IT Professional, Author / Researcher                       E. Terrell 
Internet Draft                                             August 1999 
draft-terrell-math-ipaddr-ipv4-00.txt
Category: Informational
Expires February 22, 2000 




The Mathematical Reality of IP Addressing in IPv4 Questions 
the need for Another IP System of Addressing


Status of this Memo

This document is an Internet-Draft and is 
in full conformance with all provisions of Section 10 of RFC2026.

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Contents


Abstract


Introduction


Chapter I    'The 32 Bit Structure and the Layout of the 4 Octets'


Chapter II  'Subnetting the Past, its future a boon for Squeezing Out 
             more IP's'


Chapter III  'Addressing the Future, taking notes on the IP's past'


Chapter IV  'The Questions Raised: Is there Tenure for a Better 
             Allocation Plan?'


Chapter V Conclusion; 'An Alternate View of the IPv4 Addressing Scheme'


Security


Appendix I : Previous IPv4 Class Addressing System Schematic


References













Abstract 

This paper was necessitated by an overwhelming desire; an attempt to 
end the apparent disparity in the dissemination of information absent 
of the logical and thoroughness in rendering an explanation of the IP 
Addressing Scheme. To render a more pointed fact, I needed to pass a 
CISCO Certification Examination. However, this can never be 
accomplished, if the information that is needed and used in the 
preparation thereof, lacks continuity and propagates errors pertaining 
to foundational information. Needless to say, my endeavors were not in 
vein. That is,as a direct result of this undertaking, I corrected  the 
underlining errors, derived a possible alternative approach to the 
IPv4 Addressing Scheme, and expanded its Class system ( that is no 
longer in use ). In other words, I was indeed successful in the 
elimination of the problems associated with IP Address Flooding 
inherent in IPv4 and the complexities of IPv6. In short, small 
business and single family dwellings can now have the option of 
having their own private IP Addressing Scheme, without the disparity 
resulting from the steep learning curve presented in IPv6. While the 
Internet Community at large, will not suffer a shortage of the 
availability IP Addresses for assigned distribution. Especially since, 
while the number available IP Addresses do not exceed the amount 
reported to be provided, if IPv6 is implemented. It does indeed, 
provide enough IP Addresses to cover their continued issuance for at 
least another 100 years or so. Which is dependent upon the adoption 
of an adequate scheme for its allocation and distribution. 
  
  
  
Introduction 

The DARPA mission was quite clear from its inception. However, the 
demand for simplification and ease of implementation promoted the 
need for a further ' break down ' of the Addressing Scheme regarding 
IPv4 configuration. Needless to say, while this simplicity and ease 
of use / implementation will be lost with the establishment of the 
next generation; IPv6. It is nonetheless, the objective of this 
paper to rekindle an interest for a further exploitation of IPv4 prior 
to any implementation of IPv6. Moreover, it is indeed worthy of 
reiteration. That is, it was the lack of Interest after the 5 classes 
or Divisions of the IPv4 Addressing Scheme evolved. That it can clearly 
be surmised; with its inception and obvious lack of attention from the 
Computer Science Community, there arose an inherent Mathematical 
problem associated with the determination of  the Address Ranges for 
each Subnet for a given IP Network Address. Nevertheless, the current 
Classification of the IPv4 Addressing Scheme yields; 



1. Class A: 1 - 126, with 24 Bit Host count ; Where 0 ( Zero ) and 
   127 reserved   unknown Network and loopback

2. Class B: 128 - 191, with 16 Bit Host count 

3. Class C: 192 - 223, with 8 Bit Host count 

4. Class D: 224 - 239 ; Used for Multicasting, Host count not 
   applicable 

5. Class E: 240 - 254 ; Denoting Experimental, Host counts not 
   applicable 


Note: There is no Division of Classes D or E, in fact they are not 
      capable of being Subnetted. 


The problem however, is that, throughout every explanation ( read thus 
far ) of this IPv4 Addressing Scheme. It is a given that, the Network 
Address is determined by the ( or Should be determined by ) First Octet 
in the Address Scheme. However, this method is reserved for the Class 
A. In other words, if the Addressing Scheme ( or Pattern ) that 
resolves the Class A, were used to determine the number of Networks 
and Hosts within all Classes of IPv4 Addressing Scheme. Then, one 
would find that the number of Networks and their associated Hosts, 
which could be associated with every Class less than Class A, would 
increase. Especially if the Subnet were defined and employed as an 
additional Identifier, not only of the number of Networks, but the 
number of Hosts as well. Nonetheless, the logical discrepancy within 
the Mathematical Calculation of the present IPv4 Addressing Scheme 
shall become even more apparent. Especially when one is required to 
determine the number of Networks that would result from the 
determination of the Address Range for each Subnet of a given Network 
Address. That is, if one were to continue to employ the current methods 
as outlined and defined for the IPv4 Addressing Scheme, then they 
would also discover that many Authors have not distinguished any 
difference between the mathematical calculation of  the Decimal and 
Binary numbers associated with the IP Addresses. 

Nevertheless, it is from the discovery of these minor discrepancies 
and their solution, which prompted the need for this Internet Draft. 


Please note: Only IP Addressing and its related Mathematical 
             Calculations, shall be discussed here. All other quires 
             inherent to IP Addressing and Messaging should be sought 
             within the Subject Matter of their respective RFC's. 



Chapter I  'The 32 Bit Structure and the Layout of the 4 Octets' 

We currently utilize the 32 Bit, 4 Octet Rule as the instrument of 
choice in the IPv4 Addressing Scheme. Furthermore, while the Class 
System has been eliminated for the current IP Addressing Scheme. It 
will, because of my continued belief in its importance, use the IP 
Class Addressing Scheme and the methods derived herein, which the 
presentation of an overview. 

Nevertheless, while the Classes of the IPv4's Addressing Scheme is 
noted above. There are two additional points that warrant mention 
in this deliberation: 


1. ' The Octet Rule

2. The Laws of Ones and Zeros in the IP Address format. 


Let's examine number 2 first. Where the Laws of IP Addressing States: 

1. The Network Address portion of an IP address cannot be Set to 
   either all Binary Ones or All Binary Zeros 

2. The Subnet portion of an IP address cannot be Set to either All 
   Binary Ones or All Binary Zeros 

3. The Host portion of an IP address cannot be Set to All 
   Binary Ones or All Binary Zeros 

4. The IP address 127.x.x.x can never be assigned as a Network Address 


Asides from noting the fact, that these Laws govern only Binary 
representation ( with the exception of number 4, of course ). It 
should be quite clear that there is a marked difference between the 
results and the mathematical calculation of decimal and binary 
numeration. Nonetheless, from the analysis of number one above, one 
observes that if it is violated in either case. Then the violations 
would interfere with the predefined assignment of All Binary Ones; 
being the Broadcast Address. Or if the IP address were All Zeros, then 
it would interfere with the predefined IP address indicating that the 
Network in question has an Unknown Address. Nevertheless, with the 
defining premise or argument of the basic IP Address of All Binary 
Ones being given as the Broadcast Address. Then it stands to reason 
that; for every IP Address prefixed with the Address of a Host  
followed by Octets of Binary Ones. This would also be a Broadcast 
Address, whose broadcasts would be directed to, and defined by the 
Octets preceding those containing All Binary Ones. Where any IP 
Address of a Subnet whose remaining Octets contain All Binary Ones, 
then this would be a Broadcast Address for every Subnet maintaining 
this Address. Needless to say, this same line of reasoning can be 
applied to Host Addressing as well. However, while the definition 
for an IP Address containing All Binary Zeros is well established, 
its logical reasoning does not map as easily as that containing the 
Binary of All Ones. Nevertheless, for every IP Address of any Host, 
whose remaining Octets are All Binary Zeros, then the IP Address 
resolves to the Host Address as defined by the Octets preceding the 
Octets having All Binary Zeros assigned. In other words, if the 
remaining Octets in a Subnet IP Address are All Binary Zeros, 
then this Address location is defined as being Address for the Subnet 
in question. This rules applies for the Host Address as well, except 
that the IP Address would be the location of the Host having this IP 
Address assignment.

The OCTET Rule, while less cumbersome to explain, maintains a 
reputation for confusion, because its definition concerns 
two different systems of Counting: Decimal and Binary. 
Nevertheless, IPv4 IP Addressing Scheme is a 32 Bit implementation 
containing 4 sections for its addressing space. However, for ease of 
translation for electronic communication, each of these address spaces 
has 8 Binary numerical representations. That can only maintain either 
one of two states, that is represented as either a 1 or a 0. However, 
these Binary digits, collectively called an OCTET, that represent a 
Binary Number, can also be translated into Integer. The mathematical 
numeration more easily understood by every casual person, and known 
as the Counting Numbers. Needless to say, it is the Translation or 
Numerical conversion of these different Systems of Enumeration where 
the Equivalency between them, begins and ends. In other words, Binary 
is not equal in any way to Decimal, and any translation a variable 
pertaining to one of the other, requires another mathematical operation 
for this equivalency to realize. Given by the example Equations:


                     1. xxxxxxx    /=    yyyyyyy

         
                     2. ( xxxxxxx ) * v   =   ( yyyyyyy )

     { Where /= means "NOT EQUAL TO", and " x " is Binary, " y ' is 
      Decimal. The ' * ' is used to denote the operation of 
       multiplication. }



In words, the Octet Rule states: " There must exist 8 Bits for every 
Byte of Address Space used in an IP Address Format, and this Address 
Space is called an Octet. " However, while the method, specification, 
and procedures for the Internet Protocol is defined in RFC 791. Which 
defines the boundary of the Address Space as being from ' 0 - 255 ', 
in a 32 Bit logical Address Format. It can be proven mathematically 
that the Octet Rule is indeed valid, even if the size of the Logical 
Address Format changes. In other words, under the IPv4 system there 
is 4 Octets in the Logical Address Format, which renders a 32 Bit 
Address Format. While the IPv6 system provides a 128 Bit Address 
Format, and there is a marked difference in their structures. Where 
by, the former maintains a two tier system, while the latter is 3 
tier system comprising HEX notation. Nonetheless, neither system 
violates the Octet Rule, which is more clearly understood when the 
former explanation used with the pictorial given by Figure A.

----------------------------------------------
                                             -
                                             -
Figuer A                                     -
                                             -
                                             -
IP Addressing Structure                      -
.....................................        -
. 1st   . 2nd     . 3rd   . 4th     .        -
. Octet . Octet   . Octet . Octet   .        -
.....................................        -
                                             --------------------
                                                                -
Binary Addressing                                               -
.....................................                           -
. XXXX  . XXXX    . XXXX  . XXXX    .                           -
. XXXX  . XXXX    . XXXX  . XXXX    .                           -
.....................................                           - 
                                                                -
Decimal Addressing                                              -
.....................................                           -
. Y Y Y . Y Y Y   . Y Y Y . Y Y Y   .                           -
.       .         .       .         .                           -
.....................................                           -
............................................................... -
. These are two distinct Systems of Enumeration, Binary and   . -                                                       .
. Decimal, that are only Equivalentin the Results they yield. . -
. Where by, their representation and methods,in the Logic of  . -
. their respective Mathematical Operations, are indeed        . -
. different.This is even more apparent from their Numerical   . -
. Representatioms:                                            . -
.                                                               -
. Where the Binary Addressing method, provides for an 8 Bit   . -
. Digit Displacement, and each individual digit can only be   . -
. either a 1 or a 0. Nonetheless, these 8 digits of 1's and   . -
. 0's combined, represent only one number.                    . -
.                                                             . -
. However, the Decimal Addressing method, provides a 3 digit  . -
. Displacement, but they can only be Integers that represent  . -
. only one number. Which is also an Integer.                  . -
.                                                             . -
............................................................... -
-----------------------------------------------------------------





Chapter II  'Subnetting the Past, its future a boon for Squeezing Out
             more IP's' 

The Subnetting features of IPv4 did not offer much through options and 
choice regarding IP Address assignment, allocation, or Networking in 
general. And while Subnetting the Network ( The sub-division of the 
Parent Network IP Address ) did relieve congestion, provided 
performance gains, and improved management. Needless to say, these 
were indeed significant benefits for the groping beginnings. Still, 
it did nothing to increase the number of IP Addresses for allocation 
to establish a new Network, that is, offer another outside connection: 
the Parent Network. However, it did provide the IETF with a foundation, 
if exploited, would have avoided the necessity of an urgency fostered 
by explosive growth, to implement a new IP Addressing Scheme. 

Nevertheless, by exploiting the Default Subnet Mask, that is, 
understanding its real purpose as used in BITWISE ANDING. That being, 
IP Network Address Resolution, Octet by Octet. Then anyone could 
easily visualize that, the former IPv4 Class Addressing Scheme, as 
depicted in Appendix I, Table 6, could be expanded to that rendered 
in Table 4. Where the Default Subnet Mask, now the Subnet Identifier, 
assumes the duties of its actual definition. That is, it remains the 
Default Subnet Mask, which when used in Bitwise Anding serves to 
resolve the Parent Network IP Address. This working definition itself, 
commands the expansion and the results as depicted in Table 4. In 
other words, the Default Subnet Mask, is indeed a Subnet, and it is 
also an Identifier of the Parent Network.

It should be understood, that the explosive growth of the Internet 
and Internetworking environments fostered a serious burden upon the 
IETF, as well as every other IP Professional. Notwithstanding the 
hurried imposition of market demands, the crying voice of the consumer, 
and the shortage of the insightful. It is easy to understand why the 
nearly Doubled Number of Available IP Addresses, from an expansion of 
this 32 Bit Addressing Scheme was over looked. As a point of fact, 
this could quite easily be the underlining reason for 2 digit coding 
error, that the resulting Y2K problem. In other words, the original 
IPv4 Class Addressing Scheme yield the possible number of available 
IP Addresses as being approximately 3.12 * 10^9. While the expansion 
given be Table 4, renders the number of available IP Addresses as 
being approximately 5.46 * 10^9. Which, to say the very least, is 
nearly double the original value, while the Address Range remained 
Constant; i.e. 32 Bits. Now, just consider what could be achieved if 
the Address Bit Range was increased to 128 Bit, which is the case for 
IPv6. It is worthy of note, to mention that, while IPv6 has a reported 
address range of approximately Y.YY * 10^39 at an 128 Bit Address 
Space. Well- Only a test of Class A-1, of IPv4 was performed. 
Nonetheless, the value of the number of available IP Addresses in 
IPv4 was approximately Y.YY * 10^22, however this was only a 64 Bit 
Address Space.

Moreover, the great benefit of this analysis is that, IPv4 was shown 
to have a life, retained its simplicity, and would prove to be cost 
effective if its expansion were implemented. Furthermore, from 
Figure 1, it can be easily seen that, tools can be created to assist 
the professional, that is, if IPv4 were implemented as either a 64 or 
128 Bit IP Address Range, that would facilitate IP resolution, and 
expedite IP deployment in any Network.




========================================================================
=  Octets     2st   3nd   4rd                  Figure 1                =
=              |     |  .......                                        =
=              |     |  .     .                                        =
=  -----       v     |  . 001 .  The IP Addressing Slide Ruler clearly =
=    ^      .......  |  .......  establishes the Differences between   =
=    |      . **  .  |  .     .  Decimal and Binary Calculations.      =
=    |      . 001 .  v  . 160 .  Where, in this case, the Number of    =
=    |      ...................  Rulers or Slides, represents the      =
=    |      ...................  Maximum number of Hosts available in  =
=    |      .     .     .     .  an IP Address Range having an         =
=           . 160 . 001 . 188 .  Exponental Power of 3. That is, if    =
=   IP      ...................  the First Octet is Defined by the     =
= Address   ...................  "Subnet Identifier", as providing     =
= Range     .     .     .     .  a Network within the IP Address       =
=           . 188 . 160 . 223 .  Range assigned to this Class. That is,=
= 1 - 254   ...................  the individual Ruler or Slide, has a  =
=    |      ...................  one-to-one correspondence with the    =
=    |      .     .     .     .  OCTET it represents, and is equal to  =
=    |      . 223 . 188 . 239 .  an Exponental Power of 1. Which also  =
=    |      ...................  maintains this one-to-one             =
=    |      ...................  relationship. In any case, it should  =
=    |      .     .     .     .  be understood that the Decimal is an  =
=    |      . 239 . 223 . 254 .  Integer representing the IP Address,  =
=    |      ...................  and has only 1 value that occupies    =                                                                    
=    |      ...................  the given Octet. However, the Binary  =
=    |      .     .     .        representation for the IP Address, is =
=    |      . 254 . 239 .        an 8 digit Logical Expression         =
=    v      .............        occuping one Octet. Where each digit  =                      
=  -----          .......        has a 2 state representation of either=
=                 .     .        a 1 or a 0. The distinction is that,  =                                                               
=                 . 254 .        this is a Logical expression, that has=
=                 .......        no Equivalence. However, there is a   =
=                                Mathematical Method which resolves    =
= The ( ** ) indicates           this distinction, and allows for the  =
= the Reference point            Translation of each into the other.   =
= of the IP Side Ruler.          In other words, one System can never  =
=                                be used to interpret any given value  =
=                                of the other, at least, not without   =
=                                the Mathematical Method used for      =                                                                      
=                                Translation. But each, can separately =
=                                be mapped to the structure of the 'IP =
=                                Slide Ruler ', rendering a translation=
=                                for one of the two representations.   =
=                                (Noting that the Binary Translation of=
=                                 its Decimal equivalent must be known =
=                                 first.)                              =
========================================================================







Chapter III  'Addressing the Future, taking notes on the IP's past'

The IP Addressing Scheme of IPv4 served, some actually thought, the 
limits of its rational purpose. It was indeed successful in meeting 
the challenges of the growing Internetworking community. However, it 
has been shown and proven that, it limits may not possess a boundary. 
Nevertheless, while there exist the possibility of expanding the 
number of Bits of any IP Addressing Scheme either previously or 
currently employed. At this junction, one would perhaps question the 
aggression of this endeavor. In other words, prior to any major 
change, it now seem more rational to consider directing ones thoughts 
towards the simple idea of exploitation that which is already in 
place. 

Just imagine for a moment, the cost of implementing another more 
radical, or different IP Addressing Scheme. 'If it ain't broke, don't 
fix it! ' Clearly, this phrase has no application here, because, even 
with backward compatibility were employed. Would that suffice? Would 
it allow the millions of software applications, or hardware for that 
matter, and the purchaser's to run their course of time through its 
normal use and associated life expectancy? Or is this some sort proof, t
hat we have not learned to fully use and exploit our available 
resources? Nonetheless, one of the most obvious implications of 
the exploitation of IPv4 was that an Addressing beyond the 32 Bit now 
employed is not necessary. Furthermore, the analogy of the Telephone 
Area Codes, renders an excellent example of measures that might be 
employed to separate the current IPv4 Addressing Scheme so that every 
country could used the same scheme, differing only by the Code of the 
Country that uses it. While clearly, the Class System has been 
erroneously blamed, guilty of providing too many IP Addresses because 
the Classes were too large in size. However, this class system 
provided a greater structure to the IP Addressing System, one that I 
fear, will be lost when IPv6 is employed.



Chapter IV  'The Questions Raised: Is there Tenure for a Better 
             Allocation Plan?' 

When entertaining the thoughts of a tenure, I think of Einstein and 
Newton, whose works will never be forgotten, and will forever retain a 
significance in natural science and mathematics. Even when reading the 
works dealing with Internetworking, I noticed the popular catch 
phrases of its underlining philosophy: ' Careful Planning ' , 
' Attention to Detail ', ' Thorough Analysis 'or ' Planned Expansion '. 
In other words, it is possible to plan or devise an IP Address 
Allocation Scheme , with the appropriate IP Address System, that will 
not have a tenure. Because, the right IP Addressing System, with an 
accompanying Allocation Scheme could possibly last forever, and never
would be an exhaustion of the available IP Addresses. In other words, 
the foundation has been provided, all that is needed now are the 
careful collective thoughts for its use and implementation.




Chapter V Conclusion; 'An Alternate View of the IPv4 Addressing 
                       Scheme would prove Beneficial now!' 

Nevertheless, if we allowed the Pre-Defined ' Class Address Range ', 
remain as the standard classification for the division of the IPv4 
Addressing Scheme Classes, and redefined the application / used of the 
Default Subnet  as the Subnet Identifier. We could effectively extend 
the amount of Addresses available for use under our current system. 
Where by, we could  increase the number of addresses available in the 
current system ( IPv4 ) while retaining the simplicity of its ease of 
use and implementation. As has been clearly shown.

Needless to say, this can be organized as: 


                        Table 1. 
    Structure of the Decimal Representation IP Class System 


1. Class A-1, 1 - 126, 
   Subnet Identifier 255.000.000.000: 
   126 Networks and 254^3 Hosts             0 

   Class A-2, 1- 126, 
   Subnet Identifier 255.255.000.000:  
   126^2 Networks and 254^2 Hosts         10 

   Class A-3, 1 - 126, 
   Subnet Identifier 255.255.255.000: 
   126^3 Networks and 254 Hosts             110 
  

2. Class B-1, 128 - 191, 
   Subnet Identifier 255.000.000.000: 
   63 Networks and 254^3 Hosts             0 

   Class B-2, 128 - 191, 
   Subnet Identifier 255.255.000.000: 
   63^2 Networks and 254^2 Hosts         10 

   Class B-3, 128 -191, 
   Subnet Identifier 255.255.255.000:  
   63^3 Networks and 254 Hosts            110 


3. Class C-1, 192 - 223, 
   Subnet Identifier 255.000.000.000: 
   31 Networks and 254^3 Hosts             0 

   Class C-2, 192 - 223, 
   Subnet Identifier 255.255.000.000: 
   31^2 Networks and 254^2 Hosts        10 

   Class C-3, 192 - 223, 
   Subnet Identifier 255.255.255.000: 
   31^3 Networks and 254 Hosts           110 
  

4. Class D-1, 224 - 239, 
   Subnet Identifier 255.000.000.000: 
   15 Networks and 254^3 Hosts               0 

   Class D-1, 224 - 239, 
   Subnet Identifier 255.255.000.000: 
   15^2 Networks and 254^2 Hosts          10 

   Class D-3, 224 - 239, 
   Subnet Identifier 255.255.255.000: 
  15^3 Networks and 254 Hosts             110 
  

5. Class E-1, 240 - 254, 
   Subnet Identifier 255.000.000.000: 
   14 Networks and 254^3 Hosts               0 

   Class E-2, 240 - 254, 
   Subnet Identifier 255.255.000.000: 
   14^2 Networks and 254^2 Hosts          10 

   Class E-3, 240 - 254, 
   Subnet Identifier 255.255.255.000: 
   14^3 Networks and 254 Hosts             110 
  
  
Special Note: 255.255.255.255 remains' a Broadcast IP, 127.x.x.x 
              remains LoopBack IP, and 0.0.0.0 remains Network 
              Unknown. This renders the Range of possible Hosts to 
              the Value of the given range; 1 - 254. However, with the
              implementation of this minor change to the Class 
              Addressing Scheme, not only can Class D and E be given 
              their respective Multicast and Experimental IP Addresses.
              But, with this new division, even smaller Companies or 
              perhaps private homes can now have their individual IP
              Addresses without the problems associated with IP Address 
              Flooding, which plagued the former Addressing Scheme. 
              Moreover, this minor change does not usher the 
              complexities or sharpen the learn curve, as does IPv6. 
              And while it does not provide a greater number of 
              available IP Addresses for allocation to public and 
              private sectors, there is a definite increase of 
              available IP Addresses. 
  
  
Nevertheless, from the above representation ( table 1 ) it can seen 
that the calculations for the total number of Hosts ( that is the 
combined number of Networks and Hosts, from each class ) is similar 
to the layout of IPv4. And while the learning curve still exists, 
the difficulties would be miniscule when compared to that of learning 
Ipv6, at least for those acquainted with IP Addressing. 

Needless to say, a comparison between the entries of Table 1, when 
utilizing Figure 1 ( taking the " ** " of the first IP Slide Ruler 
as our reference point ), that renders a clear and concise view of 
the ease of use and possible implementation of this minor change in 
the IPv4 Addressing Scheme. Nonetheless, it should be emphasized, that 
the authoritative community as a whole; i.e. Authors of IP Addressing 
or Internetworking Fundamentals, have shown a lack of continuity and 
consistency regarding the actual methods, determination and or actual 
explanation of the processes involved in these calculations. Where by, 
it has been a consistent error regarding the confusion or inability to 
differentiate between the calculation of the Decimal Number and the 
Binary Number for their individual determination. Which, to say the 
very least, has rendered the understanding of the most significant 
part of the concept of Internetworking ( that of IP Addressing ) almost
an impossible undertaking. However, this is not, nor is it intended to 
be, a verbal lashing of the Computer Science Community. Needless to 
say, through the use of the IP Slide Rules, one can easily see the 
difference between the numeral values of the Decimal and Binary 
calculations, including their results. 

Nevertheless, to continue with the analysis and the comparison. It 
should now become easy to determine the number of Networks and Hosts 
for a given Network Address. For example, if anyone needed to know the 
number of Hosts for a given Class, they need only to observe the First 
Octet of the Network Address and its Subnet Identifier. Especially
since, it has been established that there is a distinct difference 
between the calculation of the Decimal and the Binary notations. Where 
by, the clarity of the latter is given by Table II when compared with 
that of Table1.




                             Table 2. 
        Structure of the Binary Representation IP Class System 
  
1. Class A-1, 1 - 126, 
   Subnet Identifier 255.000.000.000: 
   126 Networks and 2^24 Hosts            0 

   Class A-2, 1- 126,  
   Subnet Identifier 255.255.000.000:  
   2^14 Networks and 2^16 Hosts           10 

   Class A-3, 1 - 126, 
   Subnet Identifier 255.255.255.000:
   2^21 Networks and 2^8 Hosts             110 


2. Class B-1, 128 - 191, 
   Subnet Identifier 255.000.000.000: 
   63 Networks and 2^24 Hosts             0 

   Class B-2, 128 - 191, 
   Subnet Identifier 255.255.000.000: 
   2^14 Networks and 2^16 Hosts          10 

   Class B-3, 128 -191, 
   Subnet Identifier 255.255.255.000: 
   2^21 Networks and 2^8 Hosts           110 
  

3. Class C-1, 192 - 223,
   Subnet Identifier 255.000.000.000: 
   31 Networks and 2^24 Hosts               0 

   Class C-2, 192 - 223, 
   Subnet Identifier 255.255.000.000: 
   2^14 Networks and 2^16 Hosts         10 

   Class C-3, 192 - 223, 
   Subnet Identifier 255.255.255.000: 
   2^21 Networks and 2^8 Hosts           110 


4. Class D-1, 224 - 239, 
   Subnet Identifier 255.000.000.000: 
   15 Networks and 2^24 Hosts               0 

   Class D-21, 224 - 239, 
   Subnet Identifier 255.255.000.000:
   2^14 Networks and 2^16 Hosts          10 

   Class D-3, 224 - 239, 
   Subnet Identifier 255.255.255.000: 
   2^21 Networks and 2^8 Hosts             110 


5. Class E-1, 240 - 254, 
   Subnet Identifier 255.000.000.000: 
   14 Networks and 2^24 Hosts             0 

   Class E-2, 240 - 254, 
   Subnet Identifier 255.255.000.000: 
   2^14 Networks and 2^16 Hosts          10 

   Class E-3, 240 - 254, 
   Subnet Identifier 255.255.255.000: 
   2^21 Networks and 2^8 Hosts            110 



Note: The number of Networks in the Primary Division of each Class, is 
      the difference between the IP Address Range for each respective 
      Class Boundary's. Moreover, the Subnet Identifier, 255, has a 
      Binary Representation of; 11111111. 


>From the above however, it can be clearly seen that the Total Number 
of Hosts resulting from this 32 Bit Expansion achieves approximately 
5.xx * 10^9. However, it does, at least make provisions for 
individualized IP Addressing assignments. Nevertheless, it is far 
less than the reported X.xx * 10^39 number of Hosts promised by the 
implementation of IPv6. The conclusions of the former notwithstanding, 
however, it should be pointed out. That a 64 or more Bit Expansion of 
the current IPv4 Addressing Scheme would more closely approach, and 
possibly exceed, not only the Number of Hosts, as is the promise of 
IPv6. But, would retain its overall simplicity, in its implementation 
and ease of use.

Furthermore, this particular Addressing Scheme follows, as can be 
argued, directly from Logic of the Binary Representation and its 
inherent methods of Mathematical Reasoning. Where by, the clarity and 
support of this argument is given by Table 3. 



                         Table 3. 
  Structure of the 64 Bit Decimal Representation IP Class System 
  
1. Class A-1, 1 - 126, 
   Subnet Identifier 255.000.000.000.000.000.000.000: 
   126^1 Networks and 254^7 Hosts : 0 

  Class A-2, 1- 126, 
  Subnet Identifier 255.255.000.000.000.000.000.000:  
  126^2 Networks and 254^6 Hosts : 10 

  Class A-3, 1 - 126, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  126^3 Networks and 254^5 Hosts : 110 

  Class A-4, 1 - 126, 
  Subnet Identifier 255.255.255.255.000.000.000.000: 
  126^4 Networks and 254^4 Hosts : 1110 

  Class A-5, 1 - 126, 
  Subnet Identifier 255.255.255.255.255.000.000.000: 
  126^5 Networks and 254^3 Hosts :11110 

  Class A-6, 1 - 126, 
  Subnet Identifier 255.255.255.255.255.255.000.000: 
  126^6 Networks and 254^2 Hosts :111110 

  Class A-7, 1 - 126, 
  Subnet Identifier 255.255.255.255.255.255.255.000: 
  126^7 Networks and 254 Hosts : 1111110 


2. Class B-1, 128 - 191, 
   Subnet Identifier 255.000.000.000.000.000.000.000: 
   63^1 Networks and 254^7 Hosts : 0 

  Class B-2, 128 - 191, 
  Subnet Identifier 255.255.000.000.000.000.000.000: 
  63^2 Networks and 254^6 Hosts : 10 

  Class B-3, 128 -191, 
  Subnet Identifier 255.255.255.000.000.000.000.000:  
  63^3 Networks and 254^5 Hosts :110 

  Class B-4, 128 -191, 
  Subnet Identifier 255.255.255.255.000.000.000.000:  
  63^4 Networks and 254^4 Hosts :1110 

  Class B-5, 128 -191, 
  Subnet Identifier 255.255.255.255.255.000.000.000:  
  63^5 Networks and 254^3 Hosts:11110 

  Class B-6, 128 -191, 
  Subnet Identifier 255.255.255.255.255.255.000.000:  
  63^6 Networks and 254^2 Hosts:111110 

  Class B-7, 128 -191, 
  Subnet Identifier 255.255.255.255.255.255.255.000: 
  63^7 Networks and 254 H osts:11111110 
  


3. Class C-1, 192 - 223, 
   Subnet Identifier 255.000.000.000.000.000.000.000: 
   31^1 Networks and 254^7 Hosts : 0 

  Class C-2, 192 - 223, 
  Subnet Identifier 255.255.000.000.000.000.000.000: 
  31^2 Networks and 254^6 Hosts : 10 

  Class C-3, 192 - 223, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  31^3 Networks and 254^5 Hosts :110 

  Class C-4, 192 - 223, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  31^4 Networks and 254^4 Hosts :1110 

  Class C-5, 192 - 223, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  31^5 Networks and 254^3 Hosts :11110 

  Class C-6, 192 - 223, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  31^6 Networks and 254^2 Hosts:111110 

  Class C-7, 192 - 223, 
  Subnet Identifier 255.255.255.000.000.000.000.000: 
  31^7 Networks and 254 Hosts :1111110 


4. Class D-1, 224 - 239, 
   Subnet Identifier 255.000.000.000.000.000.000.000: 
   15^1 Networks and 254^7 Hosts : 0 

   Class D-2, 224 - 239, 
   Subnet Identifier 255.255.000.000.000.000.000.000: 
   15^2 Networks and 254^6 Hosts : 10 

   Class D-3, 224 - 239, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   15^3 Networks and 254^5 Hosts :110 

   Class D-4, 224 - 239, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   15^4 Networks and 254^4 Hosts:1110 

   Class D-5, 224 - 239, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   15^5 Networks and 254^3 Hosts:11110 

   Class D-6, 224 - 239, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   15^6 Networks and 254^2 Hosts:111110 

   Class D-7, 224 - 239, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   15^7 Networks and 254 Hosts :1111110   


5. Class E-1, 240 - 254, 
   Subnet Identifier 255.000.000.000.000.000.000.000: 
   14^1 Networks and 254^7 Hosts : 0 

   Class E-2, 240 - 254, 
   Subnet Identifier 255.255.000.000.000.000.000.000: 
   14^2 Networks and 254^6 Hosts : 10 

   Class E-3, 240 - 254, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   14^3 Networks and 254^5 Hosts : 110

   Class E-4, 240 - 254, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   14^4 Networks and 254^4 Hosts :1110 

   Class E-5, 240 - 254, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   14^5 Networks and 254^3 Hosts :11110 

   Class E-6, 240 - 254, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   14^6 Networks and 254^2 Hosts:111110 

   Class E-7, 240 - 254, 
   Subnet Identifier 255.255.255.000.000.000.000.000: 
   14^7 Networks and 254 Hosts :1111110 





                                  Table 4. 
     Structure Reality of the Decimal Representation IP Class System 

1. Class A-1, 1 - 126, Subnet Identifier 255.y.y.y: 
   2,056,641,048 Networks and 8,129,016 Hosts: 0 
  
   Class A-2, 1- 126, Subnet Identifier 255.255.y.y:  
   1,032,321,024 Networks and 32,004 Hosts:  10 

   Class A-3, 1 - 126, Subnet Identifier 255.255.255.y: 
   2,000,376 Networks and 254 Hosts:          110 
  

2. Class B-1, 128 - 191, Subnet Identifier 255.y.y.y: 
   1,028,320,524 Networks and 4,064,508 Hosts: 0 

   Class B-2, 128 - 191, Subnet Identifier 255.255.y.y: 
   256,048,002 Networks and 16,002 Hosts: 10 
 
   Class B-3, 128 -191, Subnet Identifier 255.255.255.y:  
   250,047 Networks and 254 Hosts:     110 


3. Class C-1, 192 - 223, Subnet Identifier 255.y.y.y: 
   505,998,988 Networks and 1,999,996 Hosts: 0 

   Class C-2, 192 - 223, Subnet Identifier 255.255.y.y: 
   61,992,802 Networks and 7,074 Hosts        10 

   Class C-3, 192 - 223, Subnet Identifier 255.255.255.y: 
   29,791 Networks and 254 Hosts           110 
  

4. Class D-1, 224 - 239, Subnet Identifier 255.y.y.y: 
   244,838,220 Networks and 967,740 Hosts:      0 

   Class D-1, 224 - 239, Subnet Identifier 255.255.y.y: 
   14,512,290 Networks and 3,810 Hosts: 10 

   Class D-3, 224 - 239, Subnet Identifier 255.255.255.y: 
   3,375 Networks and 254 Hosts             110 
  

5. Class E-1, 240 - 254, Subnet Identifier 255.y.y.y: 
   228,515,672 Networks and 903,224 Hosts:   0 

   Class E-2, 240 - 254, Subnet Identifier 255.255.y.y: 
   12,641,580 Networks and 3,556 Hosts:    10 

   Class E-3, 240 - 254, Subnet Identifier 255.255.255.y: 
   2,744 Networks and 254 Hosts:          110


Note: The Ternary Section of every Network Class need not be 
      Sub-Divided, and could be combined for the issuance of 
      Individual IP Addresses.


Nevertheless, the question of ponderence, is whether or not this is 
the perfect IP Addressing System. That is, does it have a tenure or 
life expectancy? I would assume that almost everyone would answer 
these questions with a no. 

In fact, an issue, while not a major problem, does indeed exist with 
the current expansion of IPv4 Class Addressing Scheme, as depicted in 
Table 1. Where by, the Mathematics Analysis reveals that the Second 
Octet of the Primary Section of Each Class maintains a Set of Values 
within each of their respective IP Address Ranges. Which can not be 
employed or used as part of the count resulting in the total number 
of available IP Addresses. This is because they are not available as 
a valid IP Address, and if they were, then there would exist a 
mathematical conflict with the calculation of the total number of 
available IP Addresses of the Secondary Section for each IP Address 
Class. In other words, there would arise an error in reporting the 
results of the calculated totals. This can easily visualized when 
compared with the results of the second Octet of the Secondary 
Section for each of the IPv4 Class Address Ranges. That is, there 
exist a barrier imposed by the use of the Subnet Identifier of the 
second Octet from the Secondary Section of each IPv4 Class Address 
Schemes, with bars the use of any of the numbers given by the IP 
Address Range for that given IP Address Class. This is seen true, 
because the 1 - 254 total Host Count, does indeed contain all of the 
numbers available to be used as IP Addresses. However, this does 
cripple the IPv4 Class Addressing System. Where by, the calculation 
of the mathematical difference between IP Address Range for each Class 
and the total Host count would yield the valid Address Range that can 
be use to calculate that total number of  available IP Addresses. 
Nonetheless, while using the pictorial of Figure A, this is given by 
the following Table of Laws / Rules:



              { The IPV4 The Laws of the Octet }         

1. By definition, there exist 3 distinct Sections or Divisions for 
   every IP Address Class. However, the number of Sections or 
   Divisions is dependent upon IP Bit Address Range defined for the 
   IP Address.

2. The Sections or Divisions of the IP Address Class are defined as: 
   Primary, Secondary, Ternary, etc...And are labeled according to 
   their respective Class Location (e.g.: Class A would be Class A-1, 
   Class A-2, Class A-3, and continued as would be necessary to 
   distinguish the remaining Classes, B - E.)

3. The Subnet Identifier assigns to any Octet in which it defines, 
   within any Section or Division of every IP Class, when not use as 
   the Default Subnet Mask, only the value of the numbers available 
   in the IP Address Range assigned to that IP Class.

4. For every OCTET within any Section or Division of any IP Class, 
   which is not defined by the Subnet Identifier, can be assigned 
   any value in the range of 1 - 254. That is, provided that there 
   is no succeeding Section or Division, or the OCTET of the 
   succeeding Section or Division, whose reference is the same Octet, 
   is not defined by the Subnet Identifier.

5. For every OCTET within any Section or Division of any IP Class, 
   that is defined by the Subnet Identifier and is preceded by a 
   Section or Division whose reference is the same Octet. Where the 
   case is such that: The Octet of the preceding Section or Division 
   is not defined by the Subnet Identifier. Then, the Octet of the 
   preceding Section or Division can not be assigned any value as 
   given by the IP Address Range assigned to that IP Class.



Needless to say, this situation can be further explored, through 
mathematical calculations. Where the given example in this case would 
be Class A-1 and Class A-2.


1.Class A-1, 1 - 126, Subnet Identifier 255.000.000.000: 126 Networks 
                      and 254^3 Hosts: 0 

2.Class A-2, 1- 126, Subnet Identifier 255.255.000.000:  126^2 Networks
                     and 254^2 Hosts: 10



Nevertheless, upon examination of these classes, it is quite obvious 
that if Class A-1's second Octet were to maintain the IP Address 
Range from 1 - 126, then it would be reporting IP Address of Class A-2 
because the second Octet of this Class is defined by the Subnet 
Identifier. However, the easiest mathematical method for the 
determination of the total number of available IP Addresses from Class 
A-1 would be to calculate the total number of IP Addresses available 
from its original configuration. Then subtract the value as would be 
determined from the calculation of the Class A-1 IP Address 
configuration that can not be used. In which case, we have: 



3. Class A-1, 1 - 126, Subnet Identifier 255.126.000.000: 126 Networks 
                       and 254^2 Hosts    0
                                                             


                           or

4.   126 * (254)^2 = 8,129,016




Where the total, would be that given by Table 1., as being: 




5.   126 * (254)^3 = 2,064,770,064







In other words, the total number of available IP Addresses in 
Class A-1, that could be assigned as a Parent Network IP Address for 
connection to the Internetwork ( That is, other than the in house 
Network ), would be the difference between these equations. As given 
by:




6.   2,064,770,064 - 8,128,016 = 2,056,641,048





This method is demonstrated and render by Table 4. Where the results 
of equation 6 equals the total number of IP Addresses available for 
assignment as a Parent Network in an Internetworking Environment, and 
the results of equation 4 yield the number of Hosts that can be 
repeatedly assigned and used as private Domain Network IP Addresses. 
In which case, one would need to access the Parent Network to have 
access to any of these internal private Networks and Hosts identified 
by these IP Addresses. Thus, there would be no conflict from there 
continued use!

Nonetheless, this line of logical reasoning can be and is applied 
throughout the expansion of the IPv4 Class Addressing Scheme, as 
denoted by Table 4. Where it can be concluded that, regardless of IP 
Address Range, that being 32, 64, 128, or 256 Bit, it does not matter. 
This method of mathematical calculation would still apply. In which 
case, this white paper concludes; There yet remains a value in the 
IPv4 Addressing Scheme, which surpasses the promises of IPv6, and 
could conceivably satisfy our needs indefinitely without an expansion 
beyond the 32 Bit address range. That is, if it were distributed with 
country and or state codes as its prefix.



Security 

There are no security considerations rendered in this document.



Appendix I : Previous IPv4 Class Addressing System Schematic 



                      Table 6. 
 Structure of the IPv4 Representation IP Class System 

Class A, 1 - 126, Default Subnet Mask 255.y.y.y: 126 Networks 
                  and 16,387,064 Hosts: 0 

Class B, 128- 191, Default Subnet Mask 255.255.y.y:  16,383 Networks 
                   and 32,004 Hosts:  10 

Class A-3, 192 - 223, Default Subnet Mask 255.255.255.y: 
                      2,097,151 Networks and 254 Hosts:110



Note: If you enjoy the exercise, feel free, find and correct the 
      Mathematical problems. 





References 

1.  E. Terrell ( not published notarized, 1979 ) " The Proof of 
    Fermat's Last Theorem: The Revolution in Mathematical Thought " 
    Outlines the significance of the need for a thorough understanding 
    of the Concept of Quantification and the Concept of the Common 
    Coefficient. These principles, as well many others, were found to 
    maintain an unyielding importance in the Logical Analysis of 
    Exponential Equations in Number Theory. 

2.  E. Terrell ( not published notarized, 1983 ) " The Rudiments of 
    Finite Algebra: The Results of Quantification " Demonstrates the 
    use of the Exponent in Logical Analysis, not only of the Pure 
    Arithmetic Functions of Number Theory, but Pure Logic as well. 
    Where the Exponent was utilized in the Logical Expansion of the 
    underlining concepts of Set Theory and the Field Postulates. The 
    results yield; another Distributive Property ( i.e. Distributive 
    Law ) and emphasized the possibility of an Alternate View of the
    Entire Mathematical field. 

3.  G Boole ( Dover publication, 1958 ) "An Investigation of The Laws 
    of Thought" On which is founded The Mathematical Theories of Logic 
    and Probabilities; and the Logic of Computer Mathematics. 

4.  R Carnap ( University of Chicago Press, 1947 / 1958 ) "Meaning and 
    Necessity" A study in Semantics and Modal Logic. 

5.  R Carnap ( Dover Publications, 1958 ) " Introduction to Symbolic 
    Logic and its Applications" 
  
6.  Authors: Arnett, Dulaney, Harper, Hill, Krochmal, Kuo, LeValley, 
    McGarvey, Mellor, Miller, Orr, Ray, Rimbey, Wang, ( New Riders 
    Publishing, 1994 ) " Inside TCP/IP " 

7.  B Graham ( AP Professional, 1996 )  " TCP/IP Addressing " 
    Lectures on the design and optimizing IP addressing. 

8.  Postel, J. (ed.), "Internet Protocol - DARPA Internet Program  
    Protocol Specification," RFC 791, USC/Information Sciences 
    Institute, September 1981.   

9.  Cisco Systems, Inc. ( Copyright 1989 - 1999 ) " Internetworking 
    Technology Overview " 

10. S. Bradner, A. Mankin, Network Working Group of Harvard University
    ( December 1993 ) " RFC 1550: IP: Next Generation (IPng) White 
    Paper Solicitation "  

11. RFC 791 



Author 
( please send comments to the address below ) 

Eugene Terrell 
24409 Soto Road  Apt. 7 
Hayward, CA.  94544-1438 
Voice: 510-537-2390 
E-Mail: eterrell00@netzero.net 

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