RFC 1071 – Computing the Internet Checksum

RFC 1071 – Computing the Internet Checksum

```Network Working Group                                         R.  Braden
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

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RFC 1071            Computing the Internet Checksum       September 1988

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]              

[A,B] +' [C,D] +' ... +' [Z,0]              

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  could be split into:

( [A,B] +' [C,D] +' ... +' [J,0] )

+' ( [0,K] +' ... +' [Y,Z] )               

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RFC 1071            Computing the Internet Checksum       September 1988

(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]                   

the result is the same as , 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 ; 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  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

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

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RFC 1071            Computing the Internet Checksum       September 1988

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
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

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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

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RFC 1071            Computing the Internet Checksum       September 1988

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

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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
*/
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;
}

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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
movl    a0@+,d2     |  Fetch 32-bit word

| ... 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)
jcc     3\$
3\$:
andl    #0xffff,d0

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RFC 1071            Computing the Internet Checksum       September 1988

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.

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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).

*              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.
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:
*
BC    12,*+6             Branch if no carry
BC    12,*+6             Branch if no carry
*
*                    ... 14 more replications ...
*
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

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RFC 1071            Computing the Internet Checksum       September 1988

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

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RFC 1071            Computing the Internet Checksum       September 1988

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

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RFC 1071            Computing the Internet Checksum       September 1988

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    and  Atlantic  Satellite  Network    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  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

- 1 -

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RFC 1071            Computing the Internet Checksum       September 1988

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 .  Of course,
there  are  many Abelian groups but not all would be satisfactory
for  use  as  checksum  operators.   (Another  operator   readily

- 2 -

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RFC 1071            Computing the Internet Checksum       September 1988

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:

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
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  uses two's complement add because space for
the loader program is extremely limited.

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RFC 1071            Computing the Internet Checksum       September 1988

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  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  and the ANSI standards for 800 and 6250  bits
per inch magnetic tapes (described in ) 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  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
(possibly different) transformation, a receiver  can  reconstruct

- 4 -

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RFC 1071            Computing the Internet Checksum       September 1988

Internet Experiment Note  45                          5 June 1978
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-
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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Braden, Borman, & Partridge                                    [Page 17]

RFC 1071            Computing the Internet Checksum       September 1988

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  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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Braden, Borman, & Partridge                                    [Page 18]

RFC 1071            Computing the Internet Checksum       September 1988

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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Braden, Borman, & Partridge                                    [Page 19]

RFC 1071            Computing the Internet Checksum       September 1988

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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Braden, Borman, & Partridge                                    [Page 20]

RFC 1071            Computing the Internet Checksum       September 1988

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    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 .  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 .  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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Braden, Borman, & Partridge                                    [Page 21]

RFC 1071            Computing the Internet Checksum       September 1988

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."

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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Braden, Borman, & Partridge                                    [Page 22]

RFC 1071            Computing the Internet Checksum       September 1988

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  
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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Braden, Borman, & Partridge                                    [Page 23]

RFC 1071            Computing the Internet Checksum       September 1988

Internet Experiment Note  45                          5 June 1978
TCP Checksum Function Design                   William W. Plummer

References

  Cerf, V.G. and Kahn, Robert E., "A Protocol for Packet Network
Communications," IEEE Transactions on Communications, vol.
COM-22, No.  5, May 1974.

  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.

  Jacobs, Irwin, et al., "CPODA - A Demand Assignment Protocol
for SatNet", Fifth Data Communications Symposium, September
27-9, 1977, Snowbird, Utah

  Bolt Beranek and Newman, Inc.  "Specifications for the
Interconnection of a Host and an IMP", Report 1822, January
1976 edition.

  Dean, Richard A., "Elements of Abstract Algebra", John Wyley
and Sons, Inc., 1966

  Peterson, W. Wesley, "Error Correcting Codes", M.I.T. Press
Cambridge MA, 4th edition, 1968.

  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.

  Kirstein, Peter, private communication

  Cerf, V. G. and Postel, Jonathan B., "Specification of
Internetwork Transmission Control Program Version 3",
University of Southern California Information Sciences
Institute, January 1978.

 Digital Equipment Corporation, "PDP-10 Reference Handbook",
1970, pp. 114-5.

 Swanson, Robert, "Understanding Cyclic Redundancy Codes",
Computer Design, November, 1975, pp. 93-99.

 Clements, Robert C., private communication.

 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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