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RFC 4471 - Derivation of DNS Name Predecessor and Successor


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Network Working Group                                          G. Sisson
Request for Comments: 4471                                     B. Laurie
Category: Experimental                                           Nominet
                                                          September 2006

            Derivation of DNS Name Predecessor and Successor

Status of This Memo

   This memo defines an Experimental Protocol for the Internet
   community.  It does not specify an Internet standard of any kind.
   Discussion and suggestions for improvement are requested.
   Distribution of this memo is unlimited.

Copyright Notice

   Copyright (C) The Internet Society (2006).

Abstract

   This document describes two methods for deriving the canonically-
   ordered predecessor and successor of a DNS name.  These methods may
   be used for dynamic NSEC resource record synthesis, enabling
   security-aware name servers to provide authenticated denial of
   existence without disclosing other owner names in a DNSSEC secured
   zone.

Table of Contents

   1. Introduction ....................................................2
   2. Notational Conventions ..........................................3
   3. Derivations .....................................................3
      3.1. Absolute Method ............................................3
           3.1.1. Derivation of DNS Name Predecessor ..................3
           3.1.2. Derivation of DNS Name Successor ....................4
      3.2. Modified Method ............................................4
           3.2.1. Derivation of DNS Name Predecessor ..................5
           3.2.2. Derivation of DNS Name Successor ....................6
   4. Notes ...........................................................6
      4.1. Test for Existence .........................................6
      4.2. Case Considerations ........................................7
      4.3. Choice of Range ............................................7
      4.4. Wild Card Considerations ...................................8
      4.5. Possible Modifications .....................................8
           4.5.1. Restriction of Effective Maximum DNS Name Length ....8
           4.5.2. Use of Modified Method with Zones Containing

                  SRV RRs .............................................8
   5. Examples ........................................................9
      5.1. Examples of Immediate Predecessors Using Absolute Method ..10
      5.2. Examples of Immediate Successors Using Absolute Method ....14
      5.3. Examples of Predecessors Using Modified Method ............19
      5.4. Examples of Successors Using Modified Method ..............20
   6. Security Considerations ........................................21
   7. Acknowledgements ...............................................21
   8. References .....................................................21
      8.1. Normative References ......................................21
      8.2. Informative References ....................................22

1.  Introduction

   One of the proposals for avoiding the exposure of zone information
   during the deployment DNSSEC is dynamic NSEC resource record (RR)
   synthesis.  This technique is described in [DNSSEC-TRANS] and
   [RFC4470], and involves the generation of NSEC RRs that just span the
   query name for non-existent owner names.  In order to do this, the
   DNS names that would occur just prior to and just following a given
   query name must be calculated in real time, as maintaining a list of
   all possible owner names that might occur in a zone would be
   impracticable.

   Section 6.1 of [RFC4034] defines canonical DNS name order.  This
   document does not amend or modify this definition.  However, the
   derivation of immediate predecessor and successor, although trivial,
   is non-obvious.  Accordingly, several methods are described here as
   an aid to implementors and a reference to other interested parties.

   This document describes two methods:

   1.  An "absolute method", which returns the immediate predecessor or
       successor of a domain name such that no valid DNS name could
       exist between that DNS name and the predecessor or successor.

   2.  A "modified method", which returns a predecessor and successor
       that are more economical in size and computation.  This method is
       restricted to use with zones consisting exclusively of owner
       names that contain no more than one label more than the owner
       name of the apex, where the longest possible owner name (i.e.,
       one with a maximum length left-most label) would not exceed the
       maximum DNS name length.  This is, however, the type of zone for
       which the technique of online signing is most likely to be used.

2.  Notational Conventions

   The following notational conventions are used in this document for
   economy of expression:

   N: An unspecified DNS name.

   P(N): Immediate predecessor to N (absolute method).

   S(N): Immediate successor to N (absolute method).

   P'(N): Predecessor to N (modified method).

   S'(N): Successor to N (modified method).

3.  Derivations

   These derivations assume that all uppercase US-ASCII letters in N
   have already been replaced by their corresponding lowercase
   equivalents.  Unless otherwise specified, processing stops after the
   first step in which a condition is met.

   The derivations make reference to maximum label length and maximum
   DNS name length; these are defined in Section 3.1 of [RFC1034] to be
   63 and 255 octets, respectively.

3.1.  Absolute Method

3.1.1.  Derivation of DNS Name Predecessor

   To derive P(N):

   1.  If N is the same as the owner name of the zone apex, prepend N
       repeatedly with labels of the maximum length possible consisting
       of octets of the maximum sort value (e.g., 0xff) until N is the
       maximum length possible; otherwise proceed to the next step.

   2.  If the least significant (left-most) label of N consists of a
       single octet of the minimum sort value (e.g., 0x00), remove that
       label; otherwise proceed to the next step.

   3.  If the least significant (right-most) octet in the least
       significant (left-most) label of N is the minimum sort value,
       remove the least significant octet and proceed to step 5.

   4.  Decrement the value of the least significant (right-most) octet
       of the least significant (left-most) label, skipping any values
       that correspond to uppercase US-ASCII letters, and then append

       the least significant (left-most) label with as many octets as
       possible of the maximum sort value.  Proceed to the next step.

   5.  Prepend N repeatedly with labels of as long a length as possible
       consisting of octets of the maximum sort value until N is the
       maximum length possible.

3.1.2.  Derivation of DNS Name Successor

   To derive S(N):

   1.  If N is two or more octets shorter than the maximum DNS name
       length, prepend N with a label containing a single octet of the
       minimum sort value (e.g., 0x00); otherwise proceed to the next
       step.

   2.  If N is one octet shorter than the maximum DNS name length and
       the least significant (left-most) label is one or more octets
       shorter than the maximum label length, append an octet of the
       minimum sort value to the least significant label; otherwise
       proceed to the next step.

   3.  Increment the value of the least significant (right-most) octet
       in the least significant (left-most) label that is less than the
       maximum sort value (e.g., 0xff), skipping any values that
       correspond to uppercase US-ASCII letters, and then remove any
       octets to the right of that one.  If all octets in the label are
       the maximum sort value, then proceed to the next step.

   4.  Remove the least significant (left-most) label.  Unless N is now
       the same as the owner name of the zone apex (this will occur only
       if N was the maximum possible name in canonical DNS name order,
       and thus has wrapped to the owner name of zone apex), repeat
       starting at step 2.

3.2.  Modified Method

   This method is for use with zones consisting only of single-label
   owner names where an owner name consisting of label of maximum length
   would not result in a DNS name that exceeded the maximum DNS name
   length.  This method is computationally simpler and returns values
   that are more economical in size than the absolute method.  It
   differs from the absolute method detailed above in the following
   ways:

   1.  Step 1 of the derivation P(N) has been omitted as the existence
       of the owner name of the zone apex never requires denial.

   2.  A new step 1 has been introduced that removes unnecessary labels.

   3.  Step 4 of the derivation P(N) has been omitted as it is only
       necessary for zones containing owner names consisting of more
       than one label.  This omission generally results in a significant
       reduction of the length of derived predecessors.

   4.  Step 1 of the derivation S(N) had been omitted as it is only
       necessary for zones containing owner names consisting of more
       than one label.  This omission results in a tiny reduction of the
       length of derived successors, and maintains consistency with the
       modification of step 4 of the derivation P(N) described above.

   5.  Steps 2 and 4 of the derivation S(N) have been modified to
       eliminate checks for maximum DNS name length, as it is an
       assumption of this method that no DNS name in the zone can exceed
       the maximum DNS name length.

3.2.1.  Derivation of DNS Name Predecessor

   To derive P'(N):

   1.  If N is two or more labels longer than the owner name of the
       apex, repeatedly remove the least significant (left-most) label
       until N is only one label longer than the owner name of the apex;
       otherwise proceed to the next step.

   2.  If the least significant (left-most) label of N consists of a
       single octet of the minimum sort value (e.g., 0x00), remove that
       label; otherwise proceed to the next step.  (If this condition is
       met, P'(N) is the owner name of the apex.)

   3.  If the least significant (right-most) octet in the least
       significant (left-most) label of N is the minimum sort value,
       remove the least significant octet.

   4.  Decrement the value of the least significant (right-most) octet,
       skipping any values that correspond to uppercase US-ASCII
       letters, and then append the label with as many octets as
       possible of the maximum sort value.

3.2.2.  Derivation of DNS Name Successor

   To derive S'(N):

   1.  If N is two or more labels longer than the owner name of the
       apex, repeatedly remove the least significant (left-most) label
       until N is only one label longer than the owner name of the apex.
       Proceed to the next step.

   2.  If the least significant (left-most) label of N is one or more
       octets shorter than the maximum label length, append an octet of
       the minimum sort value to the least significant label; otherwise
       proceed to the next step.

   3.  Increment the value of the least significant (right-most) octet
       in the least significant (left-most) label that is less than the
       maximum sort value (e.g., 0xff), skipping any values that
       correspond to uppercase US-ASCII letters, and then remove any
       octets to the right of that one.  If all octets in the label are
       the maximum sort value, then proceed to the next step.

   4.  Remove the least significant (left-most) label.  (This will occur
       only if the least significant label is the maximum label length
       and consists entirely of octets of the maximum sort value, and
       thus has wrapped to the owner name of the zone apex.)

4.  Notes

4.1.  Test for Existence

   Before using the result of P(N) or P'(N) as the owner name of an NSEC
   RR in a DNS response, a name server should test to see whether the
   name exists.  If it does, either a standard non-synthesised NSEC RR
   should be used, or the synthesised NSEC RR should reflect the RRset
   types that exist at the NSEC RR's owner name in the Type Bit Map
   field as specified by Section 4.1.2 of [RFC4034].  Implementors will
   likely find it simpler to use a non-synthesised NSEC RR.  For further
   details, see Section 2 of [RFC4470].

4.2.  Case Considerations

   Section 3.5 of [RFC1034] specifies that "while upper and lower case
   letters are allowed in names, no significance is attached to the
   case".  Additionally, Section 6.1 of [RFC4034] states that when
   determining canonical DNS name order, "uppercase US-ASCII letters are
   treated as if they were lowercase US-ASCII letters".  Consequently,
   values corresponding to US-ASCII uppercase letters must be skipped
   when decrementing and incrementing octets in the derivations
   described in Section 3.

   The following pseudo-code is illustrative:

   Decrement the value of an octet:

      if (octet == '[')       // '[' is just after uppercase 'Z'
              octet = '@';    // '@' is just prior to uppercase 'A'
      else
              octet--;

   Increment the value of an octet:

      if (octet == '@')       // '@' is just prior to uppercase 'A'
              octet = '[';    // '[' is just after uppercase 'Z'
      else
              octet++;

4.3.  Choice of Range

   [RFC2181] makes the clarification that "any binary string whatever
   can be used as the label of any resource record".  Consequently, the
   minimum sort value may be set as 0x00 and the maximum sort value as
   0xff, and the range of possible values will be any DNS name that
   contains octets of any value other than those corresponding to
   uppercase US-ASCII letters.

   However, if all owner names in a zone are in the letter-digit-hyphen,
   or LDH, format specified in [RFC1034], it may be desirable to
   restrict the range of possible values to DNS names containing only
   LDH values.  This has the effect of

   1.  making the output of tools such as `dig' and `nslookup' less
       subject to confusion,

   2.  minimising the impact that NSEC RRs containing DNS names with
       non-LDH values (or non-printable values) might have on faulty DNS
       resolver implementations, and

   3.  preventing the possibility of results that are wildcard DNS names
       (see Section 4.4).

   This may be accomplished by using a minimum sort value of 0x1f (US-
   ASCII character `-') and a maximum sort value of 0x7a (US-ASCII
   character lowercase `z'), and then skipping non-LDH, non-lowercase
   values when incrementing or decrementing octets.

4.4.  Wild Card Considerations

   Neither derivation avoids the possibility that the result may be a
   DNS name containing a wildcard label, i.e., a label containing a
   single octet with the value 0x2a (US-ASCII character `*').  With
   additional tests, wildcard DNS names may be explicitly avoided;
   alternatively, if the range of octet values can be restricted to
   those corresponding to letter-digit-hyphen, or LDH, characters (see
   Section 4.3), such DNS names will not occur.

   Note that it is improbable that a result that is a wildcard DNS name
   will occur unintentionally; even if one does occur either as the
   owner name of, or in the RDATA of an NSEC RR, it is treated as a
   literal DNS name with no special meaning.

4.5.  Possible Modifications

4.5.1.  Restriction of Effective Maximum DNS Name Length

   [RFC1034] specifies that "the total number of octets that represent a
   name (i.e., the sum of all label octets and label lengths) is limited
   to 255", including the null (zero-length) label that represents the
   root.  For the purpose of deriving predecessors and successors during
   NSEC RR synthesis, the maximum DNS name length may be effectively
   restricted to the length of the longest DNS name in the zone.  This
   will minimise the size of responses containing synthesised NSEC RRs
   but, especially in the case of the modified method, may result in
   some additional computational complexity.

   Note that this modification will have the effect of revealing
   information about the longest name in the zone.  Moreover, when the
   contents of the zone changes, e.g., during dynamic updates and zone
   transfers, care must be taken to ensure that the effective maximum
   DNS name length agrees with the new contents.

4.5.2.  Use of Modified Method with Zones Containing SRV RRs

   Normally, the modified method cannot be used in zones that contain
   Service Record (SRV) RRs [RFC2782], as SRV RRs have owner names that
   contain multiple labels.  However, the use of SRV RRs can be

   accommodated by various techniques.  There are at least four possible
   ways to do this:

   1.  Use conventional NSEC RRs for the region of the zone that
       contains first-level labels beginning with the underscore (`_')
       character.  For the purposes of generating these NSEC RRs, the
       existence of (possibly fictional) ownernames `9{63}' and `a'
       could be assumed, providing a lower and upper bound for this
       region.  Then all queries where the QNAME does not exist but
       contains a first-level label beginning with an underscore could
       be handled using the normal DNSSEC protocol.

       This approach would make it possible to enumerate all DNS names
       in the zone containing a first-level label beginning with
       underscore, including all SRV RRs, but this may be of less a
       concern to the zone administrator than incurring the overhead of
       the absolute method or of the following variants of the modified
       method.

   2.  The absolute method could be used for synthesising NSEC RRs for
       all queries where the QNAME contains a leading underscore.
       However, this re-introduces the susceptibility of the absolute
       method to denial of service activity, as an attacker could send
       queries for an effectively inexhaustible supply of domain names
       beginning with a leading underscore.

   3.  A variant of the modified method could be used for synthesising
       NSEC RRs for all queries where the QNAME contains a leading
       underscore.  This variant would assume that all predecessors and
       successors to queries where the QNAME contains a leading
       underscore may consist of two labels rather than only one.  This
       introduces a little additional complexity without incurring the
       full increase in response size and computational complexity as
       the absolute method.

   4.  Finally, a variant of the modified method that assumes that all
       owner names in the zone consist of one or two labels could be
       used.  However, this negates much of the reduction in response
       size of the modified method and may be nearly as computationally
       complex as the absolute method.

5.  Examples

   In the following examples,

      the owner name of the zone apex is "example.com.",

      the range of octet values is 0x00 - 0xff excluding values
      corresponding to uppercase US-ASCII letters, and

      non-printable octet values are expressed as three-digit decimal
      numbers preceded by a backslash (as specified in Section 5.1 of
      [RFC1035]).

5.1.  Examples of Immediate Predecessors Using Absolute Method

   Example of a typical case:

      P(foo.example.com.) =

           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255.\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.fon\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255.example.com.

      or, in alternate notation:

           \255{49}.\255{63}.\255{63}.fon\255{60}.example.com.

      where {n} represents the number of repetitions of an octet.

   Example where least significant (left-most) label of DNS name
   consists of a single octet of the minimum sort value:

      P(\000.foo.example.com.) = foo.example.com.

   Example where least significant (right-most) octet of least
   significant (left-most) label has the minimum sort value:

      P(foo\000.example.com.) =

           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255.\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.foo.example.com.

      or, in alternate notation:

           \255{45}.\255{63}.\255{63}.\255{63}.foo.example.com.

   Example where DNS name contains an octet that must be decremented by
   skipping values corresponding to US-ASCII uppercase letters:

      P(fo\[.example.com.) =

           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255.\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.fo\@\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255.example.com.

      or, in alternate notation:

           \255{49}.\255{63}.\255{63}.fo\@\255{60}.example.com.

      where {n} represents the number of repetitions of an octet.

   Example where DNS name is the owner name of the zone apex, and
   consequently wraps to the DNS name with the maximum possible sort
   order in the zone:

      P(example.com.) =

           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255.\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.example.com.

      or, in alternate notation:

           \255{49}.\255{63}.\255{63}.\255{63}.example.com.

5.2.  Examples of Immediate Successors Using Absolute Method

   Example of typical case:

      S(foo.example.com.) = \000.foo.example.com.

   Example where DNS name is one octet short of the maximum DNS name
   length:

      N =  fooooooooooooooooooooooooooooooooooooooooooooooo
           .ooooooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooooo.ooooooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooooooo.ooooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo.example.com.

      or, in alternate notation:

           fo{47}.o{63}.o{63}.o{63}.example.com.

      S(N) =

           fooooooooooooooooooooooooooooooooooooooooooooooo
           \000.ooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooooooooo.ooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooooooooooo.ooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           oooo.example.com.

      or, in alternate notation:

           fo{47}\000.o{63}.o{63}.o{63}.example.com.

   Example where DNS name is the maximum DNS name length:

      N  = fooooooooooooooooooooooooooooooooooooooooooooooo
           o.oooooooooooooooooooooooooooooooooooooooooooooo
           ooooooooooooooooo.oooooooooooooooooooooooooooooo
           ooooooooooooooooooooooooooooooooo.oooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           o.example.com.

      or, in alternate notation:

           fo{48}.o{63}.o{63}.o{63}.example.com.

      S(N) =

           fooooooooooooooooooooooooooooooooooooooooooooooo
           p.oooooooooooooooooooooooooooooooooooooooooooooo
           ooooooooooooooooo.oooooooooooooooooooooooooooooo
           ooooooooooooooooooooooooooooooooo.oooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           o.example.com.

      or, in alternate notation:

           fo{47}p.o{63}.o{63}.o{63}.example.com.

   Example where DNS name is the maximum DNS name length and the least
   significant (left-most) label has the maximum sort value:

      N =  \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.ooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooooooooo.ooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooooooooooo.ooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           oooo.example.com.

      or, in alternate notation:

           \255{49}.o{63}.o{63}.o{63}.example.com.

      S(N) =

           oooooooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooop.oooooooooooooooooooooooooooooooo
           ooooooooooooooooooooooooooooooo.oooooooooooooooo
           ooooooooooooooooooooooooooooooooooooooooooooooo.
           example.com.

      or, in alternate notation:

           o{62}p.o{63}.o{63}.example.com.

   Example where DNS name is the maximum DNS name length and the eight
   least significant (right-most) octets of the least significant
   (left-most) label have the maximum sort value:

      N  = foooooooooooooooooooooooooooooooooooooooo\255
           \255\255\255\255\255\255\255.ooooooooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooo.ooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooo.ooooooooooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooo.example.com.

      or, in alternate notation:

           fo{40}\255{8}.o{63}.o{63}.o{63}.example.com.

      S(N) =

           fooooooooooooooooooooooooooooooooooooooop.oooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           ooooooooo.oooooooooooooooooooooooooooooooooooooo
           ooooooooooooooooooooooooo.oooooooooooooooooooooo
           ooooooooooooooooooooooooooooooooooooooooo.example.com.

      or, in alternate notation:

           fo{39}p.o{63}.o{63}.o{63}.example.com.

   Example where DNS name is the maximum DNS name length and contains an
   octet that must be incremented by skipping values corresponding to
   US-ASCII uppercase letters:

      N  = fooooooooooooooooooooooooooooooooooooooooooooooo
           \@.ooooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooooooo.ooooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooooooooo.ooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           oo.example.com.

      or, in alternate notation:

           fo{47}\@.o{63}.o{63}.o{63}.example.com.

      S(N) =

           fooooooooooooooooooooooooooooooooooooooooooooooo
           \[.ooooooooooooooooooooooooooooooooooooooooooooo
           oooooooooooooooooo.ooooooooooooooooooooooooooooo
           oooooooooooooooooooooooooooooooooo.ooooooooooooo
           oooooooooooooooooooooooooooooooooooooooooooooooo
           oo.example.com.

      or, in alternate notation:

           fo{47}\[.o{63}.o{63}.o{63}.example.com.

   Example where DNS name has the maximum possible sort order in the
   zone, and consequently wraps to the owner name of the zone apex:

      N  = \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255.\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255.\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.example.com.

      or, in alternate notation:

           \255{49}.\255{63}.\255{63}.\255{63}.example.com.

      S(N) = example.com.

5.3.  Examples of Predecessors Using Modified Method

   Example of a typical case:

      P'(foo.example.com.) =

           fon\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255.example.com.

      or, in alternate notation:

           fon\255{60}.example.com.

   Example where DNS name contains more labels than DNS names in the
   zone:

      P'(bar.foo.example.com.) = foo.example.com.

   Example where least significant (right-most) octet of least
   significant (left-most) label has the minimum sort value:

      P'(foo\000.example.com.) = foo.example.com.

   Example where least significant (left-most) label has the minimum
   sort value:

      P'(\000.example.com.) = example.com.

   Example where DNS name is the owner name of the zone apex, and
   consequently wraps to the DNS name with the maximum possible sort
   order in the zone:

      P'(example.com.) =

           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255.example.com.

      or, in alternate notation:

           \255{63}.example.com.

5.4.  Examples of Successors Using Modified Method

   Example of a typical case:

      S'(foo.example.com.) = foo\000.example.com.

   Example where DNS name contains more labels than DNS names in the
   zone:

      S'(bar.foo.example.com.) = foo\000.example.com.

   Example where least significant (left-most) label has the maximum
   sort value, and consequently wraps to the owner name of the zone
   apex:

      N  = \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255\255\255\255\255\255\255\255\255\255
           \255\255\255.example.com.

      or, in alternate notation:

           \255{63}.example.com.

      S'(N) = example.com.

6.  Security Considerations

   The derivation of some predecessors/successors requires the testing
   of more conditions than others.  Consequently, the effectiveness of a
   denial-of-service attack may be enhanced by sending queries that
   require more conditions to be tested.  The modified method involves
   the testing of fewer conditions than the absolute method and
   consequently is somewhat less susceptible to this exposure.

7.  Acknowledgements

   The authors would like to thank Sam Weiler, Olaf Kolkman, Olafur
   Gudmundsson, and Niall O'Reilly for their review and input.

8.  References

8.1.  Normative References

   [RFC1034]      Mockapetris, P., "Domain names - concepts and
                  facilities", STD 13, RFC 1034, November 1987.

   [RFC1035]      Mockapetris, P., "Domain names - implementation and
                  specification", STD 13, RFC 1035, November 1987.

   [RFC2181]      Elz, R. and R. Bush, "Clarifications to the DNS
                  Specification", RFC 2181, July 1997.

   [RFC2782]      Gulbrandsen, A., Vixie, P., and L. Esibov, "A DNS RR
                  for specifying the location of services (DNS SRV)",
                  RFC 2782, February 2000.

   [RFC4034]      Arends, R., Austein, R., Larson, M., Massey, D., and
                  S. Rose, "Resource Records for the DNS Security
                  Extensions", RFC 4034, March 2005.

8.2.  Informative References

   [RFC4470]      Weiler, S. and J. Ihren, "Minimally Covering NSEC
                  Records and DNSSEC On-line Signing", RFC 4470, April
                  2006.

   [DNSSEC-TRANS] Arends, R., Koch, P., and J. Schlyter, "Evaluating
                  DNSSEC Transition Mechanisms", Work in Progress,
                  February 2005.

Authors' Addresses

   Geoffrey Sisson
   Nominet
   Sandford Gate
   Sandy Lane West
   Oxford
   OX4 6LB
   GB

   Phone: +44 1865 332211
   EMail: geoff@nominet.org.uk

   Ben Laurie
   Nominet
   17 Perryn Road
   London
   W3 7LR
   GB

   Phone: +44 20 8735 0686
   EMail: ben@algroup.co.uk

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