Adler-32

Adler-32 is a checksum algorithm which was invented by Mark Adler in 1995,^[1] and is a modification of the Fletcher checksum. Compared to a cyclic redundancy check of the same length, it trades reliability for speed (preferring the latter). Adler-32 is more reliable than Fletcher-16, and slightly less reliable than Fletcher-32.^[2]

History

The Adler-32 checksum is part of the widely used zlib compression library, as both were developed by Mark Adler. A "rolling checksum" version of Adler-32 is used in the rsync utility.

The algorithm

An Adler-32 checksum is obtained by calculating two 16-bit checksums A and B and concatenating their bits into a 32-bit integer. A is the sum of all bytes in the stream plus one, and B is the sum of the individual values of A from each step.

At the beginning of an Adler-32 run, A is initialized to 1, B to 0. The sums are done modulo 65521 (the largest prime number smaller than 2¹⁶). The bytes are stored in network order (big endian), B occupying the two most significant bytes.

The function may be expressed as

A = 1 + D1 + D2 + ... + Dn (mod 65521)
 B = (1 + D1) + (1 + D1 + D2) + ... + (1 + D1 + D2 + ... + Dn) (mod 65521)
   = n×D1 + (n−1)×D2 + (n−2)×D3 + ... + Dn + n (mod 65521)

 Adler-32(D) = B × 65536 + A

where D is the string of bytes for which the checksum is to be calculated, and n is the length of D.

Example

The Adler-32 sum of the ASCII string "Wikipedia" would be calculated as follows:

A = 920  =  398 hex  (base 16)
   B = 4582 = 11E6 hex
   Output = 4,582 × 65,536 + 920 = 300286872 = 11E60398 hex

The modulo operation had no effect in this example, since none of the values reached 65521.

Comparison with the Fletcher checksum

The first difference between the two algorithms is that Adler-32 sums are calculated modulo a prime number, whereas Fletcher sums are calculated modulo 2⁴−1, 2⁸−1, or 2¹⁶−1 (depending on the number of bits used), which are all composite numbers. Using a prime number makes it possible for Adler-32 to catch differences in certain combinations of bytes that Fletcher is unable to detect.

The second difference, which has the largest effect on the speed of the algorithm, is that the Adler sums are computed over 8-bit bytes rather than 16-bit words, resulting in twice the number of loop iterations. This results in the Adler-32 checksum taking between one-and-a-half to two times as long as Fletcher's checksum for 16-bit word aligned data. For byte-aligned data, Adler-32 is faster than a properly implemented Fletcher's checksum (e.g., one found in the Hierarchical Data Format).

Example implementation

In C, an inefficient but straightforward implementation is :

const int MOD_ADLER = 65521;
 
uint32_t adler32(unsigned char *data, size_t len) /* where data is the location of the data in physical memory and 
                                                      len is the length of the data in bytes */
{
    uint32_t a = 1, b = 0;
    size_t index;
 
    /* Process each byte of the data in order */
    for (index = 0; index < len; ++index)
    {
        a = (a + data[index]) % MOD_ADLER;
        b = (b + a) % MOD_ADLER;
    }
 
    return (b << 16) | a;
}

See the zlib source code for a more efficient implementation that requires a fetch and two additions per byte, with the modulo operations deferred with two remainders computed every several thousand bytes.

And in F# as follows

let Adler32 (input:byte[]) :uint32 =
    let adler32Accumulate (a:uint32, b:uint32) (v:byte) =
        ((a + uint32 v) % 65521ul, (b + a + uint32 v) % 65521ul)
    Array.fold adler32Accumulate (1ul, 0ul) input
    |> function
        | (a,b) -> (b <<< 16) ||| a

Advantages and disadvantages

• Like the standard CRC-32, the Adler-32 checksum can be forged easily and is therefore unsafe for protecting against intentional modification.
• It's faster than CRC-32 on many platforms.^[3]
• Adler-32 has a weakness for short messages with few hundred bytes, because the checksums for these messages have a poor coverage of the 32 available bits.

Weakness

Jonathan Stone discovered in 2001 that Adler-32 has a weakness for very short messages. He wrote "Briefly, the problem is that, for very short packets, Adler-32 is guaranteed to give poor coverage of the available bits. Don't take my word for it, ask Mark Adler.  :-)"^[4] The problem is that sum A does not wrap for short messages. The maximum value of A for a 128-byte message is 32640, which is below the value 65521 used by the modulo operation. An extended explanation can be found in RFC 3309, which mandates the use of CRC32C instead of Adler-32 for SCTP, the Stream Control Transmission Protocol. Adler-32 has also been shown to be weak for small incremental changes,^[5] and also weak for strings generated from a common prefix and consecutive numbers (like auto-generated label names by typical code generators).^[6]

See also

• List of hash functions

Notes

External links

• RFC 1950 – specification, contains example C code
• ZLib – implements the Adler-32 checksum
• RFC 3309 – information about the short message weakness and related change to SCTP
• Catalogue of parametrised CRC algorithms