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draft-irtf-cfrg-kangarootwelve.xml
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draft-irtf-cfrg-kangarootwelve.xml
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<?xml version="1.0" encoding="US-ASCII"?>
<!DOCTYPE rfc SYSTEM "rfc2629.dtd" [
<!ENTITY rfc2119 SYSTEM "http://xml.resource.org/public/rfc/bibxml/reference.RFC.2119.xml">
<!ENTITY rfc8174 SYSTEM "http://xml.resource.org/public/rfc/bibxml/reference.RFC.8174.xml">
]>
<?rfc toc="yes"?>
<?rfc symrefs="yes"?>
<?rfc compact="yes"?>
<?rfc subcompact="no"?>
<?rfc strict="no"?>
<?rfc rfcedstyle="yes"?>
<?rfc comments="yes"?>
<?rfc inline="yes"?>
<rfc category="info" docName="draft-irtf-cfrg-kangarootwelve-latest" ipr="trust200902">
<front>
<title abbrev="KangarooTwelve">KangarooTwelve and TurboSHAKE</title>
<!-- If the author is acting as editor, use the <role=editor> attribute-->
<!-- see RFC2223 for guidelines regarding author names -->
<author fullname="Benoît Viguier" initials="B" surname="Viguier">
<organization>ABN AMRO Bank</organization>
<address>
<postal>
<street>Groenelaan 2</street>
<city>Amstelveen</city>
<country>The Netherlands</country>
</postal>
<email>cs.ru.nl@viguier.nl</email>
</address>
</author>
<author fullname="David Wong" initials="D" surname="Wong" role="editor">
<organization>zkSecurity</organization>
<address>
<email>davidwong.crypto@gmail.com</email>
</address>
</author>
<author fullname="Gilles Van Assche" initials="G" surname="Van Assche" role="editor">
<organization>STMicroelectronics</organization>
<address>
<email>gilles.vanassche@st.com</email>
</address>
</author>
<author fullname="Quynh Dang" initials="Q" surname="Dang" role="editor">
<organization abbrev="NIST">National Institute of Standards and Technology</organization>
<address>
<email>quynh.dang@nist.gov</email>
</address>
</author>
<author fullname="Joan Daemen" initials="J" surname="Daemen" role="editor">
<organization>Radboud University</organization>
<address>
<email>joan@cs.ru.nl</email>
</address>
</author>
<!-- <author fullname="Stanislav V. Smyshlyaev" initials="S" surname="Smyshlyaev">
<organization>CryptoPro</organization>
<address>
<email>smyshsv@gmail.com</email>
</address>
</author> -->
<!-- <author fullname="John Mattsson" initials="J" surname="Mattsson">
<organization>Ericsson</organization>
<address>
<email>john.mattsson@ericsson.com</email>
</address>
</author> -->
<!-- month and day will be generated automatically by XL2RFC;
be sure the year is current.-->
<date year="2024" />
<!--WG name at the upperleft corner of the doc,
IETF is fine for non-WG IETF submissions -->
<workgroup>Crypto Forum</workgroup>
<keyword>Keccak</keyword>
<keyword>Sakura</keyword>
<keyword>KangarooTwelve</keyword>
<keyword>TurboSHAKE</keyword>
<keyword>Cryptographic Hash</keyword>
<keyword>eXtendable Output Function</keyword>
<abstract>
<t>This document defines four eXtendable Output Functions (XOF),
hash functions with output of arbitrary length, named TurboSHAKE128,
TurboSHAKE256, KT128 and KT256.</t>
<t>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.</t>
<t>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.</t>
</abstract>
</front>
<middle>
<section title="Introduction">
<t>This document defines the TurboSHAKE128, TurboSHAKE256 <xref target="TURBOSHAKE"></xref>,
KT128 and KT256 <xref target="KT"></xref> 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 <xref
target="FIPS202"></xref> and have a higher speed than the SHA-3 and SHAKE functions.</t>
<t>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.</t>
<t>
KangarooTwelve applies tree hashing on top of TurboSHAKE and comprises two functions, KT128 and KT256.
Note that <xref target="KT"></xref> only defined KT128 under the name KangarooTwelve.
KT256 is defined in this document.
</t>
<t>
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 <xref target="SP800-185"></xref>, 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 <xref target="SAKURA"/>.
The use of TurboSHAKE in KangarooTwelve makes it faster than ParallelHash.</t>
<t>The security of TurboSHAKE128, TurboSHAKE256, KT128 and KT256 builds on the public
scrutiny that Keccak has received since its
publication <xref target="KECCAK_CRYPTANALYSIS"/><xref target="TURBOSHAKE"/>.</t>
<t>With respect to <xref target="FIPS202"></xref> and <xref target="SP800-185"></xref>
functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 feature the following advantages:</t>
<t><list style="symbols">
<t>Unlike SHA3-224, SHA3-256, SHA3-384, SHA3-512, the TurboSHAKE and
KangarooTwelve functions have an extendable output.</t>
<t>Unlike any <xref target="FIPS202"></xref> defined function, similarly to
functions defined in <xref target="SP800-185"></xref>, KT128 and KT256
allow the use of a customization string.</t>
<t>Unlike any <xref target="FIPS202"></xref> and <xref target="SP800-185"></xref>
functions but ParallelHash, KT128 and KT256 exploit available parallelism.</t>
<t>Unlike ParallelHash, KT128 and KT256 do not have overhead when
processing short messages.</t>
<t>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 <xref target="FIPS202"></xref>.
The KangarooTwelve functions immediately benefit from the same speedup, improving over
<xref target="FIPS202"></xref> and <xref target="SP800-185"></xref>.</t>
</list></t>
<t>With respect to SHA-256 and SHA-512 and other <xref target="FIPS180"/> functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 feature the following advantages:</t>
<t><list style="symbols">
<t>Unlike <xref target="FIPS180"/> functions, the TurboSHAKE and KangarooTwelve functions have an extendable output.</t>
<t>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.</t>
<t>Unlike the SHA-256 and SHA-512 functions, TurboSHAKE128, TurboSHAKE256, KT128 and KT256 do not suffer from the length extension weakness.</t>
<t>Unlike any <xref target="FIPS180"></xref> 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.</t>
</list></t>
<t>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.</t>
<section title="Conventions">
<t>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 <xref target="RFC2119"></xref> <xref target="RFC8174"></xref>
when, and only when, they appear in all capitals, as shown here.</t>
<t>The following notations are used throughout the document:</t>
<t><list style="hanging">
<t hangText="`...`">denotes a string of bytes given in
hexadecimal. For example, `0B 80`.</t>
<t hangText="|s|">denotes the length of a byte string `s`.
For example, |`FF FF`| = 2.</t>
<t hangText="`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`.</t>
<t hangText="`00`^0">denotes the empty byte-string.</t>
<t hangText="a||b">denotes the concatenation of two strings a and b.
For example, `10`||`F1` = `10 F1`</t>
<t hangText="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`.</t>
<t hangText="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`.</t>
</list></t>
<t>In the following, x and y are byte strings of equal length:</t>
<t><list style="hanging">
<t hangText="x^=y"> denotes x takes the value x XOR y.</t>
<t hangText="x & y"> denotes x AND y.</t>
</list></t>
<t>In the following, x and y are integers:</t>
<t><list style="hanging">
<t hangText="x+=y"> denotes x takes the value x + y.</t>
<t hangText="x-=y"> denotes x takes the value x - y.</t>
<t hangText="x**y"> denotes the exponentiation of x by y.</t>
<t hangText="x mod y"> denotes reminder of the division of x by y.</t>
<t hangText="x / y"> denotes the integer dividend of the division of x by y.</t>
</list></t>
</section>
</section>
<section title="TurboSHAKE">
<section anchor="TurboSHAKE_Interface" title="Interface">
<t>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 <xref target="TURBOSHAKE"/>.
The capacity is an essential parameter of the sponge construction, see <xref target="FIPS202"/> for more details.)
</t>
<t>
An instance of TurboSHAKE takes as input parameters a byte-string M, an OPTIONAL byte D and a positive integer L
where<list style="hanging">
<t hangText="M"> byte-string, is the Message and</t>
<t hangText="D"> byte in the range [`01`, `02`, .. , `7F`], is an OPTIONAL Domain separation byte and</t>
<t hangText="L"> positive integer, is the requested number of output bytes.</t>
</list></t>
<t>
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).
</t>
<t>By default, the Domain separation byte is `1F`. For an API that
does not support a domain separation byte, D MUST be the `1F`.</t>
<t>
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.
</t>
<t>
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.
</t>
</section>
<section title="Specifications">
<t>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 <xref target="FIPS202"></xref>.
KP denotes this permutation.</t>
<t>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.</t>
<t><figure><artwork><![CDATA[
+-------------+--------------+
| Rate | Capacity |
+----------------+-------------+--------------+
| TurboSHAKE128 | 168 Bytes | 32 Bytes |
| | | |
| TurboSHAKE256 | 136 Bytes | 64 Bytes |
+----------------+-------------+--------------+]]></artwork>
</figure></t>
<t>We now describe the operations inside TurboSHAKE128.<list style="symbols">
<t>First the input M' is formed by appending the domain separation byte D to the message M.</t>
<t>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.
</t>
<t>M' never has a length of 0 bytes due to the presence of the domain separation byte.</t>
<t>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.</t>
<t>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.</t>
<t>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.</t>
</list></t>
<t>TurboSHAKE256 performs the same steps but makes use of 136-byte blocks with respect
to padding, absorbing, and squeezing phases.</t>
<t>
The definition of the TurboSHAKE functions equivalently implements the pad10*1 rule; see Section 5.1 of <xref target="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 <xref target="KT"></xref> and Section 3 of <xref target="TURBOSHAKE"></xref>.</t>
<t>The pseudocode versions of TurboSHAKE128 and TurboSHAKE256 are provided respectively in <xref target="TSHK128_PC"/> and <xref target="TSHK256_PC"/>.</t>
</section>
</section>
<section title="KangarooTwelve: Tree hashing over TurboSHAKE">
<section title="Interface">
<t>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 <list style="hanging">
<t hangText="M"> byte-string, is the Message and</t>
<t hangText="C"> byte-string, is an OPTIONAL Customization string and</t>
<t hangText="L"> positive integer, the requested number of output bytes.</t>
</list></t>
<t>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 <xref target="TurboSHAKE_Interface"/>), but with a larger range.
</t>
<t>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.</t>
<t>Note that an implementation MAY propose an interface with input and/or output incrementality as specified in <xref target="TurboSHAKE_Interface"/>.</t>
</section>
<section title="Specification of KT128">
<t>On top of the sponge function TurboSHAKE128, KT128 uses a
Sakura-compatible tree hash mode <xref target="SAKURA"></xref>.
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 <xref target="RE"/>.</t>
<t><figure><artwork><![CDATA[
S = M || C || length_encode( |C| ) ]]></artwork></figure></t>
<t>Then, split S into n chunks of 8192 bytes.</t>
<t><figure><artwork><![CDATA[
S = S_0 || .. || S_(n-1)
|S_0| = .. = |S_(n-2)| = 8192 bytes
|S_(n-1)| <= 8192 bytes ]]></artwork></figure></t>
<t>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.</t>
<t><figure><artwork><![CDATA[
CV_i = TurboSHAKE128( S_i, `0B`, 32 )]]></artwork></figure></t>
<t>Compute the final node: FinalNode.
<list style="symbols">
<t>If |S| <= 8192 bytes, FinalNode = S</t>
<t>Otherwise compute FinalNode as follows:</t>
</list></t>
<t><figure><artwork><![CDATA[
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`]]></artwork></figure></t>
<t>Finally, the KT128 output is retrieved:
<list style="symbols">
<t>If |S| <= 8192 bytes, from TurboSHAKE128( FinalNode, `07`, L )</t>
</list></t>
<t><figure>
<artwork><![CDATA[
KT128( M, C, L ) = TurboSHAKE128( FinalNode, `07`, L )]]>
</artwork></figure></t>
<t><list style="symbols">
<t>Otherwise from TurboSHAKE128( FinalNode, `06`, L )</t>
</list></t>
<t><figure>
<artwork><![CDATA[
KT128( M, C, L ) = TurboSHAKE128( FinalNode, `06`, L )]]>
</artwork></figure></t>
<t>The following figure illustrates the computation flow of KT128
for |S| <= 8192 bytes:</t>
<t><figure><artwork><![CDATA[
+--------------+ TurboSHAKE128(.., `07`, L)
| S |-----------------------------> output
+--------------+]]></artwork></figure></t>
<t>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 ) :</t>
<t><figure><artwork><![CDATA[
+--------------+
| 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]]></artwork></figure></t>
<t>A pseudocode version is provided in <xref target="KT128_PC"/>.</t>
<t>The table below gathers the values of the domain separation
bytes used by the tree hash mode:</t>
<t><figure><artwork><![CDATA[
+--------------------+------------------+
| Type | Byte |
+--------------------+------------------+
| SingleNode | `07` |
| | |
| IntermediateNode | `0B` |
| | |
| FinalNode | `06` |
+--------------------+------------------+]]></artwork>
</figure></t>
</section>
<section anchor="RE" title="length_encode( x )">
<t>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</t>
<t><figure>
<artwork><![CDATA[
x = sum of 256**i * x_i for i from 0 to n-1]]></artwork></figure></t>
<t>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.</t>
<t>As example, length_encode(0) = `00`, length_encode(12) = `0C 01` and
length_encode(65538) = `01 00 02 03`</t>
<t>A pseudocode version is as follows where { b } denotes the byte of numerical value b.</t>
<t><figure><artwork><![CDATA[
length_encode(x):
S = `00`^0
while x > 0
S = { x mod 256 } || S
x = x / 256
S = S || { |S| }
return S
end]]></artwork></figure></t>
</section>
<section title="Specification of KT256">
<t>KT256 is specified exactly like KT128, with two differences:</t>
<list style="symbols">
<t>All the calls to TurboSHAKE128 in KT128 are replaced with calls to TurboSHAKE256 in KT256.</t>
<t>The chaining values CV_1 to CV_(n-1) are 64-byte long in KT256 and are computed as follows:</t>
</list>
<t><figure><artwork><![CDATA[
CV_i = TurboSHAKE256( S_i, `0B`, 64 )]]></artwork></figure></t>
<t>A pseudocode version is provided in <xref target="KT256_PC"/>.</t>
</section>
</section>
<section title="Message authentication codes">
<t>Implementing a MAC with KT128 or KT256 MAY use a hash-then-MAC construction.
This document recommends a method called HopMAC, defined as follows:</t>
<t><figure>
<artwork><![CDATA[
HopMAC128(Key, M, C, L) = KT128(Key, KT128(M, C, 32), L)
HopMAC256(Key, M, C, L) = KT256(Key, KT256(M, C, 64), L)]]></artwork>
</figure></t>
<t>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.</t>
<t>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.</t>
<t>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).</t>
</section>
<section title="Test vectors">
<t>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.
</t>
<t><figure><artwork><![CDATA[ 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`]]></artwork></figure>
</t>
<t><figure><artwork><![CDATA[ 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`]]></artwork></figure></t>
<t><figure><artwork><![CDATA[
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`
]]></artwork></figure></t>
<t><figure><artwork><![CDATA[
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`
]]></artwork></figure></t>
<t><figure><artwork><![CDATA[ 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`]]></artwork></figure></t>
<t><figure><artwork><![CDATA[ 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`]]></artwork></figure></t>
</section>
<section anchor="Security" title="Security Considerations">
<t>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.</t>
<t><figure><artwork><![CDATA[
+-------------------------------+
| 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) |
+-----------------+-------------------------------+]]></artwork>
</figure></t>
<t>
To be more precise, KT128 is made of two layers:
<list style="symbols">
<t>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).</t>
<t>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 <xref target="SAKURA"/>.</t>
</list></t>
<t>This reasoning is detailed and formalized in <xref target="KT"/>.</t>
<t>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.</t>
<t>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 <xref target="NISTPQ"/>.</t>
<t>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 <xref target="NISTPQ"/>.</t>
<t>
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 <xref target="FIPS198"/>.