CLEAN - Calculating, Legacy, Endianness Addressing, … · Converting from decimal to binary IPv4 addresses 5. Networks Karst Koymans (UvA) CLEAN Tuesday, ... .An IPv4 address is

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CLEANCalculating, Legacy, Endianness

Addressing, Networks

Karst Koymans

Informatics InstituteUniversity of Amsterdam

(version 15.3, 2016/02/08 10:13:12)

Tuesday, February 9, 2016

Karst Koymans (UvA) CLEAN Tuesday, February 9, 2016 1 / 34

...1 CalculatingCountingMultiplication

...2 Legacy

...3 EndiannessNumeral systemsIntermezzoBig-endian and little-endian legacy

...4 AddressingThings to know by heartConverting from decimal to binaryIPv4 addresses

...5 Networks


Calculating

Outline...1 Calculating

CountingMultiplication

...2 Legacy



...5 Networks


Calculating Counting



...2 Legacy



...5 Networks



The process of counting

Counting is a process that. . .. . . starts with 0 as the initial count. . . repeats until there is nothing left to count

. . . labels an element with the count n up till now

. . . updates the count to n + 1

The final result is a number n

which labels every element with a number < n

Hence n = {0, . . . , n − 1}



The legacy of counting

Mathematicians traditionally start counting at 1which is “wrong” (in a very objective manner1)

Suggested set-range operator “...”Ruby has this as a sequence-range operatorn = 0...nSo this includes the left hand side 0but excludes the right hand side nThis eliminates in many cases the awkward n − 1Could be overloaded for sets and sequences

1EWD831, http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDFKarst Koymans (UvA) CLEAN Tuesday, February 9, 2016 6 / 34

http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF

Calculating Multiplication



...2 Legacy



...5 Networks



Legacy “bigendian” numbers.Space needed, but not present, at left margin for multiplication..

......

12345 12345

6789 6789

-----x -----x

11105 111105

8760 98760

415 86415

70 74070

-----+ --------+

10205 83810205



“Right”-endian numbers, in sequence order

123456789

x-----111105987608641574070+--------

83810205


Legacy



...2 Legacy



...5 Networks


Legacy

Legacy

.Legacy..

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The biggest obstacle and problem for progress and innovation is legacy

and the incapability of humans to cope with change.

.Suggestion..

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Imagine we could change the world by writing numbers “the other way

around”. To support the difficult transition we could use 9876543210 a as

digits in place of 0123456789.aMaybe using a font that also distinguishes 0 and 8 from their mirror images.


Endianness



...2 Legacy



...5 Networks


Endianness Numeral systems



...2 Legacy



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Mapping fixed length binary sequences to numbers.Definition..

......

A binary sequence b of length n is a map(ping) or function b : n → 2.

The “natural” assignment of a natural number to b is

n−1∑i=0

b(i)2i =∑

i∈0...nb(i)2i

.Theorem..

......

This correspondence is a 1-1, onto function between sequences

of length n and natural numbers smaller than 2n.

Because of “trailing zeroes” this does not extend to a 1-1 functionbetween sequences of arbitrary length and all natural numbers.



Mapping arbitrary length binary sequences to numbers

By shifting the assignments of sequences of length n to the right by addingthe number 2n − 1 we get a 1-1 correspondence between arbitrary lengthbinary sequences and all natural numbers.

.Theorem..

......

The map b 7→∑

i∈0...n b(i)2i + (2n − 1) = ∑i∈0...n (b(i) + 1)2i

is a 1-1, onto function between 2<ω and N.

This construction leads to the “2-adic numeration system”, also calledbijective base-2 numeral system. It also works with bases k > 2.



2-adic versus binary numbers

2-adic binary 0 1 2 3 4 5 6 7 8 9 10 11 12<> <> 01 0 0 12 1 1 211 00 0 312 01 1 421 10 2 522 11 3 6111 000 0 7112 001 1 8121 010 2 9122 011 3 10211 100 4 11212 101 5 12


Endianness Intermezzo



...2 Legacy



...5 Networks



Punycode bootstring encoding

Specified in RFC 3492

For internationalized domain names and usesa “little-endian” number representation for deltas (good!)a mixed-radix system with threshold valuesa generalisation of the k-adic numeration system

No “trailing zeroes” issueSelf-delimiting inside number sequencesAlso called “Generalized variable-length integers”



Generalized variable-length integers

.Bootstring algorithm..

......

Let a be a positive natural number and ri, ti be sequences of radices andthresholds. Usually we demand that ri + ti = k, for some fixed “base” k, sothat we may use digits in 0...k (remember this means 0, . . . , k-1).Write A0 = a and start calculating inductively, given Ai. If Ai < ti, theprocess ends with last digit ai = Ai. Otherwise, use quotient withremainder to produce Bi and Ci such that Ai = Bi + riCi. Increase Bi untilit is bigger than the threshold, as follows:Calculate pi ≤ Ci such that ti ≤ Bi + ripi < ti + ri. ThenAi = (Bi + ripi) + ri(Ci − pi). The next digit in the expansion of a isai = Bi + ripi and the process continues with Ai+1 = Ci − pi.

Interesting cases are ti = 0, ri = k; ti = 1, ri = k − 1 and ti = k − 1, ri = 1,which give normal k-ary; k-adic and “unary” presentations.


Endianness Big-endian and little-endian legacy



...2 Legacy



...5 Networks


Endianness Big-endian and little-endian legacy

Historic conventions.Writing numbers in “reverse” (right-to-left)........

The root of all evil and why we should go evil

.Increasing memory “normally” (left-to-right)..

......

Big-endian systems keep multi-byte numbers looking okMSB (Most Significant Byte) first

Little-endian systems make multi-byte numbers look weirdLSB (Least Significant Byte) firstEffectively also growing memory in reverse (to the left)

Giving at least consistency in appearance


Addressing



...2 Legacy



...5 Networks


Addressing Things to know by heart



...2 Legacy



...5 Networks


Powers of two (0-15)

20 = 1

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

26 = 64

27 = 128

28 = 256

29 = 512

210 = 1024

211 = 2048

212 = 4096

213 = 8192

214 = 16384

215 = 32768

Powers of two (16-31)2

216 = 65536

217 = 131072

218 = 262144

219 = 524288

220 = 1048576

221 = 2097152

222 = 4194304

223 = 8388608

224 = 16777216

225 = 33554432

226 = 67108864

227 = 134217728

228 = 268435456

229 = 536870912

230 = 1073741824

231 = 2147483648

2No need to know more than the first entry by heart ;)

Hexadecimal notation (with legacy binary and decimal)

0x0 = 0b0000 = 0 ( 0)

0x1 = 0b0001 = 1 ( 1)

0x2 = 0b0010 = 2 ( 2)

0x3 = 0b0011 = 3 ( 3)

0x4 = 0b0100 = 4 ( 4)

0x5 = 0b0101 = 5 ( 5)

0x6 = 0b0110 = 6 ( 6)

0x7 = 0b0111 = 7 ( 7)

0x8 = 0b1000 = 8 ( 8)

0x9 = 0b1001 = 9 ( 9)

0xa = 0b1010 = 10 ( 10)

0xb = 0b1011 = 11 ( 11)

0xc = 0b1100 = 12 ( 12)

0xd = 0b1101 = 13 ( 13)

0xe = 0b1110 = 14 ( 14)

0xf = 0b1111 = 15 ( 15)

Addressing Converting from decimal to binary



...2 Legacy



...5 Networks



The “big-endian” way

151 = 128 + 23= 128 + 16 + 7= 128 + 16 + 4 + 3= 128 + 16 + 4 + 2 + 1

151 = 1 · 128 + 0 · 64 + 0 · 32 + 1 · 16 + 0 · 8 + 1 · 4 + 1 · 2 + 1 · 1

151 = 100101112 = 0b10010111



The “little-endian” way

151 = 1 + 2 · 75= 1 + 2 · (1 + 2 · 37)= 1 + 2 · (1 + 2 · (1 + 2 · 18))= 1 + 2 · (1 + 2 · (1 + 2 · (0 + 2 · 9)))= 1 + 2 · (1 + 2 · (1 + 2 · (0 + 2 · (1 + 2 · 4))))= 1 + 2 · (1 + 2 · (1 + 2 · (0 + 2 · (1 + 2 · (0 + 2 · 2)))))= 1 + 2 · (1 + 2 · (1 + 2 · (0 + 2 · (1 + 2 · (0 + 2 · (0 + 2 · 1))))))

151 = 100101112


Addressing IPv4 addresses



...2 Legacy



...5 Networks


Addressing IPv4 addresses

IPv4 addresses.Definition (IPv4 address)........An IPv4 address is a mapping or function f : 32 → 2, where 2 = {0, 1}

The decimal dotted notation for IPv4 addresses can be seen asa base-256 numberwith a decimal notation for its “digits”

Other notations for IPv4 addresses might3 be useda.b.c with a, b 8-bits and c 16-bitsa.b with a 8-bits and b 24-bitsa with a 32-bits

3This is allowed by the POSIX API, see “man inet addr”.Karst Koymans (UvA) CLEAN Tuesday, February 9, 2016 31 / 34

Networks



...2 Legacy



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Networks

IPv4 address classes

.Definition (IP address classes)..

......

Class A: f(0) = 0; network part is f(0), . . . , f(7)

Class B: f(0) = 1; f(1) = 0, network part is f(0), . . . , f(15)

Class C: f(0) = f(1) = 1; f(2) = 0, network part is f(0), . . . , f(23)

Class D: f(0) = f(1) = f(2) = 1; f(3) = 0 multicast

Class E: f(0) = f(1) = f(2) = f(3) = 1 reserved

The division into classes is mostly historic.Classful addressing has been replaced by classless addressing.


Networks

IPv4 subnetworks and CIDR notation

.Definition (IP address classes)..

......

Given an IP address f : 32 → 2 and a prefix length k ≤ 32 the notation f/k

denotes a network (prefix) consisting of all IP addresses g : 32 → 2 with

g ↾ 0...k = f ↾ 0...k.

Usually we demand that f ↾ k...32 has only 0s.

In other words, the first k bits in the sequence determine the network partand the remaing (32 − k) bits the host part.In case there are non-0 host bits in the notation f/k this denotes acontamination of a host with the network this host resides in.


Documents

CLEAN - Calculating, Legacy, Endianness Addressing, … · Converting from decimal to binary IPv4 addresses 5. Networks Karst Koymans (UvA) CLEAN Tuesday, ... .An IPv4 address is