How to detect type of compression used on the file? (if no file extension is specified)

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You can determine that it is likely to be one of those formats by looking at the first few bytes. You should then test to see if it really is one of those, using an integrity check from the associated utility for that format, or by actually proceeding to decompress.

You can find the header formats in the descriptions:

Others:

  • zlib (.zz) format description, starts with two bytes (in bits) 0aaa1000 bbbccccc, where ccccc is chosen so that the first byte viewed as a int16 times 256 plus the second byte viewed as a int16 is a multiple of 31. e.g: 01111000(bits) = 120(int16), 10011100(bits) = 156(int16), 120 * 256 + 156 = 30876 which is a multiple of 31
  • compress (.Z) starts with 0x1f, 0x9d
  • bzip2 (.bz2) starts with 0x42, 0x5a, 0x68
  • Zstandard (.zstd) format description, frame starts with a 4 byte magic number using little-endian format 0xFD2FB528, a skipable frame starts with 0x184D2A5? (question mark is any value from 0 to F), and dictionary starts with 0xEC30A437.
  • A few more formats in the magic database from the file command

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