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Adler-32 is a checksum algorithm that hashes byte sequences to 32 bit values. It is named after its inventor, Mark Adler, who also co-invented the Gzip and Zlib compressed file formats. Amongst other differences, Gzip uses CRC-32 as its checksum and Zlib uses Adler-32.

The algorithm, described in RFC 1950, is simple. Conceptually, there are two unsigned integers `s1` and `s2` of infinite precision, initialized to `0` and `1`. These two accumulators are updated for every input byte `src[i]`. At the end of the loop, `s1` is `1` plus the sum of all source bytes and `s2` is the sum of all (intermediate and final) `s1` values:

```var s1 = 1;
var s2 = 0;
for_each i in_the_range_of src {
s1 = s1 + src[i];
s2 = s2 + s1;
}
return ((s2 % 65521) << 16) | (s1 % 65521);
```

The final `uint32_t` hash value is composed of two 16-bit values: `(s1 % 65521)` in the low 16 bits and `(s2 % 65521)` in the high 16 bits. `65521` is the largest prime number less than `(1 << 16)`.

Infinite precision arithmetic requires arbitrarily large amounts of memory. In practice, computing the Adler-32 hash instead uses a `uint32_t` typed `s1` and `s2`, modifying the algorithm to be concious of overflow inside the loop:

```uint32_t s1 = 1;
uint32_t s2 = 0;
for_each i in_the_range_of src {
s1 = (s1 + src[i]) % 65521;
s2 = (s2 + s1)     % 65521;
}
return (s2 << 16) | s1;
```

The loop can be split into two levels, so that the relatively expensive modulo operation can be hoisted out of the inner loop:

```uint32_t s1 = 1;
uint32_t s2 = 0;
for_each_sub_slice s of_length_up_to M partitioning src {
for_each i in_the_range_of s {
s1 = s1 + s[i];
s2 = s2 + s1;
}
s1 = s1 % 65521;
s2 = s2 % 65521;
}
return (s2 << 16) | s1;
```

We just need to find the largest `M` such that the inner loop cannot overflow. The worst case scenario is that `s1` and `s2` both start the inner loop at `65520` and every subsequent `src[i]` byte equals `0xFF`. A simple computation finds that the largest non-overflowing `M` is 5552.

In a happy coincidence, 5552 is an exact multiple of 16, which often works well with loop unrolling and with SIMD alignment.

## Comparison with CRC-32

Adler-32 is a very simple hashing algorithm. While its output is nominally a `uint32_t` value, it isn't uniformly distributed across the entire `uint32_t` range. The `[65521, 65535]` range of each 16-bit half of an Adler-32 checksum is never touched.

While neither Adler-32 or CRC-32 are cryptographic hash functions, there is still a stark difference in the patterns (or lack of) in their hash values of the `N`-byte string consisting entirely of zeroes, as this Go program shows:

```N  Adler-32    CRC-32      Input
0  0x00000001  0x00000000  ""
1  0x00010001  0xD202EF8D  "\x00"
2  0x00020001  0x41D912FF  "\x00\x00"
3  0x00030001  0xFF41D912  "\x00\x00\x00"
4  0x00040001  0x2144DF1C  "\x00\x00\x00\x00"
5  0x00050001  0xC622F71D  "\x00\x00\x00\x00\x00"
6  0x00060001  0xB1C2A1A3  "\x00\x00\x00\x00\x00\x00"
7  0x00070001  0x9D6CDF7E  "\x00\x00\x00\x00\x00\x00\x00"
```

Adler-32 is a simpler algorithm than CRC-32. At the time Adler-32 was invented, it had noticeably higher throughput. With modern SIMD implementations, that performance difference has largely disappeared.

# Worked Example

A worked example for calculating the Adler-32 hash of the three byte input “Hi\n”, starting from the initial state `(s1 = 1)` and `(s2 = 0)`:

```src[i]  ((s2 << 16) | s1)
----    0x00000001
0x48    0x00490049
0x69    0x00FB00B2
0x0A    0x01B700BC
```