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.
Wikipedia" would be calculated as follows:
ASCII code A B
(shown as base 10)
W: 87 1 + 87 = 88 0 + 88 = 88
i: 105 88 + 105 = 193 88 + 193 = 281
k: 107 193 + 107 = 300 281 + 300 = 581
i: 105 300 + 105 = 405 581 + 405 = 986
p: 112 405 + 112 = 517 986 + 517 = 1503
e: 101 517 + 101 = 618 1503 + 618 = 2121
d: 100 618 + 100 = 718 2121 + 718 = 2839
i: 105 718 + 105 = 823 2839 + 823 = 3662
a: 97 823 + 97 = 920 3662 + 920 = 4582
A = 920 = 398 hex (base 16)
B = 4582 = 11E6 hex
Output = 300286872 = 11E60398 hex
(The modulo operation had no effect in this example, since none of the values reached 65521).
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 properly implemented (e.g., one found in the Hierarchical_Data_Format) Fletcher's checksum.
A few tricks are used here for efficiency:
b. Any smaller value is also permissible; 4096 may be convenient in some cases.
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, Adler32 is guaranteed to give poor coverage of the available bits. Don't take my word for it, ask Mark Adler. :-)" 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 CRC32 instead of Adler-32 for SCTP, the Stream Control Transmission Protocol.