Internet-Draft KangarooTwelve May 2024
Viguier, et al. Expires 10 November 2024 [Page]
Workgroup:
Crypto Forum
Internet-Draft:
draft-irtf-cfrg-kangarootwelve-14
Published:
Intended Status:
Informational
Expires:
Authors:
B. Viguier
ABN AMRO Bank
D. Wong, Ed.
zkSecurity
G. Van Assche, Ed.
STMicroelectronics
Q. Dang, Ed.
NIST
J. Daemen, Ed.
Radboud University

KangarooTwelve and TurboSHAKE

Abstract

This document defines four eXtendable Output Functions (XOF), hash functions with output of arbitrary length, named TurboSHAKE128, TurboSHAKE256, KT128 and KT256.

All four functions provide efficient and secure hashing primitives, and the last two are able to exploit the parallelism of the implementation in a scalable way.

This document is a product of the Crypto Forum Research Group. It 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.

Status of This Memo

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.

Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."

This Internet-Draft will expire on 10 November 2024.

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Table of Contents

1. Introduction

This document defines the TurboSHAKE128, TurboSHAKE256 [TURBOSHAKE], KT128 and KT256 [KT] eXtendable Output Functions (XOF), i.e., a hash function generalization that can return an output of arbitrary length. Both TurboSHAKE128 and TurboSHAKE256 are based on a Keccak-p permutation specified in [FIPS202] and have a higher speed than the SHA-3 and SHAKE functions.

TurboSHAKE is a sponge function family that makes use of Keccak-p[n_r=12,b=1600], a round-reduced version of the permutation used in SHA-3. Similarly to the SHAKE's, it proposes two security strengths: 128 bits for TurboSHAKE128 and 256 bits for TurboSHAKE256. Halving the number of rounds compared to the original SHAKE functions makes TurboSHAKE roughly twice faster.

KangarooTwelve applies tree hashing on top of TurboSHAKE and comprises two functions, KT128 and KT256. Note that [KT] only defined KT128 under the name KangarooTwelve. KT256 is defined in this document.

The SHA-3 and SHAKE functions process data in a serial manner and are strongly limited in exploiting available parallelism in modern CPU architectures. Similar to ParallelHash [SP800-185], KangarooTwelve splits the input message into fragments. It then applies TurboSHAKE on each of them separately before applying TurboSHAKE again on the combination of the first fragment and the digests. More precisely, KT128 uses TurboSHAKE128 and KT256 uses TurboSHAKE256. They make use of Sakura coding for ensuring soundness of the tree hashing mode [SAKURA]. The use of TurboSHAKE in KangarooTwelve makes it faster than ParallelHash.

The security of TurboSHAKE128, TurboSHAKE256, KT128 and KT256 builds on the public scrutiny that Keccak has received since its publication [KECCAK_CRYPTANALYSIS][TURBOSHAKE].

With respect to [FIPS202] and [SP800-185] functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 feature the following advantages:

With respect to SHA-256 and SHA-512 and other [FIPS180] functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 feature the following advantages:

This document represents the consensus of the Crypto Forum Research Group (CFRG) in the IRTF. It is not an IETF product and is not a standard.

1.1. Conventions

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 BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.

The following notations are used throughout the document:

`...`
denotes a string of bytes given in hexadecimal. For example, `0B 80`.
|s|
denotes the length of a byte string `s`. For example, |`FF FF`| = 2.
`00`^b
denotes a byte string consisting of the concatenation of b bytes `00`. For example, `00`^7 = `00 00 00 00 00 00 00`.
`00`^0
denotes the empty byte-string.
a||b
denotes the concatenation of two strings a and b. For example, `10`||`F1` = `10 F1`
s[n:m]
denotes the selection of bytes from n (inclusive) to m (exclusive) of a string s. The indexing of a byte-string starts at 0. For example, for s = `A5 C6 D7`, s[0:1] = `A5` and s[1:3] = `C6 D7`.
s[n:]
denotes the selection of bytes from n to the end of a string s. For example, for s = `A5 C6 D7`, s[0:] = `A5 C6 D7` and s[2:] = `D7`.

In the following, x and y are byte strings of equal length:

x^=y
denotes x takes the value x XOR y.
x & y
denotes x AND y.

In the following, x and y are integers:

x+=y
denotes x takes the value x + y.
x-=y
denotes x takes the value x - y.
x**y
denotes the exponentiation of x by y.
x mod y
denotes reminder of the division of x by y.
x / y
denotes the integer dividend of the division of x by y.

2. TurboSHAKE

2.1. Interface

TurboSHAKE is a family of eXtendable Output Functions (XOF). This document focuses on only two instances, namely, TurboSHAKE128 and TurboSHAKE256. (Note that the original definition includes a wider range of instances parameterized by their capacity [TURBOSHAKE]. The capacity is an essential parameter of the sponge construction, see [FIPS202] for more details.)

An instance of TurboSHAKE takes as input parameters a byte-string M, an OPTIONAL byte D and a positive integer L where

M
byte-string, is the Message and
D
byte in the range [`01`, `02`, .. , `7F`], is an OPTIONAL Domain separation byte and
L
positive integer, is the requested number of output bytes.

Conceptually, a XOF can be viewed as a hash function with an infinitely long output truncated to L bytes. This means that calling a XOF with the same input parameters but two different lengths yields outputs such that the shorter one is a prefix of the longer one. Specifically, if L1 < L2, then TurboSHAKE(M, D, L1) is the same as the first L1 bytes of TurboSHAKE(M, D, L2).

By default, the Domain separation byte is `1F`. For an API that does not support a domain separation byte, D MUST be the `1F`.

The TurboSHAKE instance produces output that is a hash of the (M, D) couple. If D is fixed, this becomes a hash of the Message M. However, a protocol that requires a number of independent hash functions can choose different values for D to implement these. Specifically, for any distinct values D1 and D2, TurboSHAKE(M, D1, L1) and TurboSHAKE(M, D2, L2) yield independent hashes of M.

Note that an implementation MAY propose an incremental input interface where the input string M is given in pieces. If so, the output MUST be the same as if the function was called with M equal to the concatenation of the different pieces in the order they were given. Independently, an implementation MAY propose an incremental output interface where the output string is requested in pieces of given lengths. When the output is formed by concatenating the pieces in the requested order, it MUST be the same as if the function was called with L equal to the sum of the given lengths.

2.2. Specifications

TurboSHAKE makes use of the permutation Keccak-p[1600,n_r=12], i.e., the permutation used in SHAKE and SHA-3 functions reduced to its last n_r=12 rounds and specified in FIPS 202, Sections 3.3 and 3.4 [FIPS202]. KP denotes this permutation.

Similarly to SHAKE128, TurboSHAKE128 is a sponge function calling this permutation KP with a rate of 168 bytes or 1344 bits. It follows that TurboSHAKE128 has a capacity of 1600 - 1344 = 256 bits or 32 bytes. Respectively to SHAKE256, TurboSHAKE256 makes use of a rate of 136 bytes or 1088 bits, and has a capacity of 512 bits or 64 bytes.

                       +-------------+--------------+
                       |    Rate     |   Capacity   |
      +----------------+-------------+--------------+
      | TurboSHAKE128  |  168 Bytes  |   32 Bytes   |
      |                |             |              |
      | TurboSHAKE256  |  136 Bytes  |   64 Bytes   |
      +----------------+-------------+--------------+

We now describe the operations inside TurboSHAKE128.

TurboSHAKE256 performs the same steps but makes use of 136-byte blocks with respect to padding, absorbing, and squeezing phases.

The definition of the TurboSHAKE functions equivalently implements the pad10*1 rule; see Section 5.1 of [FIPS202] for a definition of pad10*1. While M can be empty, the D byte is always present and is in the `01`-`7F` range. 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 [KT] and Section 3 of [TURBOSHAKE].

The pseudocode versions of TurboSHAKE128 and TurboSHAKE256 are provided respectively in Appendix A.2 and Appendix A.3.

3. KangarooTwelve: Tree hashing over TurboSHAKE

3.1. Interface

KangarooTwelve is a family of eXtendable Output Functions (XOF) consisting of the KT128 and KT256 instances. A KangarooTwelve instance takes as input parameters two byte-strings (M, C) and a positive integer L where

M
byte-string, is the Message and
C
byte-string, is an OPTIONAL Customization string and
L
positive integer, the requested number of output bytes.

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...). It can serve the same purpose as TurboSHAKE's D input parameter (see Section 2.1), but with a larger range.

By default, the Customization string is the empty string. For an API that does not support a customization string parameter, C MUST be the empty string.

Note that an implementation MAY propose an interface with input and/or output incrementality as specified in Section 2.1.

3.2. Specification of KT128

On top of the sponge function TurboSHAKE128, KT128 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 3.3.

    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 MAY exploit the parallelism available on the platform such as SIMD instructions.

    CV_i = TurboSHAKE128( 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, the KT128 output is retrieved:

    KT128( M, C, L ) = TurboSHAKE128( FinalNode, `07`, L )

    KT128( M, C, L ) = TurboSHAKE128( FinalNode, `06`, L )

The following figure illustrates the computation flow of KT128 for |S| <= 8192 bytes:

    +--------------+  TurboSHAKE128(.., `07`, L)
    |      S       |----------------------------->  output
    +--------------+

The following figure illustrates the computation flow of KT128 for |S| > 8192 bytes and where TurboSHAKE128 and length_encode( x ) are abbreviated as respectively TSHK128 and l_e( x ) :

                                  +--------------+
                                  |     S_0      |
                                  +--------------+
                                        ||
                                  +--------------+
                                  | `03`||`00`^7 |
                                  +--------------+
                                        ||
+---------+  TSHK128(..,`0B`,32)  +--------------+
|   S_1   |---------------------->|     CV_1     |
+---------+                       +--------------+
                                        ||
+---------+  TSHK128(..,`0B`,32)  +--------------+
|   S_2   |---------------------->|     CV_2     |
+---------+                       +--------------+
                                        ||
               ..                       ..
                                        ||
+---------+  TSHK128(..,`0B`,32)  +--------------+
| S_(n-1) |----------------------->|   CV_(n-1)  |
+---------+                       +--------------+
                                        ||
                                  +--------------+
                                  |  l_e( n-1 )  |
                                  +--------------+
                                        ||
                                  +--------------+
                                  |   `FF FF`    |
                                  +--------------+
                                         | TSHK128(.., `06`, L)
                                         +-------------------->  output

A pseudocode version is provided in Appendix A.4.

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`       |
      +--------------------+------------------+

3.3. length_encode( x )

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 of 256**i * x_i for i from 0 to n-1

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 where { b } denotes the byte of numerical value b.

  length_encode(x):
    S = `00`^0

    while x > 0
        S = { x mod 256 } || S
        x = x / 256

    S = S || { |S| }

    return S
    end

3.4. Specification of KT256

KT256 is specified exactly like KT128, with two differences:

    CV_i = TurboSHAKE256( S_i, `0B`, 64 )

A pseudocode version is provided in Appendix A.5.

4. Message authentication codes

Implementing a MAC with KT128 or KT256 MAY use a hash-then-MAC construction. This document recommends a method called HopMAC, defined as follows:

    HopMAC128(Key, M, C, L) = KT128(Key, KT128(M, C, 32), L)
    HopMAC256(Key, M, C, L) = KT256(Key, KT256(M, C, 64), 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, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 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 KT128 with the key as the customization string, i.e., MAC = KT128(M, Key, L).

5. Test vectors

Test vectors are based on the repetition of the pattern `00 01 02 .. F9 FA` with a specific length. ptn(n) defines a string by repeating the pattern `00 01 02 .. F9 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`
  TurboSHAKE128(M=`00`^0, D=`1F`, 32):
    `1E 41 5F 1C 59 83 AF F2 16 92 17 27 7D 17 BB 53
     8C D9 45 A3 97 DD EC 54 1F 1C E4 1A F2 C1 B7 4C`

  TurboSHAKE128(M=`00`^0, D=`1F`, 64):
    `1E 41 5F 1C 59 83 AF F2 16 92 17 27 7D 17 BB 53
     8C D9 45 A3 97 DD EC 54 1F 1C E4 1A F2 C1 B7 4C
     3E 8C CA E2 A4 DA E5 6C 84 A0 4C 23 85 C0 3C 15
     E8 19 3B DF 58 73 73 63 32 16 91 C0 54 62 C8 DF`

  TurboSHAKE128(M=`00`^0, D=`1F`, 10032), last 32 bytes:
    `A3 B9 B0 38 59 00 CE 76 1F 22 AE D5 48 E7 54 DA
     10 A5 24 2D 62 E8 C6 58 E3 F3 A9 23 A7 55 56 07`

  TurboSHAKE128(M=ptn(17**0 bytes), D=`1F`, 32):
    `55 CE DD 6F 60 AF 7B B2 9A 40 42 AE 83 2E F3 F5
     8D B7 29 9F 89 3E BB 92 47 24 7D 85 69 58 DA A9`

  TurboSHAKE128(M=ptn(17**1 bytes), D=`1F`, 32):
    `9C 97 D0 36 A3 BA C8 19 DB 70 ED E0 CA 55 4E C6
     E4 C2 A1 A4 FF BF D9 EC 26 9C A6 A1 11 16 12 33`

  TurboSHAKE128(M=ptn(17**2 bytes), D=`1F`, 32):
    `96 C7 7C 27 9E 01 26 F7 FC 07 C9 B0 7F 5C DA E1
     E0 BE 60 BD BE 10 62 00 40 E7 5D 72 23 A6 24 D2`

  TurboSHAKE128(M=ptn(17**3 bytes), D=`1F`, 32):
    `D4 97 6E B5 6B CF 11 85 20 58 2B 70 9F 73 E1 D6
     85 3E 00 1F DA F8 0E 1B 13 E0 D0 59 9D 5F B3 72`

  TurboSHAKE128(M=ptn(17**4 bytes), D=`1F`, 32):
    `DA 67 C7 03 9E 98 BF 53 0C F7 A3 78 30 C6 66 4E
     14 CB AB 7F 54 0F 58 40 3B 1B 82 95 13 18 EE 5C`

  TurboSHAKE128(M=ptn(17**5 bytes), D=`1F`, 32):
    `B9 7A 90 6F BF 83 EF 7C 81 25 17 AB F3 B2 D0 AE
     A0 C4 F6 03 18 CE 11 CF 10 39 25 12 7F 59 EE CD`

  TurboSHAKE128(M=ptn(17**6 bytes), D=`1F`, 32):
    `35 CD 49 4A DE DE D2 F2 52 39 AF 09 A7 B8 EF 0C
     4D 1C A4 FE 2D 1A C3 70 FA 63 21 6F E7 B4 C2 B1`

  TurboSHAKE128(M=`FF FF FF`, D=`01`, 32):
    `BF 32 3F 94 04 94 E8 8E E1 C5 40 FE 66 0B E8 A0
     C9 3F 43 D1 5E C0 06 99 84 62 FA 99 4E ED 5D AB`

  TurboSHAKE128(M=`FF`, D=`06`, 32):
    `8E C9 C6 64 65 ED 0D 4A 6C 35 D1 35 06 71 8D 68
     7A 25 CB 05 C7 4C CA 1E 42 50 1A BD 83 87 4A 67`

  TurboSHAKE128(M=`FF FF FF`, D=`07`, 32):
    `B6 58 57 60 01 CA D9 B1 E5 F3 99 A9 F7 77 23 BB
     A0 54 58 04 2D 68 20 6F 72 52 68 2D BA 36 63 ED`

  TurboSHAKE128(M=`FF FF FF FF FF FF FF`, D=`0B`, 32):
    `8D EE AA 1A EC 47 CC EE 56 9F 65 9C 21 DF A8 E1
     12 DB 3C EE 37 B1 81 78 B2 AC D8 05 B7 99 CC 37`

  TurboSHAKE128(M=`FF`, D=`30`, 32):
    `55 31 22 E2 13 5E 36 3C 32 92 BE D2 C6 42 1F A2
     32 BA B0 3D AA 07 C7 D6 63 66 03 28 65 06 32 5B`

  TurboSHAKE128(M=`FF FF FF`, D=`7F`, 32):
    `16 27 4C C6 56 D4 4C EF D4 22 39 5D 0F 90 53 BD
     A6 D2 8E 12 2A BA 15 C7 65 E5 AD 0E 6E AF 26 F9`

  TurboSHAKE256(M=`00`^0, D=`1F`, 64):
    `36 7A 32 9D AF EA 87 1C 78 02 EC 67 F9 05 AE 13
     C5 76 95 DC 2C 66 63 C6 10 35 F5 9A 18 F8 E7 DB
     11 ED C0 E1 2E 91 EA 60 EB 6B 32 DF 06 DD 7F 00
     2F BA FA BB 6E 13 EC 1C C2 0D 99 55 47 60 0D B0`

  TurboSHAKE256(M=`00`^0, D=`1F`, 10032), last 32 bytes:
    `AB EF A1 16 30 C6 61 26 92 49 74 26 85 EC 08 2F
     20 72 65 DC CF 2F 43 53 4E 9C 61 BA 0C 9D 1D 75`

  TurboSHAKE256(M=ptn(17**0 bytes), D=`1F`, 64):
    `3E 17 12 F9 28 F8 EA F1 05 46 32 B2 AA 0A 24 6E
     D8 B0 C3 78 72 8F 60 BC 97 04 10 15 5C 28 82 0E
     90 CC 90 D8 A3 00 6A A2 37 2C 5C 5E A1 76 B0 68
     2B F2 2B AE 74 67 AC 94 F7 4D 43 D3 9B 04 82 E2`

  TurboSHAKE256(M=ptn(17**1 bytes), D=`1F`, 64):
    `B3 BA B0 30 0E 6A 19 1F BE 61 37 93 98 35 92 35
     78 79 4E A5 48 43 F5 01 10 90 FA 2F 37 80 A9 E5
     CB 22 C5 9D 78 B4 0A 0F BF F9 E6 72 C0 FB E0 97
     0B D2 C8 45 09 1C 60 44 D6 87 05 4D A5 D8 E9 C7`

  TurboSHAKE256(M=ptn(17**2 bytes), D=`1F`, 64):
    `66 B8 10 DB 8E 90 78 04 24 C0 84 73 72 FD C9 57
     10 88 2F DE 31 C6 DF 75 BE B9 D4 CD 93 05 CF CA
     E3 5E 7B 83 E8 B7 E6 EB 4B 78 60 58 80 11 63 16
     FE 2C 07 8A 09 B9 4A D7 B8 21 3C 0A 73 8B 65 C0`

  TurboSHAKE256(M=ptn(17**3 bytes), D=`1F`, 64):
    `C7 4E BC 91 9A 5B 3B 0D D1 22 81 85 BA 02 D2 9E
     F4 42 D6 9D 3D 42 76 A9 3E FE 0B F9 A1 6A 7D C0
     CD 4E AB AD AB 8C D7 A5 ED D9 66 95 F5 D3 60 AB
     E0 9E 2C 65 11 A3 EC 39 7D A3 B7 6B 9E 16 74 FB`

  TurboSHAKE256(M=ptn(17**4 bytes), D=`1F`, 64):
    `02 CC 3A 88 97 E6 F4 F6 CC B6 FD 46 63 1B 1F 52
     07 B6 6C 6D E9 C7 B5 5B 2D 1A 23 13 4A 17 0A FD
     AC 23 4E AB A9 A7 7C FF 88 C1 F0 20 B7 37 24 61
     8C 56 87 B3 62 C4 30 B2 48 CD 38 64 7F 84 8A 1D`

  TurboSHAKE256(M=ptn(17**5 bytes), D=`1F`, 64):
    `AD D5 3B 06 54 3E 58 4B 58 23 F6 26 99 6A EE 50
     FE 45 ED 15 F2 02 43 A7 16 54 85 AC B4 AA 76 B4
     FF DA 75 CE DF 6D 8C DC 95 C3 32 BD 56 F4 B9 86
     B5 8B B1 7D 17 78 BF C1 B1 A9 75 45 CD F4 EC 9F`

  TurboSHAKE256(M=ptn(17**6 bytes), D=`1F`, 64):
    `9E 11 BC 59 C2 4E 73 99 3C 14 84 EC 66 35 8E F7
     1D B7 4A EF D8 4E 12 3F 78 00 BA 9C 48 53 E0 2C
     FE 70 1D 9E 6B B7 65 A3 04 F0 DC 34 A4 EE 3B A8
     2C 41 0F 0D A7 0E 86 BF BD 90 EA 87 7C 2D 61 04`

  TurboSHAKE256(M=`FF FF FF`, D=`01`, 64):
    `D2 1C 6F BB F5 87 FA 22 82 F2 9A EA 62 01 75 FB
     02 57 41 3A F7 8A 0B 1B 2A 87 41 9C E0 31 D9 33
     AE 7A 4D 38 33 27 A8 A1 76 41 A3 4F 8A 1D 10 03
     AD 7D A6 B7 2D BA 84 BB 62 FE F2 8F 62 F1 24 24`

  TurboSHAKE256(M=`FF`, D=`06`, 64):
    `73 8D 7B 4E 37 D1 8B 7F 22 AD 1B 53 13 E3 57 E3
     DD 7D 07 05 6A 26 A3 03 C4 33 FA 35 33 45 52 80
     F4 F5 A7 D4 F7 00 EF B4 37 FE 6D 28 14 05 E0 7B
     E3 2A 0A 97 2E 22 E6 3A DC 1B 09 0D AE FE 00 4B`

  TurboSHAKE256(M=`FF FF FF`, D=`07`, 64):
    `18 B3 B5 B7 06 1C 2E 67 C1 75 3A 00 E6 AD 7E D7
     BA 1C 90 6C F9 3E FB 70 92 EA F2 7F BE EB B7 55
     AE 6E 29 24 93 C1 10 E4 8D 26 00 28 49 2B 8E 09
     B5 50 06 12 B8 F2 57 89 85 DE D5 35 7D 00 EC 67`

  TurboSHAKE256(M=`FF FF FF FF FF FF FF`, D=`0B`, 64):
    `BB 36 76 49 51 EC 97 E9 D8 5F 7E E9 A6 7A 77 18
     FC 00 5C F4 25 56 BE 79 CE 12 C0 BD E5 0E 57 36
     D6 63 2B 0D 0D FB 20 2D 1B BB 8F FE 3D D7 4C B0
     08 34 FA 75 6C B0 34 71 BA B1 3A 1E 2C 16 B3 C0`

  TurboSHAKE256(M=`FF`, D=`30`, 64):
    `F3 FE 12 87 3D 34 BC BB 2E 60 87 79 D6 B7 0E 7F
     86 BE C7 E9 0B F1 13 CB D4 FD D0 C4 E2 F4 62 5E
     14 8D D7 EE 1A 52 77 6C F7 7F 24 05 14 D9 CC FC
     3B 5D DA B8 EE 25 5E 39 EE 38 90 72 96 2C 11 1A`

  TurboSHAKE256(M=`FF FF FF`, D=`7F`, 64):
    `AB E5 69 C1 F7 7E C3 40 F0 27 05 E7 D3 7C 9A B7
     E1 55 51 6E 4A 6A 15 00 21 D7 0B 6F AC 0B B4 0C
     06 9F 9A 98 28 A0 D5 75 CD 99 F9 BA E4 35 AB 1A
     CF 7E D9 11 0B A9 7C E0 38 8D 07 4B AC 76 87 76`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(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`

  KT128(`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`

  KT128(`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`

  KT128(`FF FF FF`, C=ptn(41**2 bytes), 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`

  KT128(`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`

  KT128(M=ptn(8191 bytes), C=`00`^0, 32):
    `1B 57 76 36 F7 23 64 3E 99 0C C7 D6 A6 59 83 74
     36 FD 6A 10 36 26 60 0E B8 30 1C D1 DB E5 53 D6`

  KT128(M=ptn(8192 bytes), C=`00`^0, 32):
    `48 F2 56 F6 77 2F 9E DF B6 A8 B6 61 EC 92 DC 93
     B9 5E BD 05 A0 8A 17 B3 9A E3 49 08 70 C9 26 C3`

  KT128(M=ptn(8192 bytes), C=ptn(8189 bytes), 32):
    `3E D1 2F 70 FB 05 DD B5 86 89 51 0A B3 E4 D2 3C
     6C 60 33 84 9A A0 1E 1D 8C 22 0A 29 7F ED CD 0B`

  KT128(M=ptn(8192 bytes), C=ptn(8190 bytes), 32):
    `6A 7C 1B 6A 5C D0 D8 C9 CA 94 3A 4A 21 6C C6 46
     04 55 9A 2E A4 5F 78 57 0A 15 25 3D 67 BA 00 AE`
  KT256(M=`00`^0, C=`00`^0, 64):
    `B2 3D 2E 9C EA 9F 49 04 E0 2B EC 06 81 7F C1 0C
     E3 8C E8 E9 3E F4 C8 9E 65 37 07 6A F8 64 64 04
     E3 E8 B6 81 07 B8 83 3A 5D 30 49 0A A3 34 82 35
     3F D4 AD C7 14 8E CB 78 28 55 00 3A AE BD E4 A9`

  KT256(M=`00`^0, C=`00`^0, 128):
    `B2 3D 2E 9C EA 9F 49 04 E0 2B EC 06 81 7F C1 0C
     E3 8C E8 E9 3E F4 C8 9E 65 37 07 6A F8 64 64 04
     E3 E8 B6 81 07 B8 83 3A 5D 30 49 0A A3 34 82 35
     3F D4 AD C7 14 8E CB 78 28 55 00 3A AE BD E4 A9
     B0 92 53 19 D8 EA 1E 12 1A 60 98 21 EC 19 EF EA
     89 E6 D0 8D AE E1 66 2B 69 C8 40 28 9F 18 8B A8
     60 F5 57 60 B6 1F 82 11 4C 03 0C 97 E5 17 84 49
     60 8C CD 2C D2 D9 19 FC 78 29 FF 69 93 1A C4 D0`

  KT256(M=`00`^0, C=`00`^0, 10064), last 64 bytes:
    `AD 4A 1D 71 8C F9 50 50 67 09 A4 C3 33 96 13 9B
     44 49 04 1F C7 9A 05 D6 8D A3 5F 1E 45 35 22 E0
     56 C6 4F E9 49 58 E7 08 5F 29 64 88 82 59 B9 93
     27 52 F3 CC D8 55 28 8E FE E5 FC BB 8B 56 30 69`

  KT256(M=ptn(1 bytes), C=`00`^0, 64):
    `0D 00 5A 19 40 85 36 02 17 12 8C F1 7F 91 E1 F7
     13 14 EF A5 56 45 39 D4 44 91 2E 34 37 EF A1 7F
     82 DB 6F 6F FE 76 E7 81 EA A0 68 BC E0 1F 2B BF
     81 EA CB 98 3D 72 30 F2 FB 02 83 4A 21 B1 DD D0`

  KT256(M=ptn(17 bytes), C=`00`^0, 64):
    `1B A3 C0 2B 1F C5 14 47 4F 06 C8 97 99 78 A9 05
     6C 84 83 F4 A1 B6 3D 0D CC EF E3 A2 8A 2F 32 3E
     1C DC CA 40 EB F0 06 AC 76 EF 03 97 15 23 46 83
     7B 12 77 D3 E7 FA A9 C9 65 3B 19 07 50 98 52 7B`

  KT256(M=ptn(17**2 bytes), C=`00`^0, 64):
    `DE 8C CB C6 3E 0F 13 3E BB 44 16 81 4D 4C 66 F6
     91 BB F8 B6 A6 1E C0 A7 70 0F 83 6B 08 6C B0 29
     D5 4F 12 AC 71 59 47 2C 72 DB 11 8C 35 B4 E6 AA
     21 3C 65 62 CA AA 9D CC 51 89 59 E6 9B 10 F3 BA`

  KT256(M=ptn(17**3 bytes), C=`00`^0, 64):
    `64 7E FB 49 FE 9D 71 75 00 17 1B 41 E7 F1 1B D4
     91 54 44 43 20 99 97 CE 1C 25 30 D1 5E B1 FF BB
     59 89 35 EF 95 45 28 FF C1 52 B1 E4 D7 31 EE 26
     83 68 06 74 36 5C D1 91 D5 62 BA E7 53 B8 4A A5`

  KT256(M=ptn(17**4 bytes), C=`00`^0, 64):
    `B0 62 75 D2 84 CD 1C F2 05 BC BE 57 DC CD 3E C1
     FF 66 86 E3 ED 15 77 63 83 E1 F2 FA 3C 6A C8 F0
     8B F8 A1 62 82 9D B1 A4 4B 2A 43 FF 83 DD 89 C3
     CF 1C EB 61 ED E6 59 76 6D 5C CF 81 7A 62 BA 8D`

  KT256(M=ptn(17**5 bytes), C=`00`^0, 64):
    `94 73 83 1D 76 A4 C7 BF 77 AC E4 5B 59 F1 45 8B
     16 73 D6 4B CD 87 7A 7C 66 B2 66 4A A6 DD 14 9E
     60 EA B7 1B 5C 2B AB 85 8C 07 4D ED 81 DD CE 2B
     40 22 B5 21 59 35 C0 D4 D1 9B F5 11 AE EB 07 72`

  KT256(M=ptn(17**6 bytes), C=`00`^0, 64):
    `06 52 B7 40 D7 8C 5E 1F 7C 8D CC 17 77 09 73 82
     76 8B 7F F3 8F 9A 7A 20 F2 9F 41 3B B1 B3 04 5B
     31 A5 57 8F 56 8F 91 1E 09 CF 44 74 6D A8 42 24
     A5 26 6E 96 A4 A5 35 E8 71 32 4E 4F 9C 70 04 DA`

  KT256(`00`^0, C=ptn(1 bytes), 64):
    `92 80 F5 CC 39 B5 4A 5A 59 4E C6 3D E0 BB 99 37
     1E 46 09 D4 4B F8 45 C2 F5 B8 C3 16 D7 2B 15 98
     11 F7 48 F2 3E 3F AB BE 5C 32 26 EC 96 C6 21 86
     DF 2D 33 E9 DF 74 C5 06 9C EE CB B4 DD 10 EF F6`

  KT256(`FF`, C=ptn(41 bytes), 64):
    `47 EF 96 DD 61 6F 20 09 37 AA 78 47 E3 4E C2 FE
     AE 80 87 E3 76 1D C0 F8 C1 A1 54 F5 1D C9 CC F8
     45 D7 AD BC E5 7F F6 4B 63 97 22 C6 A1 67 2E 3B
     F5 37 2D 87 E0 0A FF 89 BE 97 24 07 56 99 88 53`

  KT256(`FF FF FF`, C=ptn(41**2 bytes), 64):
    `3B 48 66 7A 50 51 C5 96 6C 53 C5 D4 2B 95 DE 45
     1E 05 58 4E 78 06 E2 FB 76 5E DA 95 90 74 17 2C
     B4 38 A9 E9 1D DE 33 7C 98 E9 C4 1B ED 94 C4 E0
     AE F4 31 D0 B6 4E F2 32 4F 79 32 CA A6 F5 49 69`

  KT256(`FF FF FF FF FF FF FF`, C=ptn(41**3 bytes), 64):
    `E0 91 1C C0 00 25 E1 54 08 31 E2 66 D9 4A DD 9B
     98 71 21 42 B8 0D 26 29 E6 43 AA C4 EF AF 5A 3A
     30 A8 8C BF 4A C2 A9 1A 24 32 74 30 54 FB CC 98
     97 67 0E 86 BA 8C EC 2F C2 AC E9 C9 66 36 97 24`

  KT256(M=ptn(8191 bytes), C=`00`^0, 64):
    `30 81 43 4D 93 A4 10 8D 8D 8A 33 05 B8 96 82 CE
     BE DC 7C A4 EA 8A 3C E8 69 FB B7 3C BE 4A 58 EE
     F6 F2 4D E3 8F FC 17 05 14 C7 0E 7A B2 D0 1F 03
     81 26 16 E8 63 D7 69 AF B3 75 31 93 BA 04 5B 20`

  KT256(M=ptn(8192 bytes), C=`00`^0, 64):
    `C6 EE 8E 2A D3 20 0C 01 8A C8 7A AA 03 1C DA C2
     21 21 B4 12 D0 7D C6 E0 DC CB B5 34 23 74 7E 9A
     1C 18 83 4D 99 DF 59 6C F0 CF 4B 8D FA FB 7B F0
     2D 13 9D 0C 90 35 72 5A DC 1A 01 B7 23 0A 41 FA`

  KT256(M=ptn(8192 bytes), C=ptn(8189 bytes), 64):
    `74 E4 78 79 F1 0A 9C 5D 11 BD 2D A7 E1 94 FE 57
     E8 63 78 BF 3C 3F 74 48 EF F3 C5 76 A0 F1 8C 5C
     AA E0 99 99 79 51 20 90 A7 F3 48 AF 42 60 D4 DE
     3C 37 F1 EC AF 8D 2C 2C 96 C1 D1 6C 64 B1 24 96`

  KT256(M=ptn(8192 bytes), C=ptn(8190 bytes), 64):
    `F4 B5 90 8B 92 9F FE 01 E0 F7 9E C2 F2 12 43 D4
     1A 39 6B 2E 73 03 A6 AF 1D 63 99 CD 6C 7A 0A 2D
     D7 C4 F6 07 E8 27 7F 9C 9B 1C B4 AB 9D DC 59 D4
     B9 2D 1F C7 55 84 41 F1 83 2C 32 79 A4 24 1B 8B`

6. Security Considerations

This document is meant to serve as a stable reference and an implementation guide for the KangarooTwelve and TurboSHAKE eXtendable Output Functions. The security assurance of these functions relies on the cryptanalysis of reduced-round versions of Keccak and they have the same claimed security strength as their corresponding SHAKE functions.

                        +-------------------------------+
                        |        security claim         |
      +-----------------+-------------------------------+
      | TurboSHAKE128   |  128 bits (same as SHAKE128)  |
      |                 |                               |
      | KT128           |  128 bits (same as SHAKE128)  |
      |                 |                               |
      | TurboSHAKE256   |  256 bits (same as SHAKE256)  |
      |                 |                               |
      | KT256           |  256 bits (same as SHAKE256)  |
      +-----------------+-------------------------------+

To be more precise, KT128 is made of two layers:

This reasoning is detailed and formalized in [KT].

KT256 is structured as KT128, except that it uses TurboSHAKE256 as inner function. The TurboSHAKE256 function is exactly Keccak[r=1088, c=512] (as in SHAKE256) reduced to 12 rounds, and the same reasoning on cryptanalysis applies.

TurboSHAKE128 and KT128 aim at 128-bit security. 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 bits 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, TurboSHAKE128 and KT128 achieve NIST's post-quantum security level 2 [NISTPQ].

Similarly, TurboSHAKE256 and KT256 aim at 256-bit security. To achieve 256-bit security strength, the output L MUST be chosen long enough so that there are no generic attacks that violate 256-bit security. So for 256-bit (second) preimage security the output should be at least 256 bits, for 256 bits of security against multi-target preimage attacks with T targets the output should be at least 256+log_2(T) bits and for 256-bit collision security the output should be at least 512 bits. Furthermore, when the output length is at least 512 bits, TurboSHAKE256 and KT256 achieve NIST's post-quantum security level 5 [NISTPQ].

Unlike the SHA-256 and SHA-512 functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 do not suffer from the length extension weakness, and therefore do not require the use of the HMAC construction for instance when used for MAC computation [FIPS198]. Also, they can naturally be used as a key derivation function. The input must be an injective encoding of secret and diversification material, and the output can be taken as the derived key(s). The input does not need to be uniformly distributed, e.g., it can be a shared secret produced by the Diffie-Hellman or ECDH protocol, but it needs to have sufficient min-entropy.

Lastly, as KT128 and KT256 use TurboSHAKE with three values for D, namely 0x06, 0x07, and 0x0B. Protocols that use both KT128 and TurboSHAKE128, or both KT256 and TurboSHAKE256, SHOULD avoid using these three values for D.

7. References

7.1. Normative References

[RFC2119]
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/info/rfc2119>.
[RFC8174]
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/info/rfc8174>.
[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, .
[SP800-185]
National Institute of Standards and Technology, "NIST Special Publication 800-185 SHA-3 Derived Functions: cSHAKE, KMAC, TupleHash and ParallelHash", WWW https://doi.org/10.6028/NIST.SP.800-185, .

7.2. Informative References

[TURBOSHAKE]
Bertoni, G., Daemen, J., Hoffert, S., Peeters, M., Van Assche, G., Van Keer, R., and B. Viguier, "TurboSHAKE", WWW http://eprint.iacr.org/2023/342, .
[KT]
Bertoni, G., Daemen, J., Peeters, M., Van Assche, G., Van Keer, R., and B. Viguier, "KangarooTwelve: fast hashing based on Keccak-p", WWW https://link.springer.com/chapter/10.1007/978-3-319-93387-0_21, WWW http://eprint.iacr.org/2016/770.pdf, .
[SAKURA]
Bertoni, G., Daemen, J., Peeters, M., and G. Van Assche, "Sakura: a flexible coding for tree hashing", WWW https://link.springer.com/chapter/10.1007/978-3-319-07536-5_14, WWW http://eprint.iacr.org/2013/231.pdf, .
[KECCAK_CRYPTANALYSIS]
Keccak Team, "Summary of Third-party cryptanalysis of Keccak", WWW https://www.keccak.team/third_party.html, .
[XKCP]
Bertoni, G., Daemen, J., Peeters, M., Van Assche, G., and R. Van Keer, "eXtended Keccak Code Package", WWW https://github.com/XKCP/XKCP, .
[NISTPQ]
National Institute of Standards and Technology, "Submission Requirements and Evaluation Criteria for the Post-Quantum Cryptography Standardization Process", WWW https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/call-for-proposals-final-dec-2016.pdf, .
[FIPS180]
National Institute of Standards and Technology (NIST), "Secure Hash Standard (SHS)", FIPS PUB 180-4, WWW https://doi.org/10.6028/NIST.FIPS.180-4, .
[FIPS198]
National Institute of Standards and Technology (NIST), "The Keyed-Hash Message Authentication Code (HMAC)", FIPS PUB 198-1, WWW https://doi.org/10.6028/NIST.FIPS.198-1, .

Appendix A. Pseudocode

The sub-sections of this appendix contain pseudocode definitions of TurboSHAKE128, TurboSHAKE256 and KangarooTwelve. Standalone Python versions are also available in the Keccak Code Package [XKCP] and in [KT]

A.1. Keccak-p[1600,n_r=12] 

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 y from 0 to 4
    for x 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.

A.2. TurboSHAKE128 

TurboSHAKE128(message, separationByte, outputByteLen):
  offset = 0
  state = `00`^200
  input = message || separationByte

  # === 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

A.3. TurboSHAKE256 

TurboSHAKE256(message, separationByte, outputByteLen):
  offset = 0
  state = `00`^200
  input = message || separationByte

  # === Absorb complete blocks ===
  while offset < |input| - 136
      state ^= input[offset : offset + 136] || `00`^64
      state = KP(state)
      offset += 136

  # === Absorb last block and treatment of padding ===
  LastBlockLength = |input| - offset
  state ^= input[offset:] || `00`^(200-LastBlockLength)
  state ^= `00`^135 || `80` || `00`^64
  state = KP(state)

  # === Squeeze ===
  output = `00`^0
  while outputByteLen > 136
      output = output || state[0:136]
      outputByteLen -= 136
      state = KP(state)

  output = output || state[0:outputByteLen]

  return output

A.4. KT128 

KT128(inputMessage, customString, outputByteLen):
  S = inputMessage || customString
  S = S || length_encode( |customString| )

  if |S| <= 8192
      return TurboSHAKE128(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 = TurboSHAKE128(S[offset : offset + blockSize], `0B`, 32)
          FinalNode = FinalNode || CV
          numBlock += 1
          offset   += blockSize

      FinalNode = FinalNode || length_encode( numBlock ) || `FF FF`

      return TurboSHAKE128(FinalNode, `06`, outputByteLen)
  end

A.5. KT256 

KT256(inputMessage, customString, outputByteLen):
  S = inputMessage || customString
  S = S || length_encode( |customString| )

  if |S| <= 8192
      return TurboSHAKE256(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 = TurboSHAKE256(S[offset : offset + blockSize], `0B`, 64)
          FinalNode = FinalNode || CV
          numBlock += 1
          offset   += blockSize

      FinalNode = FinalNode || length_encode( numBlock ) || `FF FF`

      return TurboSHAKE256(FinalNode, `06`, outputByteLen)
  end

Authors' Addresses

Benoît Viguier
ABN AMRO Bank
Groenelaan 2
Amstelveen
David Wong (editor)
zkSecurity
Gilles Van Assche (editor)
STMicroelectronics
Quynh Dang (editor)
National Institute of Standards and Technology
Joan Daemen (editor)
Radboud University