| Internet-Draft | KangarooTwelve | August 2022 |
| Viguier, et al. | Expires 20 February 2023 | [Page] |
This document defines the KangarooTwelve eXtendable Output Function (XOF), a hash function with output of arbitrary length. It provides an efficient and secure hashing primitive, which is able to exploit the parallelism of the implementation in a scalable way. It uses tree hashing over a round-reduced version of SHAKE128 as underlying primitive.¶
This document builds up on the definitions of the permutations and of the sponge construction in [FIPS 202], and is meant to serve as a stable reference and an implementation guide.¶
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This document defines the KangarooTwelve eXtendable Output Function (XOF) [K12], i.e. a generalization of a hash function that can return an output of arbitrary length. KangarooTwelve is based on a Keccak-p permutation specified in [FIPS202] and has a higher speed than SHAKE and SHA-3.¶
The SHA-3 functions process data in a serial manner and are unable to optimally exploit parallelism available in modern CPU architectures. Similar to ParallelHash [SP800-185], KangarooTwelve splits the input message into fragments to exploit available parallelism. It then applies an inner hash function F on each of them separately before applying F again on the concatenation of the digests. It makes use of Sakura coding for ensuring soundness of the tree hashing mode [SAKURA]. The inner hash function F is a sponge function and uses a round-reduced version of the permutation Keccak-f used in SHA-3, making it faster than ParallelHash. Its security builds up on the scrutiny that Keccak has received since its publication [KECCAK_CRYPTANALYSIS].¶
With respect to [FIPS202] and [SP800-185] functions, KangarooTwelve features the following advantages:¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119 [RFC2119].¶
The following notations are used throughout the document:¶
In the following, x and y are byte strings of equal length:¶
In the following, x and y are integers:¶
KangarooTwelve is an eXtendable Output Function (XOF). It takes as input two byte-strings (M, C) and a positive integer L where¶
The Customization string MAY serve as domain separation. It is typically a short string such as a name or an identifier (e.g. URI, ODI...)¶
By default, the Customization string is the empty string. For an API that does not support a customization string input, C MUST be the empty string.¶
The inner function F makes use of the permutation Keccak-p[1600,n_r=12], i.e., a version of the permutation Keccak-f[1600] used in SHAKE and SHA-3 instances reduced to its last n_r=12 rounds and specified in FIPS 202, sections 3.3 and 3.4 [FIPS202]. KP denotes this permutation.¶
F is a sponge function calling this permutation KP with a rate of 168 bytes or 1344 bits. It follows that F has a capacity of 1600 - 1344 = 256 bits or 32 bytes.¶
The sponge function F takes:¶
First non-multiple of 168-bytes-length inputs are padded with zeroes to the next multiple of 168 bytes while inputs multiple of 168 bytes are kept as is. Then a byte `80` is XORed to the last byte of the padded message and the resulting string is split into a sequence of 168-byte blocks.¶
Inputs of length 0 bytes do not happen as a result of the tree hashing mode defined in section 2.2.¶
As defined by the sponge construction, the process operates on a state and consists of two phases: the absorbing phase that processes the input and the squeezing phase that produces the output.¶
In the absorbing phase the state is initialized to all-zero. The message blocks are XORed into the first 168 bytes of the state. Each block absorbed is followed with an application of KP to the state.¶
In the squeezing phase output is formed by taking the first 168 bytes of the state, repeated as many times as necessary until outputByteLen bytes are obtained, interleaved with the application of KP to the state.¶
The definition of the function F equivalently implements the pad10*1 rule. It assumes an at least one-byte-long input where the last byte is in the `01`-`7F` range, and this is the case in KangarooTwelve. This last byte serves as domain separation and integrates the first bit of padding of the pad10*1 rule (hence it cannot be `00`). Additionally, it must leave room for the second bit of padding (hence it cannot have the MSB set to 1), should it be the last byte of the block. For more details, refer to Section 6.1 of [K12].¶
A pseudocode version is available as follows:¶
F(input, outputByteLen):
offset = 0
state = `00`^200
# === Absorb complete blocks ===
while offset < |input| - 168
state ^= input[offset : offset + 168] || `00`^32
state = KP(state)
offset += 168
# === Absorb last block and treatment of padding ===
LastBlockLength = |input| - offset
state ^= input[offset:] || `00`^(200-LastBlockLength)
state ^= `00`^167 || `80` || `00`^32
state = KP(state)
# === Squeeze ===
output = `00`^0
while outputByteLen > 168
output = output || state[0:168]
outputByteLen -= 168
state = KP(state)
output = output || state[0:outputByteLen]
return output
end¶
On top of the sponge function F, KangarooTwelve uses a Sakura-compatible tree hash mode [SAKURA]. First, merge M and the OPTIONAL C to a single input string S in a reversible way. length_encode( |C| ) gives the length in bytes of C as a byte-string. See Section 2.3.¶
S = M || C || length_encode( |C| )¶
Then, split S into n chunks of 8192 bytes.¶
S = S_0 || .. || S_(n-1)
|S_0| = .. = |S_(n-2)| = 8192 bytes
|S_(n-1)| <= 8192 bytes¶
From S_1 .. S_(n-1), compute the 32-byte Chaining Values CV_1 .. CV_(n-1). In order to be optimally efficient, this computation SHOULD exploit the parallelism available on the platform such as SIMD instructions.¶
CV_i = F( S_i||`0B`, 32 )¶
Compute the final node: FinalNode.¶
FinalNode = S_0 || `03 00 00 00 00 00 00 00`
FinalNode = FinalNode || CV_1
..
FinalNode = FinalNode || CV_(n-1)
FinalNode = FinalNode || length_encode(n-1)
FinalNode = FinalNode || `FF FF`¶
Finally, KangarooTwelve output is retrieved:¶
KangarooTwelve( M, C, L ) = F( FinalNode||`07`, L )¶
KangarooTwelve( M, C, L ) = F( FinalNode||`06`, L )¶
The following figure illustrates the computation flow of KangarooTwelve for |S| <= 8192 bytes:¶
+--------------+ F(..||`07`, L)
| S |-----------------> output
+--------------+¶
The following figure illustrates the computation flow of KangarooTwelve for |S| > 8192 bytes and where length_encode( x ) is abbreviated as l_e( x ):¶
+--------------+
| S_0 |
+--------------+
||
+--------------+
| `03`||`00`^7 |
+--------------+
||
+---------+ F(..||`0B`,32) +--------------+
| S_1 |----------------->| CV_1 |
+---------+ +--------------+
||
+---------+ F(..||`0B`,32) +--------------+
| S_2 |----------------->| CV_2 |
+---------+ +--------------+
||
... ...
||
+---------+ F(..||`0B`,32) +--------------+
| S_(n-1) |----------------->| CV_(n-1) |
+---------+ +--------------+
||
+--------------+
| l_e( n-1 ) |
+--------------+
||
+--------------+ F(..||`06`, L)
| `FF FF` |-----------------> output
+--------------+¶
A pseudocode version is provided in Appendix A.2.¶
The table below gathers the values of the domain separation bytes used by the tree hash mode:¶
+--------------------+------------------+
| Type | Byte |
+--------------------+------------------+
| SingleNode | `07` |
| | |
| IntermediateNode | `0B` |
| | |
| FinalNode | `06` |
+--------------------+------------------+¶
The function length_encode takes as inputs a non negative integer x < 256**255 and outputs a string of bytes x_(n-1) || .. || x_0 || n where¶
x = sum from i=0..n-1 of 256**i * x_i¶
and where n is the smallest non-negative integer such that x < 256**n. n is also the length of x_(n-1) || .. || x_0.¶
As example, length_encode(0) = `00`, length_encode(12) = `0C 01` and length_encode(65538) = `01 00 02 03`¶
A pseudocode version is as follows.¶
length_encode(x):
S = `00`^0
while x > 0
S = x mod 256 || S
x = x / 256
S = S || length(S)
return S
end¶
Test vectors are based on the repetition of the pattern `00 01 .. FA` with a specific length. ptn(n) defines a string by repeating the pattern `00 01 .. FA` as many times as necessary and truncated to n bytes e.g.¶
Pattern for a length of 17 bytes:
ptn(17) =
`00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10`¶
Pattern for a length of 17**2 bytes:
ptn(17**2) =
`00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F
30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F
40 41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F
50 51 52 53 54 55 56 57 58 59 5A 5B 5C 5D 5E 5F
60 61 62 63 64 65 66 67 68 69 6A 6B 6C 6D 6E 6F
70 71 72 73 74 75 76 77 78 79 7A 7B 7C 7D 7E 7F
80 81 82 83 84 85 86 87 88 89 8A 8B 8C 8D 8E 8F
90 91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F
A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 AA AB AC AD AE AF
B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 BA BB BC BD BE BF
C0 C1 C2 C3 C4 C5 C6 C7 C8 C9 CA CB CC CD CE CF
D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 DA DB DC DD DE DF
E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 EA EB EC ED EE EF
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 FA
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
20 21 22 23 24 25`¶
KangarooTwelve(M=`00`^0, C=`00`^0, 32):
`1A C2 D4 50 FC 3B 42 05 D1 9D A7 BF CA 1B 37 51
3C 08 03 57 7A C7 16 7F 06 FE 2C E1 F0 EF 39 E5`
KangarooTwelve(M=`00`^0, C=`00`^0, 64):
`1A C2 D4 50 FC 3B 42 05 D1 9D A7 BF CA 1B 37 51
3C 08 03 57 7A C7 16 7F 06 FE 2C E1 F0 EF 39 E5
42 69 C0 56 B8 C8 2E 48 27 60 38 B6 D2 92 96 6C
C0 7A 3D 46 45 27 2E 31 FF 38 50 81 39 EB 0A 71`
KangarooTwelve(M=`00`^0, C=`00`^0, 10032), last 32 bytes:
`E8 DC 56 36 42 F7 22 8C 84 68 4C 89 84 05 D3 A8
34 79 91 58 C0 79 B1 28 80 27 7A 1D 28 E2 FF 6D`
KangarooTwelve(M=ptn(1 bytes), C=`00`^0, 32):
`2B DA 92 45 0E 8B 14 7F 8A 7C B6 29 E7 84 A0 58
EF CA 7C F7 D8 21 8E 02 D3 45 DF AA 65 24 4A 1F`
KangarooTwelve(M=ptn(17 bytes), C=`00`^0, 32):
`6B F7 5F A2 23 91 98 DB 47 72 E3 64 78 F8 E1 9B
0F 37 12 05 F6 A9 A9 3A 27 3F 51 DF 37 12 28 88`
KangarooTwelve(M=ptn(17**2 bytes), C=`00`^0, 32):
`0C 31 5E BC DE DB F6 14 26 DE 7D CF 8F B7 25 D1
E7 46 75 D7 F5 32 7A 50 67 F3 67 B1 08 EC B6 7C`
KangarooTwelve(M=ptn(17**3 bytes), C=`00`^0, 32):
`CB 55 2E 2E C7 7D 99 10 70 1D 57 8B 45 7D DF 77
2C 12 E3 22 E4 EE 7F E4 17 F9 2C 75 8F 0D 59 D0`
KangarooTwelve(M=ptn(17**4 bytes), C=`00`^0, 32):
`87 01 04 5E 22 20 53 45 FF 4D DA 05 55 5C BB 5C
3A F1 A7 71 C2 B8 9B AE F3 7D B4 3D 99 98 B9 FE`
KangarooTwelve(M=ptn(17**5 bytes), C=`00`^0, 32):
`84 4D 61 09 33 B1 B9 96 3C BD EB 5A E3 B6 B0 5C
C7 CB D6 7C EE DF 88 3E B6 78 A0 A8 E0 37 16 82`
KangarooTwelve(M=ptn(17**6 bytes), C=`00`^0, 32):
`3C 39 07 82 A8 A4 E8 9F A6 36 7F 72 FE AA F1 32
55 C8 D9 58 78 48 1D 3C D8 CE 85 F5 8E 88 0A F8`
KangarooTwelve(M=`00`^0, C=ptn(1 bytes), 32):
`FA B6 58 DB 63 E9 4A 24 61 88 BF 7A F6 9A 13 30
45 F4 6E E9 84 C5 6E 3C 33 28 CA AF 1A A1 A5 83`
KangarooTwelve(M=`FF`, C=ptn(41 bytes), 32):
`D8 48 C5 06 8C ED 73 6F 44 62 15 9B 98 67 FD 4C
20 B8 08 AC C3 D5 BC 48 E0 B0 6B A0 A3 76 2E C4`
KangarooTwelve(M=`FF FF FF`, C=ptn(41**2), 32):
`C3 89 E5 00 9A E5 71 20 85 4C 2E 8C 64 67 0A C0
13 58 CF 4C 1B AF 89 44 7A 72 42 34 DC 7C ED 74`
KangarooTwelve(M=`FF FF FF FF FF FF FF`, C=ptn(41**3 bytes), 32):
`75 D2 F8 6A 2E 64 45 66 72 6B 4F BC FC 56 57 B9
DB CF 07 0C 7B 0D CA 06 45 0A B2 91 D7 44 3B CF`¶
None.¶
This document is meant to serve as a stable reference and an implementation guide for the KangarooTwelve eXtendable Output Function. It relies on the cryptanalysis of Keccak and provides with the same security strength as SHAKE128, i.e., 128 bits of security against all attacks.¶
To be more precise, KangarooTwelve is made of two layers:¶
This reasoning is detailed and formalized in [K12].¶
To achieve 128-bit security strength, the output L must be chosen long enough so that there are no generic attacks that violate 128-bit security. So for 128-bit (second) preimage security the output should be at least 128 bits, for 128-bit of security against multi-target preimage attacks with T targets the output should be at least 128+log_2(T) bits and for 128-bit collision security the output should be at least 256 bits.¶
Furthermore, when the output length is at least 256 bits, KangarooTwelve achieves NIST's post-quantum security level 2 [NISTPQ].¶
Implementing a MAC with KangarooTwelve SHOULD use a HASH-then-MAC construction. This document recommends a method called HopMAC, defined as follows:¶
HopMAC(Key, M, C, L) = K12(Key, K12(M, C, 32), L)¶
Similarly to HMAC, HopMAC consists of two calls: an inner call compressing the message M and the optional customization string C to a digest, and an outer call computing the tag from the key and the digest.¶
Unlike HMAC, the inner call to KangarooTwelve in HopMAC is keyless and does not require additional protection against side channel attacks (SCA). Consequently, in an implementation that has to protect the HopMAC key against SCA only the outer call does need protection, and this amounts to a single execution of the underlying permutation.¶
In any case, KangarooTwelve MAY be used to compute a MAC with the key reversibly prepended or appended to the input. For instance, one MAY compute a MAC on short messages simply calling KangarooTwelve with the key as the customization string, i.e., MAC = K12(M, Key, L).¶
The sub-sections of this appendix contain pseudocode definitions of KangarooTwelve. A standalone Python version is also available in the Keccak Code Package [XKCP] and in [K12]¶
KP(state):
RC[0] = `8B 80 00 80 00 00 00 00`
RC[1] = `8B 00 00 00 00 00 00 80`
RC[2] = `89 80 00 00 00 00 00 80`
RC[3] = `03 80 00 00 00 00 00 80`
RC[4] = `02 80 00 00 00 00 00 80`
RC[5] = `80 00 00 00 00 00 00 80`
RC[6] = `0A 80 00 00 00 00 00 00`
RC[7] = `0A 00 00 80 00 00 00 80`
RC[8] = `81 80 00 80 00 00 00 80`
RC[9] = `80 80 00 00 00 00 00 80`
RC[10] = `01 00 00 80 00 00 00 00`
RC[11] = `08 80 00 80 00 00 00 80`
for x from 0 to 4
for y from 0 to 4
lanes[x][y] = state[8*(x+5*y):8*(x+5*y)+8]
for round from 0 to 11
# theta
for x from 0 to 4
C[x] = lanes[x][0]
C[x] ^= lanes[x][1]
C[x] ^= lanes[x][2]
C[x] ^= lanes[x][3]
C[x] ^= lanes[x][4]
for x from 0 to 4
D[x] = C[(x+4) mod 5] ^ ROL64(C[(x+1) mod 5], 1)
for y from 0 to 4
for x from 0 to 4
lanes[x][y] = lanes[x][y]^D[x]
# rho and pi
(x, y) = (1, 0)
current = lanes[x][y]
for t from 0 to 23
(x, y) = (y, (2*x+3*y) mod 5)
(current, lanes[x][y]) =
(lanes[x][y], ROL64(current, (t+1)*(t+2)/2))
# chi
for y from 0 to 4
for x from 0 to 4
T[x] = lanes[x][y]
for x from 0 to 4
lanes[x][y] = T[x] ^((not T[(x+1) mod 5]) & T[(x+2) mod 5])
# iota
lanes[0][0] ^= RC[round]
state = `00`^0
for x from 0 to 4
for y from 0 to 4
state = state || lanes[x][y]
return state
end
¶
where ROL64(x, y) is a rotation of the 'x' 64-bit word toward the bits with higher indexes by 'y' positions. The 8-bytes byte-string x is interpreted as a 64-bit word in little-endian format.¶
KangarooTwelve(inputMessage, customString, outputByteLen):
S = inputMessage || customString
S = S || length_encode( |customString| )
if |S| <= 8192
return F(S || `07`, outputByteLen)
else
# === Kangaroo hopping ===
FinalNode = S[0:8192] || `03` || `00`^7
offset = 8192
numBlock = 0
while offset < |S|
blockSize = min( |S| - offset, 8192)
CV = F(S[offset : offset + blockSize] || `0B`, 32)
FinalNode = FinalNode || CV
numBlock += 1
offset += blockSize
FinalNode = FinalNode || length_encode( numBlock ) || `FF FF`
return F(FinalNode || `06`, outputByteLen)
end
¶