    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

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            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]

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       (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

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            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

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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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       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
                *         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;
       }

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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
           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

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    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).

       *      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

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                 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  [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

                                       - 1 -

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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

                                       - 2 -

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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.

                                       - 3 -

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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 [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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    Braden, Borman, & Partridge                                    [Page 16]

    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-
         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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    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 [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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    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  [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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    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."

         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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    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  [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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    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

         [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.

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