Internet DRAFT - draft-irtf-cfrg-kangarootwelve

draft-irtf-cfrg-kangarootwelve







Crypto Forum                                                  B. Viguier
Internet-Draft                                             ABN AMRO Bank
Intended status: Informational                              D. Wong, Ed.
Expires: 9 August 2024                                        zkSecurity
                                                      G. Van Assche, Ed.
                                                      STMicroelectronics
                                                            Q. Dang, Ed.
                                                                    NIST
                                                          J. Daemen, Ed.
                                                      Radboud University
                                                         6 February 2024


                     KangarooTwelve and TurboSHAKE
                   draft-irtf-cfrg-kangarootwelve-13

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 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
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   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 9 August 2024.





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Copyright Notice

   Copyright (c) 2024 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Conventions . . . . . . . . . . . . . . . . . . . . . . .   4
   2.  TurboSHAKE  . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.1.  Interface . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.2.  Specifications  . . . . . . . . . . . . . . . . . . . . .   6
   3.  KangarooTwelve: Tree hashing over TurboSHAKE  . . . . . . . .   7
     3.1.  Interface . . . . . . . . . . . . . . . . . . . . . . . .   7
     3.2.  Specification of KT128  . . . . . . . . . . . . . . . . .   8
     3.3.  length_encode( x )  . . . . . . . . . . . . . . . . . . .  11
     3.4.  Specification of KT256  . . . . . . . . . . . . . . . . .  11
   4.  Message authentication codes  . . . . . . . . . . . . . . . .  11
   5.  Test vectors  . . . . . . . . . . . . . . . . . . . . . . . .  12
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .  20
   7.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  22
     7.1.  Normative References  . . . . . . . . . . . . . . . . . .  22
     7.2.  Informative References  . . . . . . . . . . . . . . . . .  22
   Appendix A.  Pseudocode . . . . . . . . . . . . . . . . . . . . .  23
     A.1.  Keccak-p[1600,n_r=12] . . . . . . . . . . . . . . . . . .  23
     A.2.  TurboSHAKE128 . . . . . . . . . . . . . . . . . . . . . .  25
     A.3.  TurboSHAKE256 . . . . . . . . . . . . . . . . . . . . . .  25
     A.4.  KT128 . . . . . . . . . . . . . . . . . . . . . . . . . .  26
     A.5.  KT256 . . . . . . . . . . . . . . . . . . . . . . . . . .  27
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  28

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.



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   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:

   *  Unlike SHA3-224, SHA3-256, SHA3-384, SHA3-512, the TurboSHAKE and
      KangarooTwelve functions have an extendable output.

   *  Unlike any [FIPS202] defined function, similarly to functions
      defined in [SP800-185], KT128 and KT256 allow the use of a
      customization string.

   *  Unlike any [FIPS202] and [SP800-185] functions but ParallelHash,
      KT128 and KT256 exploit available parallelism.

   *  Unlike ParallelHash, KT128 and KT256 do not have overhead when
      processing short messages.

   *  The permutation in the TurboSHAKE functions has half the number of
      rounds compared to the one in the SHA-3 and SHAKE functions,
      making them faster than any function defined in [FIPS202].  The
      KangarooTwelve functions immediately benefit from the same
      speedup, improving over [FIPS202] and [SP800-185].



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

   *  Unlike [FIPS180] functions, the TurboSHAKE and KangarooTwelve
      functions have an extendable output.

   *  The TurboSHAKE functions produce output at the same rate as they
      process input, whereas SHA-256 and SHA-512 produce output half as
      fast as they process input.

   *  Unlike the SHA-256 and SHA-512 functions, TurboSHAKE128,
      TurboSHAKE256, KT128 and KT256 do not suffer from the length
      extension weakness.

   *  Unlike any [FIPS180] functions, TurboSHAKE128, TurboSHAKE256,
      KT128 and KT256 use a round function with algebraic degree 2,
      which makes them more suitable to masking techniques for
      protections against side-channel attacks.

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 RFC 2119 [RFC2119].

   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



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      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).



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   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.

   *  First the input M' is formed by appending the domain separation
      byte D to the message M.





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   *  Non-multiple of 168-bytes-length M' are padded with zeroes to the
      next multiple of 168 bytes while M' with length multiple of 168
      bytes are kept as is.  Then a byte `80` is XORed to the last byte
      of the padded input M' and the resulting string is split into a
      sequence of 168-byte blocks.

   *  M' never has a length of 0 bytes due to the presence of the domain
      separation byte.

   *  As defined by the sponge construction, the process operates on a
      state and consists of two phases: the absorbing phase that
      processes the padded input M' 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.

   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



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   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.

   *  If |S| <= 8192 bytes, FinalNode = S

   *  Otherwise compute FinalNode as follows:






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       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:

   *  If |S| <= 8192 bytes, from TurboSHAKE128( FinalNode, `07`, L )

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

   *  Otherwise from TurboSHAKE128( FinalNode, `06`, 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 ) :
























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





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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:

   *  All the calls to TurboSHAKE128 in KT128 are replaced with calls to
      TurboSHAKE256 in KT256.

   *  The chaining values CV_1 to CV_(n-1) are 64-byte long in KT256 and
      are computed as follows:

       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 SHOULD use a HASH-then-MAC
   construction.  This document recommends a method called HopMAC,
   defined as follows:



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       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`




















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       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`




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     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



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        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



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        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



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        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



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        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



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        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



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        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:



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   *  The inner function TurboSHAKE128.  The security assurance of this
      layer relies on cryptanalysis.  The TurboSHAKE128 function is
      exactly Keccak[r=1344, c=256] (as in SHAKE128) reduced to 12
      rounds.  Any cryptanalysis of reduced-round Keccak is also
      cryptanalysis of reduced-round TurboSHAKE128 (provided the number
      of rounds attacked is not higher than 12).

   *  The tree hashing over TurboSHAKE128.  This layer is a mode on top
      of TurboSHAKE128 that does not introduce any vulnerability thanks
      to the use of Sakura coding proven secure in [SAKURA].

   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].












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   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, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [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.

   [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, December
              2016.

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, March 2023.








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   [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, July 2018.

   [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, June 2014.

   [KECCAK_CRYPTANALYSIS]
              Keccak Team, "Summary of Third-party cryptanalysis of
              Keccak", WWW https://www.keccak.team/third_party.html,
              2022.

   [XKCP]     Bertoni, G., Daemen, J., Peeters, M., Van Assche, G., and
              R. Van Keer, "eXtended Keccak Code Package",
              WWW https://github.com/XKCP/XKCP, December 2022.

   [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, December 2016.

   [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, August 2015.

   [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,
              July 2008.

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]






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   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



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       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.

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






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   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






















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  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




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Authors' Addresses

   BenoƮt Viguier
   ABN AMRO Bank
   Groenelaan 2
   Amstelveen
   Email: cs.ru.nl@viguier.nl


   David Wong (editor)
   zkSecurity
   Email: davidwong.crypto@gmail.com


   Gilles Van Assche (editor)
   STMicroelectronics
   Email: gilles.vanassche@st.com


   Quynh Dang (editor)
   National Institute of Standards and Technology
   Email: quynh.dang@nist.gov


   Joan Daemen (editor)
   Radboud University
   Email: joan@cs.ru.nl
























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