HTTP/1.1 200 OK Date: Tue, 09 Apr 2002 11:50:34 GMT Server: Apache/1.3.20 (Unix) Last-Modified: Tue, 31 Aug 1999 19:07:33 GMT ETag: "2e6e59-e17f-37cc27f5" Accept-Ranges: bytes Content-Length: 57727 Connection: close Content-Type: text/plain IT Professional, Author / Researcher E. Terrell Internet Draft August 1999 Document: draft-terrell-math-ipaddr-ipv4-02.txt Category: Informational expires March 1, 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 RFC 2026. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet-Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. Conventions Please note the font size of the Tables contained in this white paper are smaller than the expected 12 pts. However, if you are using the most current Web Browser, the View Section of the Title bar provides you with the option to either increase or decrease the font size for comfort level of viewing. (Provided that this is the HTML version.) 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 Appendix II : The Mathematical Anomaly Explained Appendix III : A peak into the Classless System (Are the Advantages Real?) References Abstract This paper was necessitated by an overwhelming desire. An attempt to end the disparity in the dissemination of information which is 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. 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 in its ease of use and 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. 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 other 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 the 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. I will, because of my continued belief in its importance, and the significance of the resulting information, begin with an overview of the IP Class Addressing Scheme. Nevertheless, 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 numbers. 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, has presented a difference, 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, which 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 or the other, requires another mathematical operation for this equivalency to be 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, and 'v' represents some Mathematical Expression used to Translate the Banary to its Decimal equivalent. } 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 IT Professional. Notwithstanding the hurried imposition of market demands, the 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. 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, this is a Logical expression, = . 254 . that has no Equivalence. However, is = ....... a there Mathematical Method which = The ( ** ) indicates resolves this distinction, and allows = the Reference point for the Translation of each into the = of the IP Side Ruler. other. 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: 64 Networks and 254^3 Hosts 0 Class B-2, 128 - 191, Subnet Identifier 255.255.000.000: 64^2 Networks and 254^2 Hosts 10 Class B-3, 128 -191, Subnet Identifier 255.255.255.000: 64^3 Networks and 254 Hosts 110 3. Class C-1, 192 - 223, Subnet Identifier 255.000.000.000: 32 Networks and 254^3 Hosts 0 Class C-2, 192 - 223, Subnet Identifier 255.255.000.000: 32^2 Networks and 254^2 Hosts 10 Class C-3, 192 - 223, Subnet Identifier 255.255.255.000: 32^3 Networks and 254 Hosts 110 4. Class D-1, 224 - 239, Subnet Identifier 255.000.000.000: 16 Networks and 254^3 Hosts 0 Class D-1, 224 - 239, Subnet Identifier 255.255.000.000: 16^2 Networks and 254^2 Hosts 10 Class D-3, 224 - 239, Subnet Identifier 255.255.255.000: 16^3 Networks and 254 Hosts 110 5. Class E-1, 240 - 254, Subnet Identifier 255.000.000.000: 15 Networks and 254^3 Hosts 0 Class E-2, 240 - 254, Subnet Identifier 255.255.000.000: 15^2 Networks and 254^2 Hosts 10 Class E-3, 240 - 254, Subnet Identifier 255.255.255.000: 15^3 Networks and 254 Hosts 110 Note: The Equation for Determining the IP Address Range for any IP Class is; (REN - RBN) + 1 = Total of Available IP Addresses for the given Class. (Where, R = Range, E = End, B = Beginning, N = Number) 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: 64 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: 32 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: 16 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: 15 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 'Quantified difference between the IP Address Range Plus 1', for each respective Class Boundary's. [(REN - RBN) + 1)] 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: 64^1 Networks and 254^7 Hosts : 0 Class B-2, 128 - 191, Subnet Identifier 255.255.000.000.000.000.000.000: 64^2 Networks and 254^6 Hosts : 10 Class B-3, 128 -191, Subnet Identifier 255.255.255.000.000.000.000.000: 64^3 Networks and 254^5 Hosts :110 Class B-4, 128 -191, Subnet Identifier 255.255.255.255.000.000.000.000: 64^4 Networks and 254^4 Hosts :1110 Class B-5, 128 -191, Subnet Identifier 255.255.255.255.255.000.000.000: 64^5 Networks and 254^3 Hosts:11110 Class B-6, 128 -191, Subnet Identifier 255.255.255.255.255.255.000.000: 64^6 Networks and 254^2 Hosts:111110 Class B-7, 128 -191, Subnet Identifier 255.255.255.255.255.255.255.000: 64^7 Networks and 254 H osts:11111110 3. Class C-1, 192 - 223, Subnet Identifier 255.000.000.000.000.000.000.000: 32^1 Networks and 254^7 Hosts : 0 Class C-2, 192 - 223, Subnet Identifier 255.255.000.000.000.000.000.000: 32^2 Networks and 254^6 Hosts : 10 Class C-3, 192 - 223, Subnet Identifier 255.255.255.000.000.000.000.000: 32^3 Networks and 254^5 Hosts :110 Class C-4, 192 - 223, Subnet Identifier 255.255.255.255.000.000.000.000: 32^4 Networks and 254^4 Hosts :1110 Class C-5, 192 - 223, Subnet Identifier 255.255.255.255.255.000.000.000: 32^5 Networks and 254^3 Hosts :11110 Class C-6, 192 - 223, Subnet Identifier 255.255.255.255.255.255.000.000: 32^6 Networks and 254^2 Hosts:111110 Class C-7, 192 - 223, Subnet Identifier 255.255.255.255.255.255.255.000: 32^7 Networks and 254 Hosts :1111110 4. Class D-1, 224 - 239, Subnet Identifier 255.000.000.000.000.000.000.000: 16^1 Networks and 254^7 Hosts : 0 Class D-2, 224 - 239, Subnet Identifier 255.255.000.000.000.000.000.000: 16^2 Networks and 254^6 Hosts : 10 Class D-3, 224 - 239, Subnet Identifier 255.255.255.000.000.000.000.000: 16^3 Networks and 254^5 Hosts :110 Class D-4, 224 - 239, Subnet Identifier 255.255.255.255.000.000.000.000: 16^4 Networks and 254^4 Hosts:1110 Class D-5, 224 - 239, Subnet Identifier 255.255.255.255.255.000.000.000: 16^5 Networks and 254^3 Hosts:11110 Class D-6, 224 - 239, Subnet Identifier 255.255.255.255.255.255.000.000: 16^6 Networks and 254^2 Hosts:111110 Class D-7, 224 - 239, Subnet Identifier 255.255.255.255.255.255.255.000: 16^7 Networks and 254 Hosts :1111110 5. Class E-1, 240 - 254, Subnet Identifier 255.000.000.000.000.000.000.000: 15^1 Networks and 254^7 Hosts : 0 Class E-2, 240 - 254, Subnet Identifier 255.255.000.000.000.000.000.000: 15^2 Networks and 254^6 Hosts : 10 Class E-3, 240 - 254, Subnet Identifier 255.255.255.000.000.000.000.000: 15^3 Networks and 254^5 Hosts : 110 Class E-4, 240 - 254, Subnet Identifier 255.255.255.255.000.000.000.000: 15^4 Networks and 254^4 Hosts :1110 Class E-5, 240 - 254, Subnet Identifier 255.255.255.255.255.000.000.000: 15^5 Networks and 254^3 Hosts :11110 Class E-6, 240 - 254, Subnet Identifier 255.255.255.255.255.255.000.000: 15^6 Networks and 254^2 Hosts:111110 Class E-7, 240 - 254, Subnet Identifier 255.255.255.255.255.255.255.000: 15^7 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: 1,044,643,524 Networks and 4,129,024 Hosts: 0 Class B-2, 128 - 191, Subnet Identifier 255.255.y.y: 264,241,280 Networks and 16,256 Hosts: 10 Class B-3, 128 -191, Subnet Identifier 255.255.255.y: 226,144 Networks and 254 Hosts: 110 3. Class C-1, 192 - 223, Subnet Identifier 255.y.y.y: 522,321,536 Networks and 2,064,512 Hosts: 0 Class C-2, 192 - 223, Subnet Identifier 255.255.y.y: 66,056,256 Networks and 8,128 Hosts 10 Class C-3, 192 - 223, Subnet Identifier 255.255.255.y: 32,768 Networks and 254 Hosts 110 4. Class D-1, 224 - 239, Subnet Identifier 255.y.y.y: 261,160,768 Networks and 1,032,256 Hosts: 0 Class D-1, 224 - 239, Subnet Identifier 255.255.y.y: 16,512,032 Networks and 4,064 Hosts: 10 Class D-3, 224 - 239, Subnet Identifier 255.255.255.y: 4,096 Networks and 254 Hosts 110 5. Class E-1, 240 - 254, Subnet Identifier 255.y.y.y: 244,838,220 Networks and 967,740 Hosts: 0 Class E-2, 240 - 254, Subnet Identifier 255.255.y.y: 14,512,290 Networks and 3,810 Hosts: 10 Class E-3, 240 - 254, Subnet Identifier 255.255.255.y: 3,375 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 (Which in Reality is actually the Subnet Address of a Subnetted Network IP Address Range: Subnet 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: 16,384 Networks and 32,004 Hosts: 10 Class C, 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. Appendix II : The Mathematical Anomaly Explained Nonetheless, this mathematical issue is an argument concerning, whether or not there exist a 'One-to-One' Correspondence between the Mathematical Calculations involving the Decimals (represented as Intehers) and those concerning the Binary Operators (Logical Expressions; the Truth Table values of 1's and 0's). Needless to say, this Mathematical Anomaly becomes even more apparent when one observes the Class B situation. Where by: 1. Class B; 128 -191, IP Address Range Default Subnet Mask; 255.255.000.000 (Which yields: 2^14 Networks and 2^16 Hosts; that is, 16,384 Networks and 65,536 Hosts.) However, this total is not the correct method of enumeration, and it is not the actual number (Integer Number) of available networks. And this FACT becomes even more apparent when the Binary Translation of the Decimal (Integers) Numbers is completed. That is, the result would yield 64 Binary Numerical Representations, ONE for each of the Decimal numbers (Integers) that are available in the IP Address for the Class B. Where Class B should maintain the representation (Which provides the actual Integer enumeration for the calculation of the total IP Addresses available. In other words, their independent count, of their respective totals for the Actual Number of Available IP Addresses in the Class B should Equal 64.) given by: 2. Class B: 128 -191, (Which equal the total of 64 possible IP Addresses for the given Address Range) Default Subnet Mask: 255.255.000.000 9Which results in 64^2 Networks and 254^2 Hosts; that is, 4,096 Networks and 64,516 Hosts.) Nevertheless, an enumeration or break down count association, of each representation, that is, Binary and Decimal. Would indeed, provide a greater support for the conclusion presented thus far. Where by, given the Classes noted in 1 & 2 above. We have: 1a. (128 + 128 + 128 + 128 + ...+ 128) = 128 x 128 = 2^14 1 2 3 4 ... - 128 = Total Count Which equal the Total number of Networks for the Given Address Range. and 1b. (255 + 255 + 255 + 255 +...+ 255) = 255 x 255 = 2^16 1 2 3 4 ... - 255 = Total Count Which equals the Total Number of Hosts for the Given Address Range. While noting that these equations represent the Binary Method for determining the number of Networks and Hosts for the given Address Range of Class B. However, keeping this in mind, notice the difference that exist when this same calculation is used for the Decimal (Integer) representation. 2a. (64 + 64 + 64 + 64 +...+ 64) = 64 x 64 = 64^2 1 2 3 4 ... - 64 = Total Count Where this number equals the number of Networks for the Given Address Range assigned to Class B. And 2b. (254 + 254 + 254 + 254 +...+ 254) = 254 x 254 = 254^2 1 2 3 4 ... - 254 = Total Count Where this equation represent the Total Number of Hosts for the Given Address Range of Class B. In other words, given the equation (191 -128) + 1 = 64. We are then presented with the Total Number of Addresses available for the given Address Range, 128 - 191, for the Class B. Where it can be seen that, any One-to-One mapping of the Numbers in the Address Range and the Counting Numbers (Integers), beginning with 1. Should yield the Total Number of Addresses available in any Count, for the Determination of the Total Number of Networks. And this same line of reasoning applies to the Host count, as well. ['Where the Subscript Number equals the Value of the Total Number of Availabe IP Addresses (a One-to-One Correspondence between the Enumeration of, and the Address Ranges given) for the Network and Host Ranges in Class B. Where both Binary and Decimal Number representations are the given examples.'] Nevertheless, when the Decimal and Binary conversion is completed. That is, when you establish a One-to-One relationship between the Binary and Decimal Numbers. You would discover that the their respective totals would be the same. That is, there can only be 64 Binary numbers and 64 Decimal numbers for the calculation of the Total Number of Networks. And there can only be 254 Binary Numbers and 254 Decimal Numbers for the calculation of the Total Number of Hosts. The difference is that, the former method reveals the Binary calculation, while the latter is the Integer (called the Decimal) Calculation. Needless to say, it should be very clear that the Binary method is a Logical Expression, and does see the Integer Count, that is the 'Difference between the Range Boundaries Plus 1'. Which yields the total number of available IP Addresses to be used to determine the actual number of Hosts within a given IP Address Class Range. Clearly, the Decimal method is indeed a Mathematical Expression representing the operations involving the Integers. Needless to say, if you are confused or are in doubt of these conclusions. Then my suggestion, would be to present my findings to a Professor of Mathematics at some well established university. Appendix III : A Peak into the Classless IP System (Are the Advantages Real?) The main point of CLASSLESS ADDRESSING (CIDR), concerns Routing, and the Router's Route Table Size. However, while this does speak of the so-called "Classless System as being a Boon for IP Address Management. The reality is that, it is nothing more than a conformation in reason/ justification, of the Error that the IPv4 Class System maintained. In other words, even with CIDR, the problem of understanding the difference between the Binary and the Decimal Mathematical Operations still exist. Needless to say, this system established a means to combine several smaller networks into one larger network, and provided a 'One (Internet) Route Path with a Mulit-IP-Network-Address-Thoroughfare'. Nonetheless, the BAND-AID is clearly short lived, because it still uses the same IP System of Addressing. So- What is the Advantage? 1. Housing several Smaller Networks under One Large Network. (Big Boy Little Boy. The Bully Scenario, because we know who pay's who! (CIDR) 2. Since the Mathematical Error was never corrrected, the Calculated total Number of available IP Addresses is still wrong. And it is for this reason, that someone has to pay someone else. (IPv4 Expansion) In other words, the only real advantage of CIDR is that of improved Router performance. Surely it can be clearly seen, that the UNADDRESSED Mathematical Error was the cause for the restructuring of the IP Addressing Scheme, under the guise of CLASSLESS SYSTEM. But who's Fooled? Especially since, the IP Class Address Range still governs this Invisible Class: called "CLASSLESS". Nonetheless, if you are in doubt of this conclusion, then ignore the recommendations. However, if it is the Internet you are connecting to- You will not obtain a connection! In short- Stay within the assigned boundaries of the assigned IP Address Class Range. Nevertheless, the conclusions of this white paper remain unchanged. (For a more detailed explanation of CIDR, See Reference Section, # 7) [Special Note: My original discovery of the Mathematical Anomaly dealt with 'Determining the Address Ranges for each Subnet'. This meant that the Subnet IP Address was not counted in the calculation of the Total Number of available IP Addresses, within the given IP Address Range for any given IP Class. It was not counted because it was being used. Which meant an error of +1 in calculating the total number of IP Addresses available. 'Mr. Alun Jones' informed me of this error in my calculations. That change is reflected in this writing.] 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 E-Mail address below only! ) Eugene Terrell 24409 Soto Road Apt. 7 Hayward, CA. 94544-1438 Voice: 510-537-2390 E-Mail: eterrell00@netzero.net ["Copyright (C) [ The Internet Society (1999). All Rights Reserved. This document and translations of it may be copied and furnished to others, and derivative works that comment on or otherwise explain it or assist in its implementation may be prepared, copied, published and distributed, in whole or in part, without restriction of any kind, provided that the above copyright notice and this paragraph are included on all such copies and derivative works. However, this document itself may not be modified in any way, such as by removing the copyright notice or references to the Internet Society or other Internet organizations, except as needed for the purpose of developing Internet standards in which case the procedures for copyrights defined in the Internet Standards process must be followed, or as required to translate it into."]