Punycode

First, all ASCII characters in the string are copied from input to output, skipping over any other characters. For example, "bücher" is copied to "bcher". If any characters were copied, i.e. there was at least one ASCII character in the input, an ASCII hyphen is added to the output next (e.g., "bücher" → "bcher-", but "ü" → ""). Since the ASCII hyphen is an ASCII character, the hyphen may itself appear in the output before this additional hyphen. However, the additional hyphen does not cause any ambiguity when reading the output, as no later part of the encoding process can introduce another ASCII hyphen; if there are one or more ASCII hyphens in the output, the last one always signifies the end of the ASCII characters.

The next part of the encoding process first requires an understanding of the decoder, which is a finite-state machine with two state variables i and n. i is an index into the string ranging from zero (representing a potential insertion at the start) to the current length of the extended string (representing a potential insertion at the end).

i starts at zero, and n starts at 128 (the first non-ASCII code point). The state progression is a monotonic function. A state change either increments i or, if i is at its maximum, resets i to zero and increments n by 1, then goes back to incrementing i in the following state change. At each state change, either the code point denoted by n is inserted or it is not inserted.

The code numbers generated by the encoder represent how many possibilities to skip before an insertion is made. There are six possible places to insert a character in the current string "bcher" (including before the first character and after the last one). There are 124 code points between the last one considered (127, the end of ASCII) and "ü" (code point 252). Also there is one position to insert a "ü" that needs to be skipped (at position zero before the 'b'). That is why it is necessary to tell the decoder to skip a total of (6 × 124) + 1 = 745 possible insertions before getting to the one required. Once the character is inserted there are now seven possible places to insert another character.

Punycode uses generalized variable-length integers to represent these values. For example, this is how "kva" is used to represent the code number 745:

A number system with little-endian ordering is used which allows variable-length codes without separate delimiters: a digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number and also on previous insertions, to increase efficiency. Correspondingly the weights of the digits vary.

In this case a number system with 36 symbols is used, with the case-insensitive 'a' through 'z' equal to the decimal numbers 0 through 25, and '0' through '9' equal to the decimal numbers 26 through 35. Thus "kva", corresponds to the decimal number string "10 21 0".

To decode this string of symbols, a sequence of thresholds will be needed, in this case (1, 1, 26). The threshold starts out as 1 and the weight is 1. The first symbol is the units place value; 'k' (=10) with a weight of 1 equals 10. After this, the threshold value is adjusted; in this case the threshold is again 1. The second symbol has a place value of 36 minus the previous threshold value, in this case, 35. Therefore, the sum of the first two symbols 'k' (=10) and 'v' (=21) is 10 × 1 + 21 × 35. Since the second symbol is not less than the threshold value of 1, there is more to come. The weight for the third symbol is the previous weight times 36 minus the second threshold value; 35 × 35. The third symbol in this example is 'a' (=0), which is less than the third threshold 26, meaning that it is the last (most significant) part of the number. Therefore, "kva" represents the decimal number (10 × 1) + (21 × 35) + (0 × 35 × 35) = 745.

The thresholds themselves are determined by an algorithm keeping them between 1 and 26 inclusive, meaning the last character of an encoding will always be alphabetic. The case can then be used to provide information about the original case of the string.

For the insertion of a second special character in "bücher", the first possibility is "büücher" with code "bcher-kvaa", the second "bücüher" with code "bcher-kvab", etc. After "bücherü" with code "bcher-kvae" comes codes representing insertion of ý, the character following ü, starting with "ýbücher" with code "bcher-kvaf" (different from "übücher" coded "bcher-jvab"), etc.

To make the encoding and decoding algorithms simple, no attempt has been made to prevent some encoded values from encoding inadmissible Unicode values: however, these should be checked for and detected during decoding.

Punycode is designed to work across all scripts, and to be self-optimizing by attempting to adapt to the character set ranges within the string as it operates. It is optimized for the case where the string is composed of zero or more ASCII characters and in addition characters from only one other script system, but will cope with any arbitrary Unicode string. Note that for DNS use, the domain name string is assumed to have been normalized using Nameprep and (for top-level domains) filtered against an officially registered language table before being punycoded, and that the DNS protocol sets limits on the acceptable lengths of the output Punycode string.