
while( count > 1 ) {
/* This is the inner loop */
sum += * (unsigned short) addr++;
count -= 2;
}
/* Add left-over byte, if any */
if( count > 0 )
sum += * (unsigned char *) addr;
/* Fold 32-bit sum to 16 bits */
while (sum>>16)
sum = (sum & 0xffff) + (sum >> 16);
checksum = ~sum;
}
4.2 Motorola 68020
The following algorithm is given in assembler language for a Motorola
68020 chip. This algorithm performs the sum 32 bits at a time, and
unrolls the loop with 16 replications. For clarity, we have omitted
the logic to add the last fullword when the length is not a multiple
of 4. The result is left in register d0.
With a 20MHz clock, this routine was measured at 134 usec/kB summing
random data. This algorithm was developed by Van Jacobson.
movl d1,d2
lsrl #6,d1 count/64 = # loop traversals
andl #0x3c,d2 Then find fractions of a chunk
negl d2
andb #0xf,cc Clear X (extended carry flag)
jmp pc@(2$-.-2:b,d2) Jump into loop
1$: Begin inner loop...
movl a0@+,d2 Fetch 32-bit word
addxl d2,d0 Add word + previous carry
movl a0@+,d2 Fetch 32-bit word
addxl d2,d0 Add word + previous carry
... 14 more replications
2$:
dbra d1,1$ (NB- dbra doesn't affect X)
movl d0,d1 Fold 32 bit sum to 16 bits
swap d1 (NB- swap doesn't affect X)
addxw d1,d0
jcc 3$
addw #1,d0
3$:
andl #0xffff,d0
4.3 Cray
The following example, in assembler language for a Cray CPU, was
contributed by Charley Kline. It implements the checksum calculation
as a vector operation, summing up to 512 bytes at a time with a basic
summation unit of 32 bits. This example omits many details having to
do with short blocks, for clarity.
Register A1 holds the address of a 512-byte block of memory to
checksum. First two copies of the data are loaded into two vector
registers. One is vector-shifted right 32 bits, while the other is
vector-ANDed with a 32 bit mask. Then the two vectors are added
together. Since all these operations chain, it produces one result
per clock cycle. Then it collapses the result vector in a loop that
adds each element to a scalar register. Finally, the end-around
carry is performed and the result is folded to 16-bits.
EBM
A0 A1
VL 64 use full vectors
S1 <32 form 32-bit mask from the right.
A2 32
V1 ,A0,1 load packet into V1
V2 S1&V1 Form right-hand 32-bits in V2.
V3 V1>A2 Form left-hand 32-bits in V3.
V1 V2+V3 Add the two together.
A2 63 Prepare to collapse into a scalar.
S1 0
S4 <16 Form 16-bit mask from the right.
A4 16
CK$LOOP S2 V1,A2
A2 A2-1
A0 A2
S1 S1+S2
JAN CK$LOOP
S2 S1&S4 Form right-hand 16-bits in S2
S1 S1>A4 Form left-hand 16-bits in S1
S1 S1+S2
S2 S1&S4 Form right-hand 16-bits in S2
S1 S1>A4 Form left-hand 16-bits in S1
S1 S1+S2
S1 #S1 Take one's complement
CMR At this point, S1 contains the checksum.
4.4 IBM 370
The following example, in assembler language for an IBM 370 CPU, sums
the data 4 bytes at a time. For clarity, we have omitted the logic
to add the last fullword when the length is not a multiple of 4, and
to reverse the bytes when necessary. The result is left in register
RCARRY.
This code has been timed on an IBM 3090 CPU at 27 usec/KB when
summing all one bits. This time is reduced to 24.3 usec/KB if the
trouble is taken to word-align the addends (requiring special cases
at both the beginning and the end, and byte-swapping when necessary
to compensate for starting on an odd byte).
* Registers RADDR and RCOUNT contain the address and length of
* the block to be checksummed.
*
* (RCARRY, RSUM) must be an even/odd register pair.
* (RCOUNT, RMOD) must be an even/odd register pair.
*
CHECKSUM SR RSUM,RSUM Clear working registers.
SR RCARRY,RCARRY
LA RONE,1 Set up constant 1.
*
SRDA RCOUNT,6 Count/64 to RCOUNT.
AR RCOUNT,RONE +1 = # times in loop.
SRL RMOD,26 Size of partial chunk to RMOD.
AR RADDR,R3 Adjust addr to compensate for
S RADDR,=F(64) jumping into the loop.
SRL RMOD,1 (RMOD/4)*2 is halfword index.
LH RMOD,DOPEVEC9(RMOD) Use magic dope-vector for offset,
B LOOP(RMOD) and jump into the loop...
*
* Inner loop:
*
LOOP AL RSUM,0(,RADDR) Add Logical fullword
BC 12,*+6 Branch if no carry
AR RCARRY,RONE Add 1 end-around
AL RSUM,4(,RADDR) Add Logical fullword
BC 12,*+6 Branch if no carry
AR RCARRY,RONE Add 1 end-around
*
* ... 14 more replications ...
*
A RADDR,=F'64' Increment address ptr
BCT RCOUNT,LOOP Branch on Count
*
* Add Carries into sum, and fold to 16 bits
*
ALR RCARRY,RSUM Add SUM and CARRY words
BC 12,*+6 and take care of carry
AR RCARRY,RONE
SRDL RCARRY,16 Fold 32-bit sum into
SRL RSUM,16 16-bits
ALR RCARRY,RSUM
C RCARRY,=X'0000FFFF' and take care of any
BNH DONE last carry
S RCARRY,=X'0000FFFF'
DONE X RCARRY,=X'0000FFFF' 1's complement
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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Internet Experiment Note 45 5 June 1978
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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Internet Experiment Note 45 5 June 1978
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
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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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Internet Experiment Note 45 5 June 1978
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 Communications, 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.
