Internet Research Task Force (IRTF) | B. Viguier |

Internet-Draft | Radboud University |

Intended status: Informational | June 14, 2017 |

Expires: December 16, 2017 |

KangarooTwelve

draft-viguier-kangarootwelve-00

This document defines the KangarooTwelve eXtendable Output Function (XOF), a hash function with arbitrary output 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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- 1. Introduction
- 1.1. Conventions
- 2. Specifications
- 2.1. Inner function: F
- 2.2. Tree hashing over F
- 2.3. right_encode( x )
- 3. Test vectors
- 4. IANA Considerations
- 5. Security Considerations
- 6. References
- 6.1. Normative References
- 6.2. Informative References
- Appendix A. Pseudo code
- A.1. Keccak-p[1600] over 12 rounds
- A.2. Inner function F
- A.3. KangarooTwelve
- Author's Address

This document defines the KangarooTwelve eXtendable Output Function (XOF) [K12], i.e. a generalization of a hash function that can return arbitrary output length. KangarooTwelve is based on a Keccak-p permutation specified in [FIPS202] and aims at higher speed than SHAKE and SHA-3.

The SHA-3 functions process data in a serial manner and unable to optimally exploit parallelism available in modern CPU architectures. KangarooTwelve splits the input message in fragments and applies an inner hash function F on each of them separately. It then applies 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 used in Keccak. Its security builds up on the scrutiny that Keccak has received since its publication [KECCAK_CRYPTANALYSIS].

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:

- `...`
- denotes a bit-string. For example, `1010101`.

A 8 bit string `b_0 b_1 b_2 b_3 b_4 b_5 b_6 b_7` is a byte represented by an integer value v following the LSB 0 convention, i.e.

v = sum for i=0..7 of 2^i * b_i

For example, `11100000` = 7. The following diagram represents the byte "07" with value 7 (decimal).

Significance of Bits MSB 7 6 5 4 3 2 1 0 LSB +-+-+-+-+-+-+-+-+ |0 0 0 0 0 1 1 1| +-+-+-+-+-+-+-+-+ hex: 0 7

- "..."
- denotes a string of bytes given in hexadecimal. For example, "0B 80", which can be also be seen as a bit-string : `11010000 00000001`.
- |s|
- denotes the length of a byte string "s". For example, |"FF FF"| = 2.
- `0^b`
- denotes the repetition of bit `0` b times. For example, `0^4` = `0000`.
- `0^0`
- denotes the empty bit-string.
- `1^b`
- denotes the repetition of bit `1` b times. For example, `1^3` = `111`.
- "00^b"
- denotes the b times the repetition of byte "00". For example, "00^7" = "00 00 00 00 00 00 00".
- a||b
- denotes the concatenation of two strings 'a' and 'b'. For example, `10`||`01` = `1001`
- s[n:m]
- denotes the selection of bytes from n to m exclusive of a string 's'. For example, for s = "A5 C6 D7", s[0:1] = "A5" and s[1:3] = "C6 D7".

KangarooTwelve is an eXtendable Output Function (XOF). It takes as an input a pair of byte-strings (M, C) and a positive integer L where

- M
- byte-string, is the Message and
- C
- byte-string, is a Customization string and
- L
- positive integer, the length of the output in bytes.

The Customization string serves as domain separation. It is typically a short string such as a name or an identifier (e.g. URI, ODI...)

The inner function F makes use of the permutation Keccak-p[1600,n_r=12], i.e., a version of the one used in SHAKE and SHA-3 instances reduced to n_r=12 rounds and specified in FIPS 202 [FIPS202]. F is a sponge function calling this permutation, multi-rate padding pad10*1 and with a rate of 168 bytes (= 1344 bits):

F = Sponge[Keccak-p[1600,n_r=12], pad10*1, r=1344]

It follows that F has a capacity of 1600 - 1344 = 256 bits.

The sponge function F takes as an input a bit-string S and a positive integer L where

- S
- bit-string, is the input String and
- L
- positive integer, the Length of the output in bytes

The input string S SHOULD be represented as a pair (Sbytes, dS), where Sbytes contains only bytes and where dS is the delimited suffix representing the trailing bits.

First, let S = Sbytes || Sbits, where Sbytes contains only bytes and Sbits contains at most 7 bits. Then, convert Sbits into the delimited suffix dS by appending a bit `1` and as many bits `0` as necessary so that dS is a byte. The numerical value of dS is thus:

dS = 2^|Sbits| + sum for i=0..|Sbits|-1 of 2^i*Sbits_i

Notice that the most significant bit `1` of dS coincides with the first bit of padding in the multi-rate padding rule pad10*1. The implementation of F therefore SHOULD add dS to the state and then the second bit of padding. Appendix A.2 provides a pseudo code version.

In the table below, here are some examples of values, including those that are used in this document:

+---------+---------------+---------------+-------------------------+ | Sbits | bit-string | value (dec) | delimited Suffix (dS) | +---------+---------------+---------------+-------------------------+ | `` | `10000000` | 1 | "01" | | | | | | | `01` | `01100000` | 6 | "06" | | | | | | | `11` | `11100000` | 7 | "07" | | | | | | | `110` | `11010000` | 11 | "0B" | +---------+---------------+---------------+-------------------------+

On top of the sponge function F, KangarooTwelve uses a Sakura-compatible tree hash mode [SAKURA]. First, merge M and C to a single input string S in a reversible way. right_encode( |C| ) gives the length in bytes of C as a byte-string. See Section 2.3.

S = M || C || right_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-bytes hashes CV_0 .. CV_n-2. This computation SHOULD exploit the parallelism available on the platform in order to be optimally efficient.

Node_i = S_i+1 || `110` CV_i = F( Node_i, 32 )

Compute the final node: Node*.

- If |S| <= 8192 bytes, then Node* = S || `11`
- Otherwise compute Node* as follow:

Node* = S_0 || "03 00 00 00 00 00 00 00" Node* = Node* || CV_0 .. Node* = Node* || CV_n-2 Node* = Node* || right_encode(n-1) Node* = Node* || "FF FF" || `01`

Finally, KangarooTwelve output is retrieved from F( Node* ).

KangarooTwelve( M, C, L ) = F( Node*, L )

For |S| > 8192 bytes, KangarooTwelve computation flow is as follow:

+--------------+ | S_0 | +--------------+ || +--------------+ | `11`||`0^62` | +--------------+ || +-------------------+ F +--------------+ | S_1 || `110` |------>| CV_0 | +-------------------+ +--------------+ || +-------------------+ F +--------------+ | S_2 || `110` |------>| CV_1 | +-------------------+ +--------------+ || ... ... || +-------------------+ F +--------------+ | S_n-1 || `110` |------>| CV_n-2 | +-------------------+ +--------------+ || +--------------+ | r_e(n-1) | +--------------+ || +------------------+ F | "FF FF" || `01` |----------> output +------------------+

We provide a pseudo code version in Appendix A.3.

The function right_encode takes as inputs a non negative integer x < 256^255 and outputs a string of bytes x_n || .. || x_0 || n where

x = sum from i=0..n of 256^i * x_i

A pseudo code version is as follow.

right_encode(x): S = 0^0 while x > 0 S = x % 256 || S x = x / 256 S = S || length(S) return S end

Test vectors are based on the repetition of pattern the "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=0^0, C=0^0, 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=0^0, C=0^0, 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=0^0, C=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0^0, 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=0xff, 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=0xff 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=0xff 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 makes no assertion to its security and relies on the cryptanalysis of Keccak [KECCAK_CRYPTANALYSIS].

[FIPS202] |
National Institute of Standards and Technology, "FIPS PUB 202 - SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", WWW http://dx.doi.org/10.6028/NIST.FIPS.202, August 2015. |

[RFC2119] |
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997. |

[K12] |
Bertoni, G., Daemen, J., Peeters, M., Van Assche, G. and R. Van Keer, "KangarooTwelve: fast hashing based on Keccak-p", WWW http://eprint.iacr.org/2016/770.pdf, August 2016. |

[KECCAK_CRYPTANALYSIS] |
Keccak Team, "Summary of Third-party cryptanalysis of Keccak", WWW https://www.keccak.team/third_party.html, 2017. |

[SAKURA] |
Bertoni, G., Daemen, J., Peeters, M. and G. Van Assche, "Sakura: a flexible coding for tree hashing", WWW http://eprint.iacr.org/2013/231.pdf, April 2013. |

The sub-sections of this appendix contain pseudo code definitions of KangarooTwelve.

Keccak-p_1600_12(state): R = "D5" 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 12 to 23 # 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)%5] ^ ROL64(C[(x+1)%5], 1) for y from 0 to 4 for x from 0 to 4 lanes = 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)%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)%5]) & T[(x+2)%5]) # iota for j from 0 to 6 R = ((R << 1) ^ ((R >> 7)* "71")) % 256 if (R & 2) lanes[0][0] = lanes[0][0] ^ (1 << ((1<<j)-1)) state = 0^0 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' bits.

F(inputBytes, Suffix, outputByteLen): state = "00^200" blockSize = 0 offset = 0 # === Absorb inputBytes === while offset < |inputBytes| blockSize = min( |inputBytes| - offset, 168) state ^= inputBytes[offset : offset + blockSize] offset = offset + blockSize if blockSize = 168 state = Keccak-p_1600_12(state) blockSize = 0 # === Absorb Suffix === state ^= "00^blockSize" || Suffix if (Suffix & "80") != 0 and blockSize == 167 state = Keccak-p_1600_12(state) state ^= "00^167" || "80" state = Keccak-p_1600_12(state) # === Squeeze === while outputByteLen > 0 blockSize = min(outputByteLen, 168) outputBytes = outputBytes || state[0:blockSize] outputByteLen = outputByteLen - blockSize if outputByteLen > 0 state = Keccak-p_1600_12(state) return outputBytes end

KangarooTwelve(inputMessage, customString, outputByteLen): S = inputMessage || customString S = S || right_encode( |customString| ) if |S| <= 8192 return F(S, "07", outputByteLen) else # === Kangaroo hopping === Node* = S[0:8192] || "03 00^7" offset = 8192 while offset < |inputBytes| blockSize = min( |inputBytes| - offset, 8192) CV = F(inputBytes[offset : offset + blockSize], "0B", 32) Node* = Node* || CV offset = offset + blockSize Node* = Node* || right_encode( |S| / 8192 ) || "FF FF" return F(Node*, "06", outputByteLen) end

Benoît Viguier
Radboud University
Toernooiveld 212
Nijmegen,
The Netherlands
EMail: b.viguier@cs.ru.nl