Punycode

Separation of ASCII characters

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. if there was at least one ASCII character in the input, an ASCII hyphen is appended to the output (e.g., "bücher" → "bcher-", but "ü" → "").

Note that hyphens are themselves ASCII characters. Thus, they can be present in the input and, if so, they will be copied to the output. This causes no ambiguity: if the output contains hyphens, the one that got added is always the last one. It marks the end of the ASCII characters.

Encoding the non-ASCII characters

The non-ASCII characters are sorted by Unicode value, lowest first (if a character occurs more than once they are sorted by position). Each is then encoded as a single number. This single number defines both the location to insert the character at and which character to insert.

• i is an index into the result to insert the code at, starting at 0 (for insertion at the start).^{[citation needed]}
• n is the number of possible insertion points (current length of the result plus one).
• j is the Unicode code point to insert minus 127.^{[citation needed]}

The encoded number is n × j + i. By dividing by n and also getting the remainder, a decoder can determine j and i.

There are six possible places to insert a character in the string "bcher" (including before the first character and after the last one). There are 124 code points between the last ASCII code point (127 = 0x7F, the end of ASCII) and ü (code point 252 = 0xFC, see Unicode's Latin-1 Supplement). The ü is inserted at position 1, after the b. Thus the encoder will add the number (6 × 124) + 1 = 745, and the decoder can retrieve these by ⌊745 ÷ 6⌋ = 124 and 745 mod 6 = 1.

These numbers are strictly increasing. For the second and subsequent inserted characters, the difference between the number and the previous one is written.

The number is encoded using the letters a through z and the digits 0 through 9. It is not base-36 but a more complex scheme described below, which allows the numbers to be concatenated, with nothing separating them.

Variable-length number encoding

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 it's (1, 1, 26, 26, ...).^[2] The weight (or place value) of the least-significant digit is always 1: 'k' (=10) with a weight of 1 equals 10. After this, the weight of the next digit depends on the first threshold: generally, for any n, the weight of the (n+1)-th digit is w × (36 − t), where w is the previous weight and t is the threshold of the n-th digit. So in this case, the second symbol has a place value of 36 minus the previous threshold value of 1, which equals 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 its threshold value of 1, there is more to come. However, since the third symbol in this example is 'a' (=0), we may ignore calculating its weight. Therefore, "kva" represents the decimal number (10 × 1) + (21 × 35) = 745.

Number 745 will be encoded as 10 + 21 × 35 + 0 (base 35 used for second digit, the most significant digit 0 needed as terminator), 10 → 'k', 21 → 'v', 0 → 'a', so "bücher" → "bcher-kva".

The thresholds themselves are determined for each successive encoded character by an algorithm keeping them between 1 and 26 inclusive.^[3] The case can then be used to provide information about the original case of the string.^[4]

Because special characters are sorted by their code points by encoding algorithm, 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 Unicode character following ü, starting with "ýbücher" with code "bcher-kvaf" (different from "übücher" coded "bcher-jvab"), etc.

ACE prefix for internationalized domain names

To prevent hyphens in non-international domain names from triggering a Punycode decoding, the string xn-- is prepended to Punycode sequences in internationalized domain names. This is called ACE (ASCII Compatible Encoding).^[5]

Thus the domain name "bücher.tld" would be represented in a URL as "xn--bcher-kva.tld".