Adler-32 is a checksum algorithm which was invented by Mark Adler in 1995, 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.

## 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:

Character 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 =  0x398  (base 16)
B = 4582 = 0x11E6
Output = 0x11E6 << 16 + 0x398 = 0x11E60398
```

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 uint32_t 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, a technique first discovered for Fletcher checksums in 1988. `js-adler32` provides a similar optimization, with the addition of a trick that delays computing the "15" in 65536 - 65521 so that modulos become faster: it can be shown that `((a >> 16) * 15 + (a & 65535)) % 65521` is equivalent to the naive accumulation.