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Latest commit 423eda1 @uwiger Merge pull request #16 from dreyk/fix-big-neg-integer
fix big negative integer

The sext application

Authors: Ulf Wiger (

A sortable serialization library This library offers a serialization format (a la term_to_binary()) that preserves the Erlang term order.

Copyright 2014 Ulf Wiger

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
See the License for the specific language governing permissions and
limitations under the License.

1. Introduction

The idea to this library came out of the need for disk-based storage with ordered_set semantics in Erlang. One previous solution used Tokyo Cabinet, in which a C routine is used to hook into the sorting logic of TC.

I thought a more generic solution would be to be able to have a version of term_to_binary() that respected the ordering semantics of Erlang terms.

A new addition is support for 'sb32' encoding. This is my own version of Base32 encoding, with a slightly different alphabet, in order to preserve sorting properties while generating octet strings that are perfectly safe to use in file names.

Another feature is "prefix encoding", which encodes a term and truncates the result if it encounters a "wildcard" (e.g. '$1' or '_'). This is to enable a convenient and efficient mapping of Erlang match specifications to e.g. prefix matching on the external storage and subsequent match_spec matching on the found erlang terms.

The serialization format supports all Erlang types, and preserves the internal Erlang term order, with a few exceptions:

  • Floats are represented based on the IEEE 764 Binary 64 standard representation. This is the representation used by Erlang, specifically the representation used when encoding floats in binaries. To be exact, sext first normalizes the float by encoding it as an Erlang binary, then serializes it.

  • In Erlang, integers are cast to floats before comparing them to a float. This means e.g. that the relative sort order of 1 and 1.0 is undefined. It is not possible for sext to preserve this ambiguity after serialization, since it could only be done by producing identical encodings for the two terms, thereby sacrificing the property that encoding a value and then decoding it again, should produce the initial value.

2. Specification

2.1 Type tags

Each data type is encoded using a type tag (1 byte) that represents its order in the global Erlang term ordering. The number type is divided into several subtypes, to facilitate a reasonably efficient representation:

negbigNegative bignum8
neg4Negative 31-bit integer9
pos4Positive 31-bit integer10
posbigPositive bignum11
atomObj of type atom()12
referenceObj of type reference()13
portObj of type port()14
pidObj of type pid()15
tupleObj of type tuple()16
listObj of type list()17
binaryObj of type binary()18
bin_tailImproper-tail marker followed by binary or bitstring19

2.2 Tuples

Tuples are encoded as the tuple tag, followed by a 32-bit size element, denoting the number of elements in the tuple, followed by each element in the tuple individually encoded.

2.3 Lists

Lists are encoded as the list tag, followed by each element in the list individually encoded, followed by the number 2 (1 byte).

Improper lists, e.g. [1,2|3], have the number 1 inserted before the improper tail. Since this also indicates the last element in the list, no end byte is needed. This ensures that it sorts before any corresponding proper list, as long as the improper tail is not a binary (binaries are greater than the missing 'cons', or list, cell).

Improper lists that have a binary or bitstring as 'tail', e.g. [1,2|<<1>>], have a ?bin_tail (code 19) inserted before the tail. This ensures that it sorts after a corresponding proper list.

2.4 Binaries and bitstrings

A binary is basically a bitstring whose size is a multiple of 8. From a sorting perspective, binaries and bitstrings are both sorted as left-aligned bit arrays.

1> bitstring_to_list(<<11111111111:11>>).

Binaries and bitstrings are encoded as the binary tag, followed by each whole byte, each padded with a leading 1 (one bit), followed by a number of 0-bits to pad again make the size a multiple of 8 bits, followed by a byte whose value is Bits, where Bits is the number of "remainder bits"; 8 if the original binary is 8-bit aligned.


2> sext:encode(<<1,2,3>>).
3> <<18, 1:1,1, 1:1,2, 1:1,3, 0:5, 8>>.

In the example above, we inserted 3 1-bits, and therefore had to insert 5 more pad bits (zeroes) at the end. The last byte is 8, signifying that the original binary was 8-bit aligned.

If the remainder is not an even 8 bits, the remainder bits are padded with a 1-bit, just like the others, then left-aligned and padded up to a whole byte (excluding the 1-bit added in front). The value of the last byte is the bit size of the remainder.


2> sext:encode(<<1,2,3>>).
3> sext:encode(<<18, 1:1,1, 1:1,2, 1:1,3, 0:5, 8>>).

The first part of the bitstring is encoded exactly like above. The number 4:3 is first padded with 1 then padded at the end to become a whole byte. Then an additional pad, 0:4, is inserted to compensate for the fact that we have inserted 4 1-bits. Finally, the last byte is 3, to signify the size of the remainder.

2.5 Positive Numbers

Numbers are encoded as the corresponding type tag, followed by the integer part, a marker indicating the presence of a fraction part, and the fraction part, if any. The integer part is encoded differently depending on the size of the value. The fraction part is encoded as a binary (without the 'binary' type tag).

2.5.1 Positive small integers, pos4

Integers up to 31 bits are encoded as << ?pos4, I:31, F:1 >> where I is the integer value, and F is 1 if a fraction part follows; 0 otherwise.

2.5.2 Positive large integers

Larger integers are converted to a byte string and then encoded like binaries (without the 'binary' type tag), followed by a byte signifying whether a fraction part follows (1 if yes; 0 otherwise).

Bytes = encode_big(I),
<< ?pos_big, Bytes/binary, F:8 >>

2.5.3 Fraction part of positive numbers

The representation of floating point numbers is based on the IEEE 764 Binary 64 standard representation. This is also the representation used by Erlang:

<<Sign:1, Exp:11, Frac:52>> = <<F/float>>

The encoding extracts the integer part and encodes it as a positive integer (either pos4 or pos_big), flags the presence of a fraction part, and encodes the fraction part as a binary (without the binary tag).

2.6 Negative Numbers

2.6.1 Small negative numbers

<< ?neg4:8, IRep:31, F:1 >>

A negative number I is encoded as IRep = Max + I, where Max is the largest possible number that can be represented with the number of bits present for the given subtype. For example, Max for neg4 is 0x7FFF FFFF (31 bits). Keep in mind that I < 0.

The fraction flag is inverted, compared to the pos4 representation, so it will be 1 if there is no fraction part; 0 otherwise.

2.6.2 Large negative numbers

Larger negative numbers are encoded as:

encode_negbig(I) ->
    {Words, Max} = get_max(-I),
    Bin = encode_bin_elems(list_to_binary(encode_big(Max + I)),
    WordsRep = 16#FFFFffff - Words,
    << ?neg_big:8, WordsRep:32, Bin/binary, F:8 >>.

That is, get_max() figures out how many 64-bit words are needed to represent -I (the positive number), and also gives the maximum value that can be represented in so many words. WordsRep in essence becomes a sub-subtag of the negative bignum.

2.6.3 Fraction of negative numbers

The fraction is encoded almost like the inverse of the positive fraction (as a "negative binary", if such a thing existed). Each byte is padded with a 0-bit rather than a 1-bit, and the byte itself is replaced by 16#ff - Byte. The sequence is then padded with 1s to become a multiple of 8 bits.

The last byte, denoting the number of significant bits in the last byte, is similarly inverted.

2.7 Atoms

Atoms are encoded as the atom tag, followed by the string representation of the atom using the binary encoding described above (but without the binary tag).

2.8 References

The encoding of references is perhaps best described by the code:

encode_ref(R) ->
    RBin = term_to_binary(R),
    <<131,114,_Len:16,100,NLen:16,Name:NLen/binary,Rest/binary>> = RBin,
    NameEnc = encode_bin_elems(Name),
    RestEnc = encode_bin_elems(Rest),
    <<?reference, NameEnc/binary, RestEnc/binary>>.

where encode_bin_elems(B) encodes the argument B the same way as a binary (excluding the 'binary' type tag).

2.9 Ports

The encoding of ports is perhaps best described by the code:

encode_port(P) ->
    PBin = term_to_binary(P),
    <<131,102,100,ALen:16,Name:ALen/binary,Rest:5/binary>> = PBin,
    NameEnc = encode_bin_elems(Name),
    <<?port, NameEnc/binary, Rest/binary>>.

2.10 Pids

The encoding of ports is perhaps best described by the code:

encode_pid(P) ->
    PBin = term_to_binary(P),
    <<131,103,100,ALen:16,Name:ALen/binary,Rest:9/binary>> = PBin,
    NameEnc = encode_bin_elems(Name),
    <<?pid, NameEnc/binary, Rest/binary>>.


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