8.9 KiB
Keccak, SHA-3, and SHAKE-N algorithms explained, with code in Erlang
Keccak is a hashing algorithm used for the SHA-3 standard. The standard is semi-readable math clownery. Hashing algorithms by nature have to be complicated and somewhat obfuscated. But, at the end of the day, all Keccak does is take in some input bits, process them in a deterministic way, and hand you output bits.
The goal of this document is just to explain what the processing steps are as straightforwardly as possible.
Keccak is the "general case", and then SHA-3 and SHAKE-128 and so on are special cases of Keccak.
tldr
References
- Helpful lecture
- Notes for that lecture
- NIST standard (btw: the double bar notation means "concatenate")
- SHA-3 Wikipedia
Pitfall: "fast keccak" versus "clear keccak"
The main reference is the "clear" Erlang code. The "fast" version is the "clear" version with optimizations applied. The fast version was written by Hans Svensson.
In particular, fast keccak is arity 4, and clear keccak is arity 3.
There are a couple of trivial differences in code structure, so when you're reading code, you need to keep in mind which version you're looking at. When I have a code sample, I will link the GitHub permalink in a comment where the code originates from, so there shouldn't be any confusion.
%% From "clear keccak"
%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L155-L159
-spec keccak(Capacity, Message, OutputBitLength) -> Digest
when Capacity :: pos_integer(),
Message :: bitstring(),
OutputBitLength :: pos_integer(),
Digest :: bitstring().
%% From "fast Keccak"
%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek_fast.erl#L142-L147
-spec keccak(Capacity, Message, Delimiter, OutputBitLength) -> Digest
when Capacity :: pos_integer(),
Message :: bitstring(),
Delimiter :: bitstring(),
OutputBitLength :: pos_integer(),
Digest :: bitstring().
The reason for this is one of Hans's optimizations was speeding up the manner in which padding was applied, and that required factoring it out into an argument.
Pitfall: "NIST SHA-3" versus "Keccak SHA-3"
There are two versions of "SHA-3". There's the "NIST version" and the "keccak version". The only difference is that NIST adds two bits (yes bits, not bytes) of padding
%% From the erlang-sha3 lib, which uses the fast version
%% - ignore the capacity and what not for now, will be explained in a minute
%% - OutputBitLength is the N in SHA3-N. So it would be 512 for SHA3-512, 256
%% for SHA3-256, etc.
%% - Message is the input bitstring to be hashed
%% https://github.com/zxq9/erlang-sha3/blob/63193654e3c05d8031300ffcd52092f75e8b5c2f/src/sha3.erl#L107-L112
kek(OutputBitLength, Message, keccak) ->
Capacity = 2 * OutputBitLength,
keccak(Capacity, Message, <<>>, OutputBitLength);
%% ^ keccak sha3 adds no padding bits
kek(OutputBitLength, Message, nist) ->
Capacity = 2 * OutputBitLength,
keccak(Capacity, Message, <<2#01:2>>, OutputBitLength).
%% ^ nist sha3 concats these padding bits to the end of Message
The reason this problem exists is the "keccak version" was used in production code before the NIST standard was published. So now there are two versions of "SHA-3" out there.
Pitfall: SHAKE-N versus SHA3-N
Note that the padding and sha3-is-keccak-but-its-not pitfall only applies to the SHA-3 fixed-length hashing functions. These produce a fixed-length (say, 512 bit) hash.
The SHAKE-N algorithms are sort of "arbitrary length SHA-3". So instead of producing 512 bits, you tell it how long the output length should be. The SHAKE-N algorithms have their own padding rule, and I don't believe they're affected by the padding thing.
%% From the "clear" version
%% The clear version was written against the NIST standard
%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L86-L89
sha3(OutputBitLength, Message) ->
Capacity = 2*OutputBitLength,
%% SHA-3 NIST padding is used in the clear version
ShaMessage = <<Message/bitstring, (2#01):2>>,
keccak(Capacity, ShaMessage, OutputBitLength).
%% https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L131-L134
shake(ShakeNumber, Message, OutputBitLength) ->
Capacity = 2*ShakeNumber,
%% SHAKE has its own padding rule
ShakeMessage = <<Message/bitstring, (2#1111):4>>,
keccak(Capacity, ShakeMessage, OutputBitLength).
SHA-s and SHAKE-s
These are the "porcelain" functions that we show to the outside world. These are just rewrites, in front of Keccak. Another way to think about it is that Keccak has tons of settings, and that each of these "algorithms" are just different settings presets.
%% From: https://github.com/pharpend/kek/blob/8a8a655a80c26ae32763cc25f1e0df8ab0653c82/kek.erl#L26-L134
-spec sha3_224(Message) -> Digest
when Message :: bitstring(),
Digest :: <<_:224>>.
%% @doc
%% SHA-3 with an output bit length of 224 bits.
%% @end
sha3_224(Message) ->
sha3(224, Message).
-spec sha3_256(Message) -> Digest
when Message :: bitstring(),
Digest :: <<_:256>>.
%% @doc
%% SHA-3 with an output bit length of 256 bits.
%% @end
sha3_256(Message) ->
sha3(256, Message).
-spec sha3_384(Message) -> Digest
when Message :: bitstring(),
Digest :: <<_:384>>.
%% @doc
%% SHA-3 with an output bit length of 384 bits.
%% @end
sha3_384(Message) ->
sha3(384, Message).
-spec sha3_512(Message) -> Digest
when Message :: bitstring(),
Digest :: <<_:512>>.
%% @doc
%% SHA-3 with an output bit length of 512 bits.
%% @end
sha3_512(Message) ->
sha3(512, Message).
-spec sha3(OutputBitLength, Message) -> Digest
when OutputBitLength :: pos_integer(),
Message :: bitstring(),
Digest :: bitstring().
%% @doc
%% SHA-3 with an arbitrary output bit length.
%%
%% This means Keccak with Capacity = 2*OutputBitLength. Additionally, SHA3
%% concatenates the bits 01 onto the end of the input, before sending the
%% Message to keccak/3.
%% @end
sha3(OutputBitLength, Message) ->
Capacity = 2*OutputBitLength,
ShaMessage = <<Message/bitstring, (2#01):2>>,
keccak(Capacity, ShaMessage, OutputBitLength).
-spec shake128(Message, OutputBitLength) -> Digest
when Message :: bitstring(),
OutputBitLength :: pos_integer(),
Digest :: bitstring().
%% @doc
%% This is the SHAKE variable-length hash with Capacity 256 = 2*128 bits.
%% @end
shake128(Message, OutputBitLength) ->
shake(128, Message, OutputBitLength).
-spec shake256(Message, OutputBitLength) -> Digest
when Message :: bitstring(),
OutputBitLength :: pos_integer(),
Digest :: bitstring().
%% @doc
%% This is the SHAKE variable-length hash with Capacity 512 = 2*256 bits.
%% @end
shake256(Message, OutputBitLength) ->
shake(256, Message, OutputBitLength).
-spec shake(ShakeNumber, Message, OutputBitLength) -> Digest
when ShakeNumber :: pos_integer(),
Message :: bitstring(),
OutputBitLength :: pos_integer(),
Digest :: bitstring().
%% @doc
%% This is the SHAKE variable-length hash with Capacity 512 = 2*ShakeNumber bits.
%%
%% This concatenates the bitstring 1111 onto the end of the Message before
%% sending the message to keccak/3.
%% @end
shake(ShakeNumber, Message, OutputBitLength) ->
Capacity = 2*ShakeNumber,
ShakeMessage = <<Message/bitstring, (2#1111):4>>,
keccak(Capacity, ShakeMessage, OutputBitLength).
Outer Keccak
Keccak uses this "sponge" metaphor. If you think of the bits as water, the state type of Keccak is somewhat analagous to a sponge. In this analogy, the Keccak algorithm consists of "absorbing" the bits into the sponge, and then "squeezing" the bits out.
Both the absorption and squeezing phases invoke "inner keccak", which is where all the real bit-churning happens. (These are the Greek letter steps in the standard).
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Source for photo: https://www.flickr.com/photos/30478819@N08/46410395345 ↩︎