What the Heck is Base64 Encoding really?

Also,  Why most implementations of Crockford-base32 encoding are probably incorrect

TLDR: The term base64 encoding is an overloaded term which promotes ambiguous usage.  It can mean either a place-based 64-symbol encoding of a single number, or a streaming algorithm for encoding an array of octets.  In common usage, it is the latter -- the same technology and IETF RFC specification that allows us to encode arbitrary binary files and place them in USENET posts and emails.  


Examining the differences between the two meanings and two implementations will teach us something about ambiguity, online/stream algorithms, and that specifications are really really very important.
  

TLDR: The Crockford-base32 encoding proposal, as written here http://www.crockford.com/wrmg/base32.html, is not as clear as it could be and lacks a canonical implementation and/or test-cases.  


This lack of clarity has lead most people to interpret it as a place-based 32-symbol encoding of a single number.  However, the particular wording and context of his proposal implies that the Crockford-base32 encoding was meant to adhere closer to the existing base64 and base32 octet binary encodings (including padding of binary data up to a multiple of the 5-bit space).  

The Presence of An Ambiguity

I want to use this post as a way of clearing up a confusion that seems to exist in many people's understanding of base64 (and also crockford-base32). 

Because on ambiguity exists, I am going to invent my own clarifying terminology (Place-Based Single Number Encoding versus Concatenative Iterative Encoding) to attempt to disambiguate.  Along the way, we will get into the nitty gritty details of binary base64-encoding (as commonly implemented in a thousand different systems) and of the radix-placed system that most people are taught as binary, octal, decimal and hexadecimal numbers.

But before I do that, let me explain what the Crockford-base32 encoding is meant to do, for those of you who have never heard of it.

Human Readable URLs

Producing URL shortening and youtube-like resource strings seems to be the raison d'être for the Crockford and other such encodings.  Many famous web applications have something like this: a unique string of letters and numbers that when used within a URL/URI, provide a unique and identifying name for that resource.  Resource examples range from videos, to shortened URLS, to collabedit pages, to flickr accounts.  
Screen shot 2014-04-25 at 5.04.00 PM.png
With all these services, we desire to provide a unique identity to a resource whose cardinality may grow into the millions or billions.  Certainly, the identity could be something as simple as an integer increasing in value (what the database guys call a sequence).  The drawback with a sequence is that it provides people a way to guess resource identities.  To counter this, we could generate a random number with insignificant chances of repeating (and of guessing) and encoding it in a high-radix encoding to shorten its string-length.

Let's examine two ways to do this.

Radix Place-Based Single Number Encoding System
This system is the easier of the two to explain, as it is covered in most high-schools, and CompSci 101.  Nevertheless, we will go through it again.
We start with the number 101.  

What is this number?  Is it the number of dalmatians that should be brutally murdered, skinned, and fabricated into a woman's coat?  Or is the cost of a 5 dollar foot long?

The answer is that it could be either.  The number is the ultimate abstraction, that which your brain interprets, and the glyphs and the symbols are the permanent memory of that number on paper or on disk.  But in-between, we need to apply a decoding to turn interpret the glyphs from symbols through numerals to a number. In binary, the glyph/symbol/numeral string of "101" can be interpreted as "five" -- just enough dollars to buy a Subway sandwich. In decimal, which most humans (without compsci/math background) take for granted, the string "101" is interpreted as "one hundred and one."
The term radix, which is synonymous with the term base, is defined as the number of unique symbols in a place-based numeric system.  The radix-system that most humans use is a base-10 or radix-10 system.  The symbols we have today for our numbers came to us during the Renaissance from the Arab mathematicians.  We called them arabic numerals, despite the fact that the Arab mathematicians imported them from the Indians who invented the numeral 0.  This special numeral 0, along with the other nine symbols, make up the most famous and consistent symbols that the world has ever seen.  We all collectively own these symbols today: '0', '1', '2', '3', '4','5','6','7','8','9', and we assume that they are in this order:  0123456789.
To restate,  the reason that we call this a radix-10 system is because we have 10 unique symbols. However to be a base-10 system, we do not have to use these exact symbols or in this traditional order.  How weird a parallel universe there must be where, due to a transposition error, our system could have been 0123465789?  Fifty would be "60."  

Place-Based Systems

Why are the numbers made from these symbols (of any radix) called place-based systems?  The answer comes from the fact that every numeric representation is a string where the place represents a power in that radix from right to left.  
As you can see above, the simple decimal number 1853 has four places:
  1. farthest to the right, the ones place  100
  2. second to the right, the tens place  101
  3. third from the right, the hundreds place 102
  4. and fourth from the right, the thousands place 103
and the string ‘1853’ is thus a representation of a number using places to indicate powers.

A famous example of a numeral system that is not place-based is the Roman Numeral system.  A Place-Based System applies not just with decimal, but with all the radix-based systems we see below.

Binary Place-Based Single Number Encoding System:  01

Standard coursework tells us of binary, a radix-2 system using (by definition) two symbols:  '0' and '1'.  In this base, the number 1 is '1', the number 2 is '10', the number 3 is '11', etc.  This is the numeral system of computers and disks that provides the unifying clutch between voltage and symbolic logic, truth, turing machines, and computation.

Octal Place-Based Single Number Encoding System: 01234567

If we have a base-8 system, we can only use eight symbols.  Reusing our first eight symbols from the base-10 system we get:  01234567.  In this base, the number seven is 7, but the number eight is 10 because we have run out of symbols. We call these octal numbers.  We should call them octal numerals.

Hexadecimal Place-Based Single Number Encoding System: 0123456789ABCDEF

And what if we have more than 10 symbols?  Hexadecimal, a base 16 system, is a good example of one.  We commonly choose to use some letters from the latin alphabet (whether capitalized or not) as the 6 remaining symbols. Thus our alphabet of symbols in order is : 0123456789ABCDEF.   In this base, the number 15 is F, and the number 16 is 10.  Most people are familiar with hexadecimal numerals usage in RGB values.


Tetrasexagesimal Place-Based Single Number Encoding System

For base64, we can use the lower-case alphabet and the upper-case alphabet:  abcdefghijklmonpqrstuvwxyz and ABCDEFGHIJKLMNOPQRSTUVWXYZ.  But this only gives us 26*2= 52 symbols, and we need 64 symbols. The symbol alphabet of numerals 0123456789 gives us ten more for a grand total of 62 symbols.  We need to add two more.  

For now, let's just choose the plus symbol '+' and the slash symbol '/'.  Because we can order these symbols in any order we want, how about the following ordering?

    Value Encoding  Value Encoding  Value Encoding  Value Encoding
        0 A            17 R            34 i            51 z
        1 B            18 S            35 j            52 0
        2 C            19 T            36 k            53 1
        3 D            20 U            37 l            54 2
        4 E            21 V            38 m            55 3
        5 F            22 W            39 n            56 4
        6 G            23 X            40 o            57 5
        7 H            24 Y            41 p            58 6
        8 I            25 Z            42 q            59 7
        9 J            26 a            43 r            60 8
       10 K            27 b            44 s            61 9
       11 L            28 c            45 t            62 +
       12 M            29 d            46 u            63 /
       13 N            30 e            47 v
       14 O            31 f            48 w         
       15 P            32 g            49 x
       16 Q            33 h            50 y


In this base, the number seven is 'H', the number 50 is 'y', the number 62 is '+', and the number 64 is 'BA'.

But now, here is the big question:  Is this the same base64 encoding that was used to pass binaries and dirty images on USENET posts?  Is this the base64 that most languages, browsers, and email-clients support with library functions? Is this the base64 of RFC 4648?

The answer is a big fat NO.  To explain why is not the same thing, we will make a digression and consider the notions of functions and types.

Functions and Types, Domains and Ranges

Types and functions are utterly fundamental to the art, science, and craft of programming.  

One might say that types allow us to think about data, while functions allow us think about computation.  Functions and types come clothed in different syntactical and run-time forms in any of your favorite language, but they are omnipresent and never to be discounted.

So why are we bringing them up?  The answer is because we they will be helpful in explaining the difference between Place-Based Single Number Encoding and Concatenative Iterative Encoding.

Up to now, we have been describing the notion of radix and base to represent a number, a system that we called Place-Based Single Number Encoding.  To help us understand this encoding, let us describe it as a function with an input and an output.

A representation is nothing more than a string of symbols in some alphabet.
Thus, the number 101 is the decimal representation of the number “one hundred and one” while 65 is its hexadecimal representation.  Both 101 and 65 are strings of symbols.

This is simplistic, but really exposes the essence of what we have been doing thus far.  When we wanted to convert a single number into a binary representation we implicitly knew that the Domain of our function (our input type) was that of a number, and that the Range (the return type) was that of a string of zeros and ones.  We can denote this function signature like this:
Ok.  It feels like we burdened the obvious with theory.   How does this help explain what email-based base64 encoding is?  Well, the answer is to show the difference in their function signature, types and all.

What the base64 encoding of RFC 4648 does, and why it is different, is this:
It iteratively consumes an array of octets, transforms them into Base64 representations, and concatenates each representation into an output array.

In conclusion, we are saying one way of seeing the difference between these systems is by comparing their signatures

Place-Based Single Number Encoding
Concatenative Iterative Encoding System
Function
Signature


Does that clear it up?


Concatenative Iterative Encoding System
Yup, the term Concatenative Iterative Encoding sounds like a lot.  But if we look at each word and examine its function signature, we will find it to be a description that is succinct and accurate.
Let's dissect both the term and the function signature, using the following visual of a file to motivate.  
What we see above is a file (any file) represented as a long tape of zeros and ones.  I have purposely colored four different groups of octets.   Grouping bits into 8-bit chunks, called octets or bytes, makes it easier for us to think about the sheer number of zeros and ones. All files, whether text or not, are ultimately a concatenation of bits, so this is a valid way of understanding a file.  

We can imagine the encoding function as a box that iteratively works its way down the tape, turning octets into base64 atoms.  As the box/function moves its way down the tape, it slowly builds up the output tape by concatenating newly encoded base64 atoms.  We are iteratively consuming octets from an array [OCTET] and concatenating them onto an output array of base64 representations [BASE64_REPRESENTATION].
The implementation details of how we transform octets into numerals from a base64 are actually quite simple.  I will go through them now, but the following sites provide more exhaustive descriptions:


We start with a stream of octets which means that we are grouping a stream of binary bits by 8 bits each.

As you can see above, by merely choosing to put a grouping marker between every 8-bits, we can interpret this stream of 24 bits as 3 octets.  Furthermore, we can choose to interpret each octet as the decimal numbers shown above, thus we see the numbers 77, 97, and 110.  This example comes from the wikipedia article, showing how a text input could have been encoded, where the ASCII values for the word ‘Man’ are the input octets -- in ASCII ‘M’=77, ‘a’=97, and ‘n’=110.


So what does base64 encoding do? Does it transform the input at all?  No.  It merely adjusts the grouping markers.  Where before we grouped by 8 bits, now we will group by 6-bits.

As you see above, we have reinterpreted 24-bits of data as 4 groups of 6-bits (sextets) rather than 3 groups of octets.  Again, putting a decimal number for these new groups, we see the numbers 19, 22, 5, 46.   But remember that base64 symbol table we had above?  I will copy it down here again:
    Value Encoding  Value Encoding  Value Encoding  Value Encoding
        0 A            17 R            34 i            51 z
        1 B            18 S            35 j            52 0
        2 C            19 T            36 k            53 1
        3 D            20 U            37 l            54 2
        4 E            21 V            38 m            55 3
        5 F            22 W            39 n            56 4
        6 G            23 X            40 o            57 5
        7 H            24 Y            41 p            58 6
        8 I            25 Z            42 q            59 7
        9 J            26 a            43 r            60 8
       10 K            27 b            44 s            61 9
       11 L            28 c            45 t            62 +
       12 M            29 d            46 u            63 /
       13 N            30 e            47 v
       14 O            31 f            48 w         
       15 P            32 g            49 x
       16 Q            33 h            50 y


As we look at it, we can assign the number 19 to the base64 encoded numeral ‘T’, ‘W’=22, ‘F’=5, and ‘u’=46.  Thus,

a text file with the ASCII contents of ‘Man’ will result in a base64 encoding of ‘TWFu’.

Summing it all up

The Concatenative Iterative Encoding System has the following signature,
which in our particular example looks like:

or equivalently

And that is it.  That is what base64 encoding does.  It groups a stream of binary bits (grouped in 8-bit chunks) into a stream of output 6-bit sextets.  Each sextet is interpreted as a single symbol from the radix-64 alphabet, and re-emitted as the ASCII representation of that symbol.  

Is it Compressions?  No.

There always seems to be a major misapprehension of base64’s purpose, and one that we should try to clear up right now.  

Base64 Encoding as a Concatenative Iterative Encoding System permits an arbitrary binary file to be translated into a string of text and copied in an email, or posted in a Usenet post.  We have protected higher-level applications (like MIME) from arbitrary binary by choosing only letters that can exist within that application’s specification.  

We have compressed the string length of symbols but we have not compressed the overall data size.  In fact, we have actually inflated the size of the overall data payload by re-embedding the string ‘TWFu’ using ASCII.  

Let us repeat the last statement:  base64 does not compress the input data.  In fact it inflates the size. It only compresses the length of the symbol string, by using an alphabet with more symbols.

A Final Comparison

Place-Based Single Number Encoding
Concatenative Iterative Encoding System
Function
Signature
Input Type
Single Number
Array of Octets
Output Type
representation of number in different radix
Array of Base64 representations
Consumption Direction
Right to Left
Left to Right
Used For
Representing a single number in a different radix, using a different alphabet consisting of different symbols.
Turning arbitrary binary into a left-to-right array of encoded characters that may be embedded and transferred within the context of an applications.
In the above chart there is one term that we have not yet discussed called “Consumption Direction”. The point of this is to show that Place-Based Single Number Encoding works on an entire number whose place-based format requires knowledge of the rightmost digit.   To put it another way, you need to know the farthest right-most bits before you know what the number is.  

On the other hand base64 as a Concatenative Iterative Encoding System consumes binary data from left to right.  The reason for this is that base64 encoding usually operates on files which cannot always be loaded into memory.  This is why we have subtly been referring to this algorithm as a streaming algorithm -- it does not have access to the entire input, but may start generating the output with only the first three octets.  The actual physical implementation of reading bits from disk means that we always have access to the most significant bits (leftmost) before the least-significant bits (rightmost) are event loaded from disk.

The Place-Based Single Number Encoding is not appropriate for encoding a file because you would
  1. need to load the file into memory to know its numeric representation
  2. the number would be HUGE
On occasion, the two systems will produce the same answer.  These occur when the number is bit-represented in multiples of 3-octets.  If the number is slightly more than 3-octets, or only 2 or 1 octet, then the base64 specification calls for something called padding which adds zeros to the rightmost side.  The effect of this is that base64 respects the leftmost/prefix data, while allowing the rightmost to be vastly different.

Padding is one of our hints that we are dealing with a streaming algorithm.

Finally,  An Analysis of Crockford-base32 and its Implementations

The Crockford-base32 implementation has a few desirable properties that I will copy directly from the website:
The encoding scheme is required to
  • Be human readable and machine readable.
  • Be compact. Humans have difficulty in manipulating long strings of arbitrary symbols.
  • Be error resistant. Entering the symbols must not require keyboarding gymnastics.
  • Be pronounceable. Humans should be able to accurately transmit the symbols to other humans using a telephone.


Additionally, it tries to remove accidental obscenities and ambiguous looking symbols (like 0 and O).

I needed a Crockford32 implementation for Erlang but could not find one, so I looked up various implementations in other languages to try to understand what they did.  I also looked closely at the base64 implementation that comes with the Erlang runtime.  It was then that I figured out that the Crockford32 implementation implies Concatenative Iterative Encoding System but that all the implementations I could find are Place-Based Single Number Encoding.

The Conflicting Evidence



Evidence for thinking Crockford32 was meant to be a
Place-Based Single Number Encoding
Evidence for thinking Crockford32 was meant to be a
Concatenative Iterative Encoding System
All implementations use this assumption.
Zero implementations use this assumption.
The quote from the website saying,

This document describes a 32-symbol notation for expressing numbers in a form that can be conveniently and accurately transmitted between humans and computer systems.”

Emphasis mine.
Quotes from the page that compare it in context with other Concatenative Iterative Encoding Systems

"Base 64 encoding uses a large symbol set containing both upper and lower case letters and many special characters. It is more compact than Base 16, but it is difficult to type and difficult to pronounce.”

“Base 32 seems the best balance between compactness and error resistance. Each symbol carries 5 bits.”
Language that implies that the primary use-case for this encoding is Human consumption which means short strings. 

Short strings do not require streaming algorithms, as an entire number can absolutely be loaded into memory:

Be pronounceable. Humans should be able to accurately transmit the symbols to other humans using a telephone.”
The most damning quote of all:

“If the bit-length of the number to be encoded is not a multiple of 5 bits, then zero-extend the number to make its bit-length a multiple of 5.”

This quote is absolutely about padding enough bits to the rightmost digits.


Conclusion

In the end, I think this has been a nice learning experience for me, and hopefully something worth sharing.  Importantly, it drives home a few points:
  • most base32/16/64 encodings were designed to be streaming algorithms and operate only on octets and usually on a dataset that would not fit into memory
  • the recent explosion in services that want short-URLs while having unique identities for resources have increased the need for non-streaming radix-encoding algorithms
  • specifications are very important.  The Crockford32 page is ambiguous, especially in comparison with the base64 specification which even includes test cases

Finally, here is an Erlang implementation I wrote for Crockford-32 as a Place-Based Single Number Encoding.

259 comments:

  1. What a horrible theory!

    Your evidence for Crockford32 being a Concatenative Iterative Encoding System is completely unconvincing.

    When Crockford talks about base64, he explicit refers only to the "symbol set" it "uses". No comparison is made with the standard as a whole.

    "Base 10 is well known and well accepted" is also obvious proof that this is a scheme to encode integers and not arbitrary data. There is no well-known standard for transmission of binary data in base 10, as it would be computationally very expensive. However there is no doubt that people use base 10 all the time to encode integers!

    Your interpretation of "zero extension" to mean padding the rightmost digits is also flawed. It's the unsigned counterpart to "sign extension", and it is extremely common: for example the x86 has a dedicated MOVZX instruction to perform Zero eXtension (by padding the leftmost digits).

    If you want to contradict the clear and unambiguous statement that "This document describes a 32-symbol notation for expressing numbers", you should have better evidence than that. You're the only one to think that the standard is ambiguous in that regard. But I do agree that test vectors would be nice, and I would add that endianness should have been explicitly mentioned.

    ReplyDelete
    Replies
    1. A "horrible theory", like my use of the word "incorrect" might be a bit harsh, but I do appreciate your passion.

      I don't advance my interpretation with 100% certainty, and I feel like I was honest enough to present a table showing evidence for both sides. The ambiguity was the frustrating part, not any belief that implementations were "incorrect".

      Your point about base10 is well-taken and does give me pause.

      However, I want to push back about the padding argument. Since neither left nor right padding was mentioned, the ambiguity remained. I reasoned thusly:

      Why would you need to pad non 5-bit multiples to the left?

      This argument is key. The quote from the page is this: “If the bit-length of the number to be encoded is not a multiple of 5 bits, then zero-extend the number to make its bit-length a multiple of 5.”

      As an example, take a 32-bit (4-octet) number, like 01100001011000010110000101100001 (which in decimal, is 1633771873). This is NOT a multiple of 5-bits.

      Why, just because it is not a multiple of 5-bits, should I zero-pad it to the left with 3-more bits, or 8-more bits?

      Left-based zero padding, DOES NOT CHANGE THE VALUE OF THE NMUBER at all.
      crockford32NumberEncoding(01100001011000010110000101100001) --> "1GP2RB1"

      crockford32NumberEncoding(0000000001100001011000010110000101100001) --> "1GP2RB1"

      The statement from the page,
      “If the bit-length of the number to be encoded is not a multiple of 5 bits, then zero-extend the number to make its bit-length a multiple of 5”

      is quite a strong statement of the form: "If then ". I can't believe it was made with the knowledge that the action has no impact whatsoever, otherwise it is completely irrelevant and can and should be removed.

      If padding to the left does not matter, then all that remains is padding to the right, which DOES matter. It was actually this sentence in the specification that gave me pause as I was familiar with how base64 encoding right-pads and inserts "==" in the output string as indication of the padding.

      If we right-pad, we get
      crockford32NumberEncoding(0110000101100001011000010110000100000000) --> "C5GP2R80"
      and "C5GP2R80" != "1GP2RB1"

      I don't know what Douglas Crockford meant the specification to REALLY be. I would love him to find this page and answer. Most people interpreted it as a base32-radix encoding with a single number. Me too.





      Delete
    2. Sorry for the strong words, I saw it on Hacker News where the title was "Why most Crockford32 Implementations are Wrong". The hubris of this statement made me overlook the more tentative wording of the post.

      "Left-based zero padding, DOES NOT CHANGE THE VALUE OF THE NMUBER at all."

      And it's exactly the reason why it's chosen! Padding the shortest binary representation of a number to the left introduces no ambiguities, since the first digit is always a 1 (for non-zero numbers), whereas padding to the right would make decoding ambiguous ("10000" could correspond to 1 or 10 or 100...).

      Of course padding is only necessary if you start from a binary representation of the number and want to transform it into a base 32 representation. You don't need it if you compute the base 32 representation from scratch:

      1100001011000010110000101100001 (base 2)
      = 1GP2RB1 (base 32)
      = 00001 10000 10110 00010 11000 01011 00001 (base 2)

      where you see that zero-extension to a multiple of 5 bits has happened naturally. I'm guessing that's why you find this mention of padding unnecessary: it's just that it was written with binary representation as a starting point.

      And that kind of padding is completely different from base64's padding with "=" symbols, which aims to let you concatenate different blocks of base64-encoded data. It doesn't make sense to concatenate numbers, that's why Crockford32 goes as far as using "=" for a completely different purpose. If it was an alternative to base64, it would have been wise to at least keep the option of base64-style padding open!

      Delete
    3. Respectfully, I am not understanding you point here.

      It seems you are agreeing that padding to the left does not change the number (and very importantly here, we are talking about bit representations of integers in an unsigned schema, since nowhere have we opened the can of worms that is 1's-complement or 2's complement for negative numbers).

      I completely agree with your statement, "Padding the shortest binary representation of a number to the left [with zeros] introduces no ambiguities" as that was exactly my point, but I take it further and say, "zero padding to the left introduces no ambiguities, but there were no ambiguities to be introduced in the first place".

      You gave me the number: 1100001011000010110000101100001
      It is 31 digits long. Its decimal representation is 1633771873. I feel no ambiguity in understanding what number it represents -- and yet it is not a multiple of 5-bits. Why do I need to pad four extra zeros to the left (35 bits now), to make it a multiple of 5? Why is Crockford-32 telling me I should zero-pad?

      As such, I am still left with this question about the Crockford 32 specification:
      ---------------------------------
      If left-padding was implied, why does the sentence,

      “If the bit-length of the number to be encoded is not a multiple of 5 bits, then zero-extend the number to make its bit-length a multiple of 5”

      have to exist at all? What does it add to the discussion regarding implementation?
      ---------------------------------

      If it adds nothing under a left-zero-padding assumption, then it is irrelevant and should be removed. Would you agree with that?

      But since it is present, I take its very existence to mean something. Especially since in _most_ base encodings, the implementation of padding is meant to line up the N-bit-space with the 8-bit (octet) unit of data storage in all files, and is always right-padded.

      I believe that the very presence of this sentence is probitive of a "Concatenative Iterative Encoding" interpretation.

      I would prefer the specification be re-issued with these ambiguities removed.

      I see no reason to not have both: 1) a specification for encoding a single number, and 2) a specification for a data encoding that operates on octets and must be zero-padded to the right if the final bit count is not a multiple of 5 and 8 -- like regular base64, base32, or the Zooko-base32 encoding.

      Likewise, it would be neat if people acknowledged the differences when they were teaching base64 as ONLY "Concatenative Iterative Encoding". As an example of that is this blog post
      http://code.tutsplus.com/tutorials/base-what-a-practical-introduction-to-base-encoding--net-27590
      where the author starts with a "Place-Based Single Number Encoding" interpretation until he gets to base32 and base64 at which point he switches to a "Concatenative Iterative Encoding" interpretation.

      Delete
    4. "You gave me the number: 1100001011000010110000101100001
      It is 31 digits long. Its decimal representation is 1633771873. I feel no ambiguity in understanding what number it represents -- and yet it is not a multiple of 5-bits. Why do I need to pad four extra zeros to the left (35 bits now), to make it a multiple of 5? Why is Crockford-32 telling me I should zero-pad?"

      Your process in obtaining 1633771873 was the following:

      representation in base 2 "1100001011000010110000101100001"
      ->
      abstract mathematical integer
      ->
      representation in base 10 "1633771873"

      You're using algebra (multiplications, divisions) to convert back and forth to the abstract mathematical integer. It's the only way to do it, because 10 and 2 are not powers of some common radix, that is, log 10 / log 2 is irrational.

      But, if you want to convert it to base 16 or 32, there is another way:
      representation in base 2 "1100001011000010110000101100001"
      ->
      padded representation in base 2 "00001 10000 10110 00010 11000 01011 00001" (the spaces are here just for clarity)
      ->
      representation in base 32 "1GP2RB1"

      Here, each step is simple, local, and involves no algebra: the first step pads to a multiple of 5 bits, and the second step is just a lookup of the symbol corresponding to each group of 5 binary digits. The reason you need to pad to a multiple of 5 bits is simply that you couldn't do the lookup if you couldn't split the binary string into groups of exactly 5 bits!

      I can't read minds, but I think Crockford simply wrote the specification with the mental model of the second process. He just didn't write down the lookup step.

      Delete
  2. This comment has been removed by a blog administrator.

    ReplyDelete
  3. I find it shocking that all of these library authors would find it intuitive for hex (base16) and base encoders to have an API like encode([octets]) but for base32 a much more limiting encode(number) would make more sense.

    I guess this leaves us with fundamentally two different Crockford base32 standards, streaming and numeric? Maybe someone feels compelled to advertise a clarified spec with both of these as sub-codecs. Crockford fixing his web page would be nice but maybe enough harm has been done that a better way would be to accept that both are reasonable and already implemented viewpoints. We might be better off just documenting them both instead of figuring out which one is "correct".

    Thanks for the article! At least that cleared up why my work-in-progress (stream) implementation doesn't match the output of some other libraries for even a single low number/byte.

    ReplyDelete
  4. Unfortunately, the Dish anywhere is not available as a stand-alone app on Roku where. But you can screen share dish anywhere on roku. How to do this? It is simple! Download the app on your iPhone device and install it. Enable the screen sharing option on iPhone and Roku devices and connect them. Moreover, you must launch the app and start the streaming process. If you want to know the detailed steps regarding watching Dish anywhere programs on Roku, you can navigate our sites and read the articles about it. (Connecting Roku on Apple TV is must). Call +1-805-221-0330 for any doubts

    ReplyDelete
  5. Here is the article to learn Roku.com/link activation steps. Complete the device activation and select the top channel to activate. Spend your quality time reading the instructions below
    • At first, choose the best model to begin your purchase. If you are new to Roku, we recommend the top brands such as Roku Express, Express Plus, premiere, premiere plus, and streaming stick
    • The features and specification of these models differ and it’s your interest to pick the best one that suits your budget and requirements
    • Check if you have the necessary requirements to proceed with the device setup
    • If yes, begin the hardware setup
    • Sign in to the account to collect the Roku device activation code
    • This code must be provided on the portal, Roku.com/linkto complete the device activation process

    ReplyDelete
  6. Attach the go.roku.com/rss to your TV using HDMI or turn on your Roku TV. If you have a 4K-enabled Roku Streaming Stick or Streaming Stick, make sure that you attach the stick to an HDMI port that is compliant with HDCP 2.2. This is particularly important for HDR-encoded content compatibility. In case of any queries that you have, you can feel free to contact the active customer care team. just call @support number +1-805-457-0700

    ReplyDelete
  7. Paper Lords is one of the leading Academic Writing Services UK provider agency in the United Kingdom.
    We have been offering our services in the market for over 5 triumphant years.

    ReplyDelete
  8. Nice Blog! I never read this kind of content!
    Akshi Engineers has the resources to Gear Boxes Manufacturer in India at affordable and reasonable prices. To learn more contact us to discuss how we can fulfill your requirment of Gear Boxes as soon as doable. Our Company offering all the equipment realtes to rolling mill industry with 100% satisfaction or quality at best prices in india and other countries. For more information cantact us or visit our website.

    ReplyDelete
  9. AOL is one of the best online email service platforms. Which also has amazing features with more security reasons. Sometimes, when your AOL Mail Not Working on MacBook Pro, don't worry, you can consult our expert and easily resolve your issues within a few minutes. And moreover, you can dial our toll-free number and talk to our technical expertise to solve your problem. You can also visit our website.

    ReplyDelete

  10. Nice Blog! I never read this kind of content!
    Akshi Engineers has the resources to Gear Boxes Manufacturer in India at affordable and reasonable prices. To learn more contact us to discuss how we can fulfill your requirment of Gear Boxes as soon as doable. Our Company offering all the equipment realtes to rolling mill industry with 100% satisfaction or quality at best prices in india and other countries. For more information cantact us or visit our website.

    ReplyDelete
  11. Looking for Homework Help In The USA? I recommend that you should try our academic writing services which you will never regret trying, we offer writing services under all niches and streams within your budget.

    ReplyDelete
  12. Nice Post. I appreciate your hard work and your skills. Thank you for sharing.
    Akshi Engineers Pvt. Ltd. Company manufacture Hot Rolling Mill Manufacturer in India for flat products like mill stand, gear boxes, shears and pinch roll etc. The Hot Rolling Mills are normally of 2Hi or 5Hi configuration depending on the type of product being rolled. The 2Hi Hot Rolling Mills are used for materials like Aluminum, Copper, Brass and other softer metals whereas the 5Hi Hot Rolling Mills are used for harder metals like Stainless Steel, Alloy Steel etc. Get Best deal by calling us at +91-98106-62353 or you can also email at sales@akshigroup.com

    ReplyDelete
  13. Goassignmenthelp is a team of leading professional writers for Assignment Help to students all around the world. Contact us persuasive essay today for the Best Essay Writing Services at a very affordable price. Hire native seasoned experts essay typer at very affordable prices. We are the most reliable programming assignment help assignment helpers.

    ReplyDelete
  14. If you wish to add the 123.hp.com/setup printer to your list, you have to use the printers & Scanners preference. Most of the time, as you try to add a printer to the MAC operating system, then installing the driver is very important and the latest software version shall be downloaded in the process. In case if the printer is enabled with the Airprint feature, it will be compatible with the MAC OS. So you can call the support team number @+1-850-761-8950 in case of the issues you might face while adding the HP printer via 123.hp.com/setup.
    Read More:
    123.hp.com/dj3630 printer setup
    How to connect HP Deskjet 3755 to wifi

    ReplyDelete
  15. I want to always read your blogs. I love them Are you also searching for dissertation writing services? we are the best solution for you. We are best known for delivering Nursing dissertation writing services to students without having to break the bank

    ReplyDelete
  16. This is really helpful post, very informative there is no doubt about it. I found this one pretty fascinating. 경마사이트

    ReplyDelete
  17. Thanks for all the tips mentioned in this article! it’s always good to read things you have heard before and are implementing, but from a different perspective, always pick up some extra bits of information. 카지노사이트

    ReplyDelete
  18. You always comes with great stuff for your user. Thanks for being user friendly :) and Keep posting such great stuff 사설토토

    ReplyDelete
  19. If you are looking for the best CA foundation pendrive classes offered by experienced faculties coming from different fields of expertise.

    ReplyDelete
  20. Thanks for sharing a great article. You are providing wonderful information, it is very useful to us. Keep posting like this informative articles.
    CNC Roll Turning Lathe Machine is the best replacement solution of Hot Rolling Mill manufacturers, suppliers, and exporters in India. CNC roll turning lathes are available in many models, which drive enormous reliability improvements, reduce work requirements, removes manual mistakes, and reduces the space used to rotate in rolling work.

    ReplyDelete
  21. Online Terpercaya di asia hadir untuk anda semua dengan permainan permainan menarik dan bonus menarik untuk anda semua

    Bonus yang diberikan NagaQQ :
    * Bonus rollingan 0.5%,setiap senin di bagikannya
    * Bonus Refferal 10% + 10%,seumur hidup
    * Bonus Jackpot, yang dapat anda dapatkan dengan mudah
    * Minimal Depo 15.000
    * Minimal WD 20.000
    * Deposit via Pulsa TELKOMSEL
    * 6 JENIS BANK ( BCA , BNI, BRI , MANDIRI , CIMB , DANAMON )

    Memegang Gelar atau title sebagai AGEN POKER ONLINE Terbaik di masanya

    Games Yang di Hadirkan NagaQQ :
    * Poker Online
    * BandarQ
    * Domino99
    * Bandar Poker
    * Bandar66
    * Sakong
    * Capsa Susun
    * AduQ
    * Perang Bacarrat
    * Perang Dadu (New Game)


    Info Lebih lanjut Kunjungi :
    Website : NAGAQQ
    Facebook : NagaQQ official
    WHATSAPP : +855977509035
    Line : Cs_nagaQQ
    TELEGRAM :+855967014811

    BACA JUGA BLOGSPORT KAMI YANG LAIN:
    Winner NagaQQ
    Daftar NagaQQ
    nagaqq

    ReplyDelete
  22. If you wish to add the 123.hp.com/setup printer to your list, you have to use the printers & Scanners preference. Most of the time, as you try to add a printer to the MAC operating system, then installing the driver is very important and the latest software version shall be downloaded in the process. In case if the printer is enabled with the Airprint feature, it will be compatible with the MAC OS. So you can call the support team number @+1-850-761-8950 in case of the issues you might face while adding the HP printer via 123.hp.com/setup.
    Read More:
    123.hp.com/dj3630 printer setup
    How to connect HP Deskjet 3755 to wifi
    HP Envy photo 7155 setup

    ReplyDelete
  23. Let us help you to activate Crackle channel using the page, crackle.com/activate. The process of activation is easy if you learn the activation steps. You can choose a compatible device and move to the store to add Crackle channel. Check if you can find the added crackle channel. If yes, tap on the channel icon and log in with the appropriate credentials. The activation code must be typed on the page, crackle.com/activate to complete the channel activation. You can now pick the best shows on the channel to start streaming. For help and support to activate the Sony crackle channel, reach out to our team of customer support @ +1-805-221-0330

    ReplyDelete
  24. Quickbooks helps you in tracking all the businesses operations very accurately and efficiently for all small and medium sized businesses. Sometimes you may not be able to run the operations correctly due to many reasons. You need to resolve these issues as quickly as possible to avoid any losses in your work. Quickbooks may encounter some bug issues and this can result in occurrence of issue- QuickBooks unable to complete the operation and need to restart Error.

    ReplyDelete
  25.  Quickbooks helps all small and medium sized businesses in their accounting tasks and also helps in maintaining the company files and records etc. However at times, you may face an issue where Quickbooks is unable to verify Financial institution Error. And, this happens when Bank ID is not properly entered or if the program does not recognize the ID code.

    ReplyDelete
  26. Please send content elimination requests to the unique source of the video. We just embed the movies from larger sites like Porn Best Website If you remove your video from its unique supply it is going to be removed from our web site Daily Porn Update

    ReplyDelete
  27. Read the article posted on your portal, 123.hp.com/setup with the title “How to setup HP printer ‘’. It’s interesting to read. The Printer setup steps are clear. I could complete the setup process easily. Let me share the post with users who do not know how to execute HP printer setup steps.

    ReplyDelete
  28. I’m happy to provide the best review rating your blog post with the title “How to activate Fx Networks using the portal, fxnetworks activate Roku. You have explained the activation process clearly. Let me share the post with users who do not know how to activate Fx Networks channel

    ReplyDelete
  29. https://www.prakse.lv/enterprise/profile/7323/igpinstitute
    https://investimonials.com/users/sunilsharmaigp@gmail.com.aspx
    https://findery.com/sunilsharma1
    https://www.bizinfe.com/Igpinstitute
    https://www.walkscore.com/people/296500010919
    https://www.spoke.com/lists/604b687a38d37e3dbd01de1a
    https://indiabusinesstoday.in/users/detail/sunilsharma-10857
    http://www.thedesignportal.org/business/igp-institute/

    ReplyDelete
  30. Excellent article and great share
    After reading your article with the title “How to execute NatGeoTv/Activate steps, I became a big admirer. You have a great job Let me share the post with streamers who do not know how to activate Nat geo channel

    ReplyDelete
  31. Amazing write-up. It's interesting to read your article. I could understand the guidelines
    to choose go.roku.com/selectinput. You have explained
    the process clearly. Keep up the hard work. Let me share the article with users who do not
    how to select the required input on Roku.

    ReplyDelete
  32. Our philosophy dissertation help handle the dissertation on any topic. We are the leading philosophy dissertation service providers. Our expert team is dedicated to helping you achieve your dream.

    ReplyDelete
  33. Excellent and informative article. I’m impressed after reading I felt hard to activate Roku. After reading your blog post, I could activate the device easily using the portal,
    Roku Com Link.

    ReplyDelete
  34. Wonderful article! 바카라사이트 We are linking to this particularly great content on our
    website. Keep up the great writing.


    ReplyDelete
  35. 카지노사이트윈 I have been browsing online more than 3 hours lately, but I by no means found any fascinating article like yours. It is pretty price sufficient for me. In my view, if all webmasters and bloggers made excellent content as you probably did, the web might be much more helpful than ever before.

    ReplyDelete
  36. 토토사이트 We stumbled over here from a different web address and thought I might as well check things out. I like what I see so now i am following you. Look forward to looking into your web page repeatedly.

    ReplyDelete
  37. 토토 Keep up the superb work, I read few blog posts on this website
    and I conceive that your site is really interesting and contains lots
    of wonderful info.

    ReplyDelete
  38. https://www.reverbnation.com/igpinstitute
    https://www.atlasobscura.com/users/51cb9625-0ff3-4104-b6ee-44880cc5a4a9
    https://list.ly/sunil-sharma-5/lists
    https://www.tuugo.us/userProfile/igpinstitute/2366694
    https://transtats.bts.gov/exit.asp?url=https://shop.igpinstitute.org/product/CA-Inter-CMA
    https://transtats.bts.gov/exit.asp?url=https://shop.igpinstitute.org/
    https://www.magcloud.com/user/igpinstitute
    https://sunilsharma.dreamwidth.org/profile
    https://tapas.io/sunilsharmaigp
    https://ioby.org/users/sunilsharmaigp446145
    https://www.flicks.co.nz/member/sunilsharmaigp9631/
    https://speakerdeck.com/sunilsharma
    https://startupmatcher.com/p/igpinstitute
    http://forums.qrecall.com/user/edit/148334.page
    https://www.ultimate-guitar.com/u/sunilsharmaigp

    ReplyDelete
  39. https://www.flipcart.link/user/sunilsharma
    http://org.domhold.com/shop.igpinstitute.org
    https://coastalncfishing.com/community/profile/sunilsharma/
    https://www.skillshare.com/profile/Sunil-Sharma/317465937
    https://social.technet.microsoft.com/Profile/Igp%20institute
    https://www.max2play.com/en/forums/users/igpinstitute/
    http://ruspioner.ru/profile/view/45101
    https://www.nj24.pl/users/sunilsharma
    https://pinshape.com/users/1285371-sunil-sharma#designs-tab-open
    https://www.manaaki.io/forums/users/igpinstitute/
    https://www.ugazdinky.sk/forums/users/Igpinstitute
    https://www.snupps.com/Igpinstitute
    https://www.bitsdujour.com/profiles/dTNOCg
    https://www.corporatelivewire.com/profile.html?id=71baf09a74dc618d9b0679309e9bc32a44844cbc

    ReplyDelete
  40. If you don't know how to download driver and software using 123.hp.com/setup, read this article titled "HP Printer Setup." The content of the blog is outstanding. Spend your leisure time reading the page if you're ready to begin the Printer setup. If you're a first-time HP printer user, I recommend reading the blog for more information on HP printers. Also, the article was clear, and you may set up your printer on your own.

    ReplyDelete
  41. https://www.codingame.com/profile/1da09614d8264865c054387917c0bec81563244
    https://linustechtips.com/profile/836853-igpinstitute/
    https://www.glistatigenerali.com/users/sunilsharma/
    https://www.codemade.io/user/sunilsharma/
    https://naijamp3s.com/profile/rahulsharma1

    ReplyDelete
  42. There must have been many difficulties in providing this information.Bóng88 Nevertheless, thank you for providing such high-quality information.

    ReplyDelete
  43. The blog written is extremely impressive, with a great topic. However, a bit more research could have strengthened it even further. LiveWebTutors is one of the reputed Dissertation Editing Services provider in the UK. The most effective and informative contents for the students.

    ReplyDelete
  44. Today I was surfing on the internet & found this article I read it & it is really amazing articles on the internet on this topic thanks for sharing such an amazing article. Are you looking for Essay Writing Help ? so contact us soon.

    ReplyDelete
  45. It's really very detailed article on this topic. I'm sure that you spent a lot of time at its writing. This is a very worthwhile advice, thank you for that. I recently read a very good article of a professional writer on this subject. For more info you may check it by yourself here. Good luck! Online Essay Help

    ReplyDelete
  46. Thank you for posting such a great article! It contains wonderful and helpful posts. Keep up the good work


    Bihar Student Credit Card Scheme 2021 | BSCC Scheme

    ReplyDelete
  47. https://actionnetwork.org/users/aman-sharma-3/profile
    https://theomnibuzz.com/members/amansharma/profile/classic/
    https://www.sporcle.com/user/amansharmatravel/
    https://rabbitroom.com/members/amansharmatravel/
    https://www.pixiv.net/en/users/68621878

    ReplyDelete
  48. Best blog Ever! A big thumbs up for your authentic information on this blog. I will come back for more. How to Buy Data on Airtel 200 for 1gb

    ReplyDelete
  49. Looking at this article, I miss the time when I didn't wear a mask. Kèonhàcái Hopefully this corona will end soon. My blog is a blog that mainly posts pictures of daily life before Corona and landscapes at that time. If you want to remember that time again, please visit us.

    ReplyDelete
  50. It's a very nice and informative article, Thank you for sharing this topic. N-Power Login Website

    ReplyDelete
  51. My brother suggested I might like this website.
    He was totally right. This post actually made my day.
    You can not imagine simply how much time I had spent for this info!
    Thanks! 토토사이트

    ReplyDelete
  52. Hello! I just would like to offer you a huge thumbs up for
    the excellent info you have right here on this post. I am coming back to
    your website for more soon. 경마

    ReplyDelete
  53. You made some good points there. I searched for the subject matter and found most individuals will go along with your blog. Congratulations! strong love message for her

    ReplyDelete
  54. 카지노사이트 Interesting topic for a blog. I have been searching the Internet for fun and came upon your website. Fabulous post. Thanks a ton for sharing your knowledge! It is great to see that some people still put in an effort into managing their websites. I’ll be sure to check back again real soon.

    ReplyDelete
  55. 토토사이트 You did cheerfully empower me because of the ideas you have posted.

    ReplyDelete
  56. 토토사이트 I am genuinely thankful to the holder of this web page who has shared this wonderful paragraph at at this place


    ReplyDelete
  57. Independent Escorts in Dubai is the right decision. the top Dubai Escorts at your one Click. Log on to us for entertainment.

    ReplyDelete
  58. To learn how to download HP printer driver from 123.hp.com, here we explain the guidelines in detail.You can choose a compatible device and connect your device to the network. Once if the network is active, go to the software download page. Enter the Printer name and version. Choose the software and click on the option, Download. After extracting the software setup file to the required folder, tap on the setup file and follow the on-screen guide. Forassistance, please contact our HP printer support.

    ReplyDelete
  59. How do I download HP printer driver from 123.hp.com
    To learn how to download HP printer driver from  123.hp.com, here we explain the guidelines in detail.You can choose a compatible device and connect your device to the network. Once if the network is active, go to the software download page. Enter the Printer name and version. Choose the software and click on the option, Download. After extracting the software setup file to the required folder, tap on the setup file and follow the on-screen guide. For assistance, please contact our HP printer support.

    ReplyDelete
  60. https://canodrom.decidim.barcelona/profiles/romitsingh/activity
    https://www.adproceed.com/author/sunilsharma/
    https://forum.urbanite-project.eu/profiles/romitsingh/activity
    https://assises.cc-seignanx.fr/profiles/romitsingh/activity
    https://www.proko.com/profile/sunilsharma/activity
    https://patly.org/profiles/romitsingh/activity
    https://europeancitizensbank.eu/profiles/romitsingh/activity
    https://osallistu.riihimaki.fi/profiles/sunil_sharma/activity
    https://besayaeuropa.es/profiles/romitsingh/activity
    https://www.participons-a-verrieres.fr/profiles/sunilsharma/activity

    ReplyDelete
  61. Are you ready to start 123.hp.com/setup? Let me suggest an article to read.
    Read the post a few days back. I could find 123.hp.com/setup steps,
    software installation procedure, troubleshooting guide to fix setup issues. It will be useful if you can suggest the latest Printer models available in the market today.

    ReplyDelete
  62. Howdy, that is striking material. I truly like the subject. Would you have the choice to uncover to me something about it ... I'd love to learn. The water damage restoration company. Best offers 80% off. Best Price available in this platform.

    ReplyDelete
  63. Absolutely this blog is unique and informative. You are a good blog creator and writer. You readily explained every topic! Looking forward to reading more posts from your blog…this site is equally good, just check it out...check my post on overwatch ranks

    ReplyDelete
  64. Thanks for the interesting knowledge and ideas given to your viewers. I enjoyed coming around this blog and I am looking forward to read more of your article. nmu post utme past question

    ReplyDelete
  65. https://www.guidetraining.xyz/community/profile/sunilsharma/
    https://highenergyhomes.com/community/profile/romitsingh/
    https://www.librarygirl.net/profile/sunilsharmaigp/profile
    https://lazismukalbar.org/forum/profile/sunilsharma/
    https://53963.peda.univ-lille.fr/community/profile/sunilsharma

    ReplyDelete
  66. Are you ready to start 123.hp.com/setup? Let me suggest an article to read. Read the post a few days back. I could find 123.hp.com/setup steps, software installation procedure, troubleshooting guide to fix setup issues. It will be useful if you can suggest the latest Printer models available in the market today.

    ReplyDelete
  67. Amazing write-up and great share
    I’m impressed and have no other words to comment on your blog. I could understand the Roku.com/link activation procedure after reading your blog. Kindly post the features of the top Roku device models available in the market today. My review rating your post would be 100 Starz
    Keep posting more creative blogs. I can share the post with users who do not know how to activate Roku using Roku.com/link

    ReplyDelete
  68. https://openlyvoluntary.com/Lets-think-forum/profile/sunilsharma/
    https://wefight.gouette.com/community/profile/sunilsharma/
    https://www.freerunmom.com/forum/profile/romitsingh/
    https://winter-modular.com/community/profile/sunilsharma/
    https://venus.coolforum.info/profile.php?mode=viewprofile&u=1065

    ReplyDelete
  69. To activate the Travel channel, here we explain the channel activation steps. Select your device and access the device app store to add Travel channel. Launch the channel app. Then proceed with the settings to collect the Travel channel activation code. Enter the code by visiting the portal, Travelchannel.com/activate to complete the channel activation. If you require any help to activate the channel, dial the toll-free number provided on our portal

    ReplyDelete
  70. you have explain streaming algorithm for encoding an array of octets and based 64 encoding in very simple ways. Look in this article
    www quickbooks com support update html error 1603 I adopt a very simple and intrective way so that we can provide the best solution of the error-1603

    ReplyDelete
  71. When will I get married? Will it be an arranged marriage or a love marriage? These are some of the questions that usually come to mind when someone reaches to marriageable age. If you are able to get the answers to your questions, then you will lead a happy and prosperous married life. You know what! Marriage prediction is one of the best tools of Vedic Astrology that can help you to get the answers to these questions. A pioneer astrologer like Dr. Vinay Bajrangi can help you in getting things done symmetrically. If you need any kind of help, then contact him. He can give you Marriage predictions based on Vedic Astrology.

    ReplyDelete
  72. Fabiflu Tablet is an anti-viral medicine used to treat mild to moderate COVID-19 disease under emergency conditions. This medicine full name is fabiflu 400 mg tablet

    ReplyDelete
  73. As I am looking at your writing, keonhacai I regret being unable to do outdoor activities due to Corona 19, and I miss my old daily life. If you also miss the daily life of those days, would you please visit my site once? My site is a site where I post about photos and daily life when I was free.

    ReplyDelete
  74. The contents of this blog are always very interesting, educative and informative, I must commend you for the good work you are doing here while I urge you to keep it up romantic good morning message for her to fall in love

    ReplyDelete
  75. Awesome post. It really shows your immense knowledge and research on this topic. Please keep sharing. Also, Check death quotes

    ReplyDelete
  76. At HAND IN HAND REMODEL AND REPAIR LLC, we have helped thousands of victims of flood and storm damage throughout the state of Washington and the nation. Our team is equipped to restore properties of all sizes and all degrees of damage. When you contact us, we respond quickly and will arrive on-site within 30 minutes or less. We get right to work, employing our proven Flood Damage Restoration process so that you can get back to enjoying your home or running your business.

    ReplyDelete
  77. I found so many exciting matters in this particular content, I would like to request please keep posting such informative content. Check out microcenter price match

    ReplyDelete
  78. quickbooks file doctor is a tool for those who are in accounts field quickbooks file doctor download

    ReplyDelete
  79. https://www.digital.hms.org.il/profile/sunilsharmaigp/profile
    https://www.fanshaweconservationarea.ca/profile/sunilsharmaigp/profile
    https://www.rsg.gg/profile/sunilsharmaigp/profile
    https://www.justicedesk.org/profile/sunilsharmaigp/profile
    https://www.imeresthalassas.gr/profile/sunilsharmaigp/profile

    ReplyDelete
  80. I'm so happy to finally find a post with what I want. 메이저토토사이트 You have inspired me a lot. If you are satisfied, please visit my website and leave your feedback.

    ReplyDelete
  81. We are an astrology company incepted in 1999 as Bajrangi Dham, and now we are known as Bajrangi Astro. If you are dreaded about Love Marriage Yoga, or other marriage predictions, contact us via mail or call. You can also book a direct appointment to bail out of all the issues.

    ReplyDelete
  82. As I do not know how to complete 123-hp-com-dj3755 setup and resolve setup issues, read your blog post titled, How to begin 123-hp-com-dj3755 setup. You have explained the setup process clearly. I like the blog format as well. Let me share the blog on my social media profile

    ReplyDelete
  83. Howdy! Do you know if they make any plugins to assist with SEO? I’m trying to get my blog to rank for some targeted keywords but I’m not seeing very good results. If you know of any please share. Cheers! 먹튀검증커뮤니티

    ReplyDelete
  84. Thanks for sharing. I found a lot of interesting information here. A really good post, very thankful and hopeful that you will write many more posts like this one. You may want to see prayer for the dead islam 

    ReplyDelete
  85. Outstanding Information. Thanks for sharing with us. Keep sharing again.
    Akshi Engineers has a outstanding Mill Stand Solution available. We exportes, manufacturers and supplies Mill Stand & TMT Bar for Rolling Mill. If you have need mill stand for Rolling Mill, pls contact us via a toll-free number or visit the website.

    ReplyDelete
  86. Excellent Post. I appreciate your knowledge and your hard work. Thanks for sharing with us.
    If you are looking for the Best Personal Loan Provider In India. do not go anywhere. Easy Loan Mart is one of the leading loan service providers with low-interest rates. We provide loan services based on your unique circumstances. We have an online personal loan facility, you can also apply for a personal loan online. To get an instant approval for a low EMI personal loan, visit our website.

    ReplyDelete
  87. This is very interesting, You are a very skilled blogger. I've joined your rss feed and look forward to seeking more of your wonderful 메이저토토. Also, I have shared your website in my social networks!

    ReplyDelete
  88. Thanks for another great post. Where else could anyone get this kind of information in such a perfect and well-written way? You did a great job. Am coming back to read more. Checkout: Green Card Marriage Cost

    ReplyDelete
  89. I have just read this blog and I’ll surely come back for more posts, and also this article gives the light in which we can observe the reality of the topic. Thanks for this nice article! Now read this too...happy doctors day gif

    ReplyDelete
  90. I like the helpful info you provide in your articles. I’ll bookmark your blog and check again here frequently. I’m quite sure I’ll learn plenty of new stuff right here! Good luck for the next. 먹튀검증업체

    ReplyDelete
  91. I just want to say I’m new to blogs and certainly savored this blog site. Very likely I’m planning to bookmark your blog . You amazingly have fantastic articles. Thanks for sharing your blog.야동

    ReplyDelete
  92. Very good article! We are linking to this particularly great content on our site. Keep up the great writing. Pretty valuable material, overall I consider this is worth a bookmark, thanks 대딸방

    ReplyDelete
  93. "It has a good meaning. If you always live positively, someday good things will happen. Let’s believe in the power of positivity. Have a nice day.횟수 무제한 출장

    ReplyDelete
  94. Nice post. I learn something totally new and challenging on sites I stumble upon every day. 타이마사지

    ReplyDelete
  95. Recently, I have started to read a lot of unique articles on different sites, and I am enjoying that a lot. Although, I must tell you that I still like the articles here a lot. They are also unique in their own way. 먹튀검증업체


    ReplyDelete

  96. I amazed with the research you made to make this actual submit incredible.
    Great activity!

    Have a look at my homepage; 사설경마

    ReplyDelete
  97. It is very well written, and your points are well-expressed. I request you warmly, please, don’t ever stop writing.
    카지노사이트

    ReplyDelete
  98. Your article is very interesting. I think this article has a lot of information needed, looking forward to your new posts.
    스포츠토토

    ReplyDelete
  99. Thank you for sharing this information. I read your blog and I can't stop my self to read your full blog. Again Thanks and Best of luck to your next Blog in future.
    온라인카지노

    ReplyDelete
  100. Hello, I read the post well. 안전놀이터추천 It's a really interesting topic and it has helped me a lot. In fact, I also run a website with similar content to your posting. Please visit once


    ReplyDelete
  101. Hi there, I simply hopped over in your website by way of StumbleUpon. Now not one thing I’d typically learn, but I favored your emotions none the less. Thank you for making something worth reading. 메이저사이트순위


    ReplyDelete
  102. Read your blog post recently as I do not know how to execute Roku TV setup steps. You have explained the setup steps clearly. I’m happy that I could complete Roku tv setup quickly. Kindly post a blog explaining the guidelines to fix setup issues that popup
    Keep up the good work
    My rating for your blog post would be 100 stars

    ReplyDelete
  103. Hello! I just would like to offer you a huge thumbs up for
    the excellent info you have right here on this post. I am coming back to
    your website for more soon. 메이저사이트

    ReplyDelete
  104. Your style is very unique compared to other people I’ve read stuff from. Thanks for posting when you have the opportunity, Guess I will just bookmark this page 카지노사이트

    ReplyDelete
  105. It as really a great and helpful piece of info. I am glad that you shared this helpful info with us. Please keep us up to date like this. Thank you for sharing. 토토

    ReplyDelete
  106. 토토사이트 I really enjoy the blog.Really looking forward to read more. Really Great.


    ReplyDelete
  107. It's interesting to read your blog post titled, How to set up hp 123.hp.com/setup and resolve setup issues that popup. I could find the instructions to connect the necessary cables insert the cartridge, connect hp dj3630 printer to the network and add Windows, Mac computers. Let me share the blog with users who do not know how to execute hp dj3630 setup

    ReplyDelete
  108. Thank you for posting such a great article. Keep it up mate.

    EEDS Login

    ReplyDelete
  109. Thank you for posting such a great article. Keep it up mate.

    UP URise Portal

    ReplyDelete
  110. Thank you for posting such a great article. Keep it up mate.

    ECHS 64kb Card Status

    ReplyDelete
  111. Someone necessarily help to make severely articles I might state. That is the first time I frequented your web page and up to now? I surprised with the analysis you made to create this actual publish extraordinary. Fantastic process 안전놀이터

    ReplyDelete
  112. Greetings ! Very useful advice in this particular article! Thanks a lot for sharing! Feel free to visit my site Stem Subjects List

    ReplyDelete
  113. If you are interested to learn how to activate Roku using Roku.com/link, let me suggest the blog post titled, How to activate Roku. Read the post a few days back. I could find clear guidelines to activate Roku. Spend your free time reading the post to learn Roku.com/link activation guidelines
    Also please do not forget to share your feedback after reading. The post can help new Roku users who do not know how to activate roku

    ReplyDelete
  114. Quality articles is the secret to attract
    the visitors to go to see the site, that’s what this website is providing. 스포츠토토 하는법

    ReplyDelete
  115. 스포츠토토 Hurrah! In the end I got a weblog from where
    I be able to actually take helpful data concerning my study
    and knowledge.

    ReplyDelete
  116. 카지노사이트추천 See the website and the information is very interesting, good work!Thank you for providing information from your website. On of the good website in search results.

    ReplyDelete
  117. This site seems to inspire me a lot. Thank you so much for organizing and providing this quality information in an easy to understand way. I think that a healthy era of big data can be maintained only when such high-quality information is continuously produced. And I, too, are working hard to organize and provide such high-quality information. It would be nice to come in once and get information.

    Also visit my site:스포츠토토


    ReplyDelete
  118. hellow~~I was surprised to see your writing. This is a really useful resource~
    I would like to see these materials often~
    I will come here often. Thank you!!!토토사이트

    ReplyDelete
  119. Thanks for sharing a nice article really such a wonderful site
    you have done a great job once more thanks a lot!!먹튀검증

    ReplyDelete
  120. I think this is an informative post and it is very beneficial and knowledgeable.
    Therefore, I would like to thank you for the endeavors that you have made in
    writing this article. All the content is absolutely well-researched. Thank!!먹튀사이트

    ReplyDelete
  121. Hello, I'm here to give you all the information about verification. Thank you for visiting our website!!먹튀사이트

    ReplyDelete
  122. When you find out your child is lying to you, ask him: Did this really happen or did you wish it happened?
    Give your child an extra consequence مشکلات زمان خواب در نوزادان as a result of the lie he told. This is done, for example, giving the child a homework to do instead of taking electronics from him.

    ReplyDelete
  123. Thank you for posting such a great article. Keep it up mate.

    पीडीएस बिहार

    ReplyDelete
  124. Set up Microsoft 365 through microsoft365.com/setup and use Microsoft apps such as Word, PowerPoint, and Excel, online storage, cloud-connected features, and more. If you have a Microsoft account associated with Outlook, OneDrive, Xbox Live, or Skype, you can proceed with Microsoft 365 login
    | microsoft365.com/setup | www.hp.com/go/wirelessprinting

    ReplyDelete
  125. A Quick Guide to Canon Printer SetupMake sure your printer hardware is configured.
    On your PC browser, go to cannon.com/ijsetup
    Click Set Up.
    Using your Canon IJ printer model, the download printer software.
    Double click to install it.
    Connect a network to your printer and PC.
    Finish the Canon IJ printer setup.
    HP LaserJet Printer Setup
    Configure your HP Printer hardware.
    Visit 123.hp.com/laserjet from the browser.Using your LaserJet model name, download software.Double click to start the installation.Connect the printer via wireless network or USB.Add the LaserJet printer to your system devices.Finish the HP LaserJet setup.

    ReplyDelete
  126. It's very interesting. And it's fun. This is a timeless article. I also write articles related to , and I run a community related to 메이저놀이터. For more information, please feel free to visit !!

    ReplyDelete
  127. Good information. Lucky me I recently found your site by
    accident (stumbleupon). I have book-marked it for later!
    토토사이트



    ReplyDelete
  128. Thanks for such a fantastic blog. Where else could anyone get that kind of info written in such a perfect way? I have a presentation that I am presently writhing on, and I have been on the look out for such great information. 먹튀검증사이트

    ReplyDelete
  129. I'm reading it well. This is something that Your post has really helped me a lot 안전공원추천

    ReplyDelete
  130. NBA superstar Stephen Curry participated in 사설토토사이트 the game wearing basketball shoes engraved with Hong Kong's famous actor's face

    ReplyDelete
  131. Wohh exactly what I was searching for, 토토검증업체 it for I should state and much obliged for the data. Instruction is unquestionably a sticky subject

    ReplyDelete
  132. Hello, I am one of the most impressed people in your article. 토토사이트 I'm very curious about how you write such a good article. Are you an expert on this subject? I think so. Thank you again for allowing me to read these posts, and have a nice day today. Thank you.

    ReplyDelete
  133. I think that is among the so much significant info for me.
    And i am satisfied reading your article. However should remark on few basic issues, The
    website style is perfect, the articles is
    truly nice : D. Good process, cheers경마

    ReplyDelete
  134. Thanks for the blog filled with so many information. Stopping by your blog helped me to get what I was looking for. Now my task has become as easy as ABC. 안전놀이터


    ReplyDelete
  135. Many thanks for the article, I have a lot of spray lining knowledge but always learn something new. Keep up the good work and thank you again. 온라인슬롯


    ReplyDelete
  136. You actually make it look so easy with your performance but I find this matter to be actually something which I think I would never comprehend. It seems too complicated and extremely broad for me. I'm looking forward for your next post, I’ll try to get the hang of it! Feel free to visit my website;
    야설

    ReplyDelete
  137. It is perfect time to make some plans for the future and it is time to be happy. I've read this post and if I could I desire to suggest you some interesting things or suggestions. Perhaps you could write next articles referring to this article. I want to read more things about it! Feel free to visit my website; 일본야동

    ReplyDelete
  138. I want you to thank for your time of this wonderful read!!! I definately enjoy every little bit of it and I have you bookma rked to check out new stuff of your blog a must read blog! Feel free to visit my website;

    국산야동

    ReplyDelete
  139. Its an amazing website, really enjoy your articles. Helpful and interesting too. Keep doing this in future. I will support you. Feel free to visit my website; 일본야동



    ReplyDelete
  140. Such a very useful article. Very interesting to read this article.I would like to thank you for the efforts you had made for writing this awesome article. Feel free to visit my website; 한국야동

    ReplyDelete
  141. It seems like I've never seen an article of a kind like . It literally means the best thorn. It seems to be a fantastic article. It is the best among articles related to 먹튀검증업체. seems very easy, but it's a difficult kind of article, and it's perfect.

    ReplyDelete
  142. What I was thinking about was solved thanks to your writing. I have written on my blog to express my gratitude to you.토토사이트My site is We would be grateful if you visit us.


    ReplyDelete
  143. I finally found what I was looking for! I'm so happy. 메이저사이트


    ReplyDelete
  144. 사설토토사이트November 25, 2021 at 1:17 AM

    Of course, your article is good enough, 사설토토사이트 but I thought it would be much better to see professional photos and videos together. There are articles and photos on these topics on my homepage, so please visit and share your opinions.


    ReplyDelete
  145. Sometimes an error pops up on your desktop with the printer is in an error state hp state which is caused by the updation of a new window and then these types of errors occur, but these types of errors can be fixed by taking steps mentioned in the above sites.

    ReplyDelete
  146. Hi, I read your blogs like every week. Your story-telling style is awesome, keep doing what you’re doing!


    https://www.betmantoto.pro

    ReplyDelete
  147. Hello Dear, are you really visiting this web site regularly, if so after that you will definitely obtain pleasant experience.
    경마사이트

    ReplyDelete
  148. I’m thinking some of my readers might find a bit of this interesting. Do you mind if I post a clip from this and link back? Thanks 안전놀이터


    ReplyDelete
  149. Your ideas inspired me very much. 바카라사이트 It's amazing. I want to learn your writing skills. In fact, I also have a website. If you are okay, please visit once and leave your opinion. Thank you.


    ReplyDelete
  150. Amazing blog ever come across on the internet. In case you need online assignment help, then sourceessay.com will be the best place to learn the amazing writing techniques from assignment writers. Write that essay Adelaide

    ReplyDelete
  151. When I read your article on this topic, the first thought seems profound and difficult. There is also a bulletin board for discussion of articles and photos similar to this topic on my site, but I would like to visit once when I have time to discuss this topic. 온라인슬롯


    ReplyDelete
  152. I figure this article can be enhanced a tad. There are a couple of things that are dangerous here, and if you somehow managed to change these things, this article could wind up a standout amongst your best ones. I have a few thoughts with respect to how you can change these things. 메이저놀이터


    ReplyDelete
  153. In this guide, we show you the steps to use the How To Install Sound Drivers drivers, and the steps to install the device on your Windows 10 device.

    ReplyDelete
  154. Awesome article! I want people to know just how good this information is in your article. It’s interesting, compelling content. Your views are much like my own concerning this subject 먹튀검증 It touched me a lot. I would love to hear your opinion on my site. Please come to the site I run once and leave a comment. Thank you.


    ReplyDelete
  155. 안전놀이터추천December 5, 2021 at 2:26 AM

    Nice to meet you. Your website is full of really interesting topics. It helps me a lot. I have a similar site. We would appreciate it if you visit once and leave your opinion. 안전놀이터추천


    ReplyDelete
  156. Traditional bookstores have always existed on high streets, but in the digital age, the internet is proving to become a serious competitor to traditional brick and mortar stores. This article examines both sides of the coin and provides an appropriate insight into the phenomenon of shopping of books online. 메이저사이트추천

    ReplyDelete
  157. Great Info. Thanks for sharing. It was really helpful. hat's a great value bomb you are dropping here. Nice to see this post. Nice info. I was getting an error code called Quickbooks update error 1328
    in the popular accounting tool called Quickbooks. Hope this helps.

    ReplyDelete
  158. Hello, I am one of the most impressed people in your article. sòng bạc I'm very curious about how you write such a good article. Are you an expert on this subject? I think so. Thank you again for allowing me to read these posts, and have a nice day today. Thank you.


    ReplyDelete
  159. Wow! This was an incredibly wonderful post. Many thanks for providing these details. best car insurance in california

    ReplyDelete
  160. When I read an article on this topic, 메가슬롯 the first thought was profound and difficult, and I wondered if others could understand.. My site has a discussion board for articles and photos similar to this topic. Could you please visit me when you have time to discuss this topic?

    ReplyDelete
  161. حتماً تا کنون دانستی‌های جالبی در مورد درب ضد سرقت شنیده‌اید و همچنین به احتمال زیاد این نکات مهیج نقش پر رنگی در ترغیب شما برای خرید و استفاده از درب‌های ضد سرقت داشته‌اند. این درب‌ها ساختار متفاوتی نسبت به درب‌های معمولی دارند و این موضوع بر کسی پوشیده نیست اما این که چه پارامترهایی از یک درب چنین مقاومت و استحکامی باقی می‌گذارد، جالب توجه است. نکات بسیاری در مورد درب ضد سرقت وجود دارد ممکن است تا کنون نشنیده باشید.

    ReplyDelete
  162. really nice blog. Quickbook is an accounting software it really very excellent working u can go through and check but few days before i found an error on it quickbooks error 404 but now its resolved.

    ReplyDelete
  163. Your information was very useful to me. That's exactly what I've been looking for 샌즈카지노!

    ReplyDelete
  164. 메이저토토사이트January 4, 2022 at 11:27 PM

    This is the perfect post.메이저토토사이트 It helped me a lot. If you have time, I hope you come to my site and share your opinions. Have a nice day.

    ReplyDelete
  165. Many thanks for the article, I have a lot of spray lining knowledge but always learn something new. Keep up the good work and thank you again. 먹튀사이트

    ReplyDelete
  166. Thank you for your post, I look for such article along time, today i find it finally. this post give me lots of advise it is very useful for me. nice i like it.
    카지노사이트

    ReplyDelete
  167. I just want to let you know that I just check out your site and I find it very interesting and informative. 온라인카지노

    ReplyDelete
  168. I want you to thank for your time of this wonderful read!!! I definately enjoy every little bit of it and I have you bookmarked to check out new stuff of your blog a must read blog! 온라인카지노

    ReplyDelete
  169. Very nice article and straight to the point. I don't know if this is truly the best place to ask but do you folks have any idea where to get some professional writers? Thank you. Feel free to visit my website; 온라인카지노

    ReplyDelete
  170. I am glad to discover this page. I have to thank you for the time I spent on this especially great reading !! I really liked each part and also bookmarked you for new information on your site. Feel free to visit my website; 스포츠토토

    ReplyDelete
  171. Excellent work! One thing I’d really like to reply to is that fat burning plan fast is possible by the suitable diet and exercise. A person’s size not just affects appearance, but also the quality of life. Self-esteem, depressive disorders, health risks, as well as physical abilities are affected in weight gain. It is possible to do everything right but still gain. In such a circumstance, a problem may be the offender. While a lot of food instead of enough exercise are usually the culprit, common health conditions and traditionally used prescriptions can greatly increase size. I am grateful for your post here.

    무료야설
    대딸방
    타이마사지
    안마
    바카라사이트

    ReplyDelete
  172. 토토사이트추천January 9, 2022 at 2:01 AM

    That's a great article! The neatly organized content is good to see. Can I quote a blog and write it on my blog? My blog has a variety of communities including these articles. Would you like to visit me later? 토토사이트추천

    ReplyDelete
  173. Many thanks for the article, I have a lot of spray lining knowledge but always learn something new. Keep up the good work and thank you again. How much Money does Soccer Players Make

    ReplyDelete
  174. I came to this site with the introduction of a friend around me and I was very impressed when I found your writing. I'll come back often after bookmarking! 먹튀검증

    ReplyDelete
  175. I think a lot of articles related to are disappearing someday. That's why it's very hard to find, but I'm very fortunate to read your writing. When you come to my site, I have collected articles related to 크레이지슬롯 .

    ReplyDelete
  176. Hello its my first time to commenting anywhere, when i read this article i thought i could also create comment due to this good 토토검증

    ReplyDelete
  177. How to Activate the Roku Channel or other Free Channels and Premium Channels to your Roku TV, Roku stick.You first need to Create an Roku Account and Activate it then you can search for your favourite channel and favourite shows, then you can download it to your roku account and activate it. If you need any help in process we can provide you the assistance on Activating the Roku and the Channels on Roku.

    ReplyDelete
  178. Thank you for the information provided! Maintain the good performance of your site. You can also check my article 스포츠토토사이트

    ReplyDelete
  179. I appreciate your information in this article. It’s smart, well-written and easy to understand. You have my attention on this subject. I will be 토토

    ReplyDelete