RFC 1982
Serial Number Arithmetic

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Network Working Group                                             R. Elz
Request for Comments: 1982                       University of Melbourne
Updates: 1034, 1035                                              R. Bush
Category: Standards Track                                    RGnet, Inc.
                                                             August 1996

Serial Number Arithmetic

Status of this Memo

   This document specifies an Internet standards track protocol for the
   Internet community, and requests discussion and suggestions for
   improvements.  Please refer to the current edition of the "Internet
   Official Protocol Standards" (STD 1) for the standardization state
   and status of this protocol.  Distribution of this memo is unlimited.


   This memo defines serial number arithmetic, as used in the Domain
   Name System.  The DNS has long relied upon serial number arithmetic,
   a concept which has never really been defined, certainly not in an
   IETF document, though which has been widely understood.  This memo
   supplies the missing definition.  It is intended to update RFC1034
   and RFC1035.

1. Introduction

   The serial number field of the SOA resource record is defined in
   RFC1035 as

   SERIAL   The unsigned 32 bit version number of the original copy of
            the zone.  Zone transfers preserve this value.  This value
            wraps and should be compared using sequence space

   RFC1034 uses the same terminology when defining secondary server zone
   consistency procedures.

   Unfortunately the term "sequence space arithmetic" is not defined in
   either RFC1034 or RFC1035, nor do any of their references provide
   further information.

   This phrase seems to have been intending to specify arithmetic as
   used in TCP sequence numbers [RFC793], and defined in [IEN-74].

   Unfortunately, the arithmetic defined in [IEN-74] is not adequate for
   the purposes of the DNS, as no general comparison operator is

   To avoid further problems with this simple field, this document
   defines the field and the operations available upon it.  This
   definition is intended merely to clarify the intent of RFC1034 and
   RFC1035, and is believed to generally agree with current
   implementations.  However, older, superseded, implementations are
   known to have treated the serial number as a simple unsigned integer,
   with no attempt to implement any kind of "sequence space arithmetic",
   however that may have been interpreted, and further, ignoring the
   requirement that the value wraps.  Nothing can be done with these
   implementations, beyond extermination.

2. Serial Number Arithmetic

   Serial numbers are formed from non-negative integers from a finite
   subset of the range of all integer values.  The lowest integer in
   every subset used for this purpose is zero, the maximum is always one
   less than a power of two.

   When considered as serial numbers however no value has any particular
   significance, there is no minimum or maximum serial number, every
   value has a successor and predecessor.

   To define a serial number to be used in this way, the size of the
   serial number space must be given.  This value, called "SERIAL_BITS",
   gives the power of two which results in one larger than the largest
   integer corresponding to a serial number value.  This also specifies
   the number of bits required to hold every possible value of a serial
   number of the defined type.  The operations permitted upon serial
   numbers are defined in the following section.

3. Operations upon the serial number

   Only two operations are defined upon serial numbers, addition of a
   positive integer of limited range, and comparison with another serial

3.1. Addition

   Serial numbers may be incremented by the addition of a positive
   integer n, where n is taken from the range of integers
   [0 .. (2^(SERIAL_BITS - 1) - 1)].  For a sequence number s, the
   result of such an addition, s', is defined as

                   s' = (s + n) modulo (2 ^ SERIAL_BITS)
   where the addition and modulus operations here act upon values that
   are non-negative values of unbounded size in the usual ways of
   integer arithmetic.

   Addition of a value outside the range
   [0 .. (2^(SERIAL_BITS - 1) - 1)] is undefined.

3.2. Comparison

   Any two serial numbers, s1 and s2, may be compared.  The definition
   of the result of this comparison is as follows.

   For the purposes of this definition, consider two integers, i1 and
   i2, from the unbounded set of non-negative integers, such that i1 and
   s1 have the same numeric value, as do i2 and s2.  Arithmetic and
   comparisons applied to i1 and i2 use ordinary unbounded integer

   Then, s1 is said to be equal to s2 if and only if i1 is equal to i2,
   in all other cases, s1 is not equal to s2.

   s1 is said to be less than s2 if, and only if, s1 is not equal to s2,

        (i1 < i2 and i2 - i1 < 2^(SERIAL_BITS - 1)) or
        (i1 > i2 and i1 - i2 > 2^(SERIAL_BITS - 1))

   s1 is said to be greater than s2 if, and only if, s1 is not equal to
   s2, and

        (i1 < i2 and i2 - i1 > 2^(SERIAL_BITS - 1)) or
        (i1 > i2 and i1 - i2 < 2^(SERIAL_BITS - 1))

   Note that there are some pairs of values s1 and s2 for which s1 is
   not equal to s2, but for which s1 is neither greater than, nor less
   than, s2.  An attempt to use these ordering operators on such pairs
   of values produces an undefined result.

   The reason for this is that those pairs of values are such that any
   simple definition that were to define s1 to be less than s2 where
   (s1, s2) is such a pair, would also usually cause s2 to be less than
   s1, when the pair is (s2, s1).  This would mean that the particular
   order selected for a test could cause the result to differ, leading
   to unpredictable implementations.

   While it would be possible to define the test in such a way that the
   inequality would not have this surprising property, while being
   defined for all pairs of values, such a definition would be
   unnecessarily burdensome to implement, and difficult to understand,
   and would still allow cases where

        s1 < s2 and (s1 + 1) > (s2 + 1)

   which is just as non-intuitive.

   Thus the problem case is left undefined, implementations are free to
   return either result, or to flag an error, and users must take care
   not to depend on any particular outcome.  Usually this will mean
   avoiding allowing those particular pairs of numbers to co-exist.

   The relationships greater than or equal to, and less than or equal
   to, follow in the natural way from the above definitions.

4. Corollaries

   These definitions give rise to some results of note.

4.1. Corollary 1

   For any sequence number s and any integer n such that addition of n
   to s is well defined, (s + n) >= s.  Further (s + n) == s only when
   n == 0, in all other defined cases, (s + n) > s.

4.2. Corollary 2

   If s' is the result of adding the non-zero integer n to the sequence
   number s, and m is another integer from the range defined as able to
   be added to a sequence number, and s" is the result of adding m to
   s', then it is undefined whether s" is greater than, or less than s,
   though it is known that s" is not equal to s.

4.3. Corollary 3

   If s" from the previous corollary is further incremented, then there
   is no longer any known relationship between the result and s.

4.4. Corollary 4

   If in corollary 2 the value (n + m) is such that addition of the sum
   to sequence number s would produce a defined result, then corollary 1
   applies, and s" is known to be greater than s.

5. Examples

5.1. A trivial example

   The simplest meaningful serial number space has SERIAL_BITS == 2.  In
   this space, the integers that make up the serial number space are 0,
   1, 2, and 3.  That is, 3 == 2^SERIAL_BITS - 1.

   In this space, the largest integer that it is meaningful to add to a
   sequence number is 2^(SERIAL_BITS - 1) - 1, or 1.

   Then, as defined 0+1 == 1, 1+1 == 2, 2+1 == 3, and 3+1 == 0.
   Further, 1 > 0, 2 > 1, 3 > 2, and 0 > 3.  It is undefined whether
   2 > 0 or 0 > 2, and whether 1 > 3 or 3 > 1.

5.2. A slightly larger example

   Consider the case where SERIAL_BITS == 8.  In this space the integers
   that make up the serial number space are 0, 1, 2, ... 254, 255.
   255 == 2^SERIAL_BITS - 1.

   In this space, the largest integer that it is meaningful to add to a
   sequence number is 2^(SERIAL_BITS - 1) - 1, or 127.

   Addition is as expected in this space, for example: 255+1 == 0,
   100+100 == 200, and 200+100 == 44.

   Comparison is more interesting, 1 > 0, 44 > 0, 100 > 0, 100 > 44,
   200 > 100, 255 > 200, 0 > 255, 100 > 255, 0 > 200, and 44 > 200.

   Note that 100+100 > 100, but that (100+100)+100 < 100.  Incrementing
   a serial number can cause it to become "smaller".  Of course,
   incrementing by a smaller number will allow many more increments to
   be made before this occurs.  However this is always something to be
   aware of, it can cause surprising errors, or be useful as it is the
   only defined way to actually cause a serial number to decrease.

   The pairs of values 0 and 128, 1 and 129, 2 and 130, etc, to 127 and
   255 are not equal, but in each pair, neither number is defined as
   being greater than, or less than, the other.

   It could be defined (arbitrarily) that 128 > 0, 129 > 1,
   130 > 2, ..., 255 > 127, by changing the comparison operator
   definitions, as mentioned above.  However note that that would cause
   255 > 127, while (255 + 1) < (127 + 1), as 0 < 128.  Such a
   definition, apart from being arbitrary, would also be more costly to

6. Citation

   As this defined arithmetic may be useful for purposes other than for
   the DNS serial number, it may be referenced as Serial Number
   Arithmetic from RFC1982.  Any such reference shall be taken as
   implying that the rules of sections 2 to 5 of this document apply to
   the stated values.

7. The DNS SOA serial number

   The serial number in the DNS SOA Resource Record is a Serial Number
   as defined above, with SERIAL_BITS being 32.  That is, the serial
   number is a non negative integer with values taken from the range
   [0 .. 4294967295].  That is, a 32 bit unsigned integer.

   The maximum defined increment is 2147483647 (2^31 - 1).

   Care should be taken that the serial number not be incremented, in
   one or more steps, by more than this maximum within the period given
   by the value of SOA.expire.  Doing so may leave some secondary
   servers with out of date copies of the zone, but with a serial number
   "greater" than that of the primary server.  Of course, special
   circumstances may require this rule be set aside, for example, when
   the serial number needs to be set lower for some reason.  If this
   must be done, then take special care to verify that ALL servers have
   correctly succeeded in following the primary server's serial number
   changes, at each step.

   Note that each, and every, increment to the serial number must be
   treated as the start of a new sequence of increments for this
   purpose, as well as being the continuation of all previous sequences
   started within the period specified by SOA.expire.

   Caution should also be exercised before causing the serial number to
   be set to the value zero.  While this value is not in any way special
   in serial number arithmetic, or to the DNS SOA serial number, many
   DNS implementations have incorrectly treated zero as a special case,
   with special properties, and unusual behaviour may be expected if
   zero is used as a DNS SOA serial number.

8. Document Updates

   RFC1034 and RFC1035 are to be treated as if the references to
   "sequence space arithmetic" therein are replaced by references to
   serial number arithmetic, as defined in this document.

9. Security Considerations

   This document does not consider security.

   It is not believed that anything in this document adds to any
   security issues that may exist with the DNS, nor does it do anything
   to lessen them.


   [RFC1034]   Domain Names - Concepts and Facilities,
               P. Mockapetris, STD 13, ISI, November 1987.

   [RFC1035]   Domain Names - Implementation and Specification
               P. Mockapetris, STD 13, ISI, November 1987

   [RFC793]    Transmission Control protocol
               Information Sciences Institute, STD 7, USC, September 1981

   [IEN-74]    Sequence Number Arithmetic
               William W. Plummer, BB&N Inc, September 1978


   Thanks to Rob Austein for suggesting clarification of the undefined
   comparison operators, and to Michael Patton for attempting to locate
   another reference for this procedure.  Thanks also to members of the
   IETF DNSIND working group of 1995-6, in particular, Paul Mockapetris.

Authors' Addresses

   Robert Elz                     Randy Bush
   Computer Science               RGnet, Inc.
   University of Melbourne        10361 NE Sasquatch Lane
   Parkville, Vic,  3052          Bainbridge Island, Washington,  98110
   Australia.                     United States.

   EMail: kre@munnari.OZ.AU       EMail: randy@psg.com