Internet Engineering Task Force V. Dolmatov, Ed.
Internet-Draft JSC "NPK Kryptonite"
Updates: 5830 (if approved) D. Eremin-Solenikov
Intended status: Informational Auriga, Inc
Expires: December 27, 2019 June 26, 2019
GOST R 34.12-2015: Block Cipher "Magma"
draft-dolmatov-magma-01
Abstract
This document is intended to be a source of information about updated
version of the block cipher with block length of n=64 bits and key
length k=256 bits (RFC5830), which is also referred as "Magma" and is
described in the Russian Federal standard GOST R 34.12-2015,
containing also the description of block cipher "Kuznechik"
(RFC7801). These algorithms are from the set of Russian
cryptographic standard algorithms (called GOST algorithms).
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on December 27, 2019.
Copyright Notice
Copyright (c) 2019 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
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 1]
Internet-Draft Abbreviated Title June 2019
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. General Information . . . . . . . . . . . . . . . . . . . . . 3
3. Definitions and Notations . . . . . . . . . . . . . . . . . . 3
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 3
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 4
4. Description of Kuznechik cipher . . . . . . . . . . . . . . . 5
5. Parameter Values . . . . . . . . . . . . . . . . . . . . . . 5
5.1. Nonlinear Bijection . . . . . . . . . . . . . . . . . . . 5
5.2. Transformations . . . . . . . . . . . . . . . . . . . . . 6
5.3. Key schedule . . . . . . . . . . . . . . . . . . . . . . 6
6. Basic encryption algorithm . . . . . . . . . . . . . . . . . 7
6.1. Encryption . . . . . . . . . . . . . . . . . . . . . . . 7
6.2. Decryption . . . . . . . . . . . . . . . . . . . . . . . 7
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 7
8. Security Considerations . . . . . . . . . . . . . . . . . . . 7
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 7
9.1. Normative References . . . . . . . . . . . . . . . . . . 7
9.2. Informative References . . . . . . . . . . . . . . . . . 8
Appendix A. Test Examples . . . . . . . . . . . . . . . . . . . 8
A.1. Transformation t . . . . . . . . . . . . . . . . . . . . 8
A.2. Transformation g . . . . . . . . . . . . . . . . . . . . 8
A.3. Key schedule . . . . . . . . . . . . . . . . . . . . . . 9
A.4. Test Encryption . . . . . . . . . . . . . . . . . . . . . 10
A.5. Test Decryption . . . . . . . . . . . . . . . . . . . . . 11
Appendix B. Background . . . . . . . . . . . . . . . . . . . . . 12
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 12
1. Introduction
The Russian Federal standard [GOSTR3412-2015] specifies basic block
ciphers used as cryptographic techniques for information processing
and information protection including the provision of
confidentiality, authenticity, and integrity of information during
information transmission, processing and storage in computer-aided
systems.
The cryptographic algorithms specified in this Standard are designed
both for hardware and software implementation. They comply with
modern cryptographic requirements, and put no restrictions on the
confidentiality level of the protected information.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 2]
Internet-Draft Abbreviated Title June 2019
2. General Information
The Russian Federal standard [GOSTR3412-2015] was developed by the
Center for Information Protection and Special Communications of the
Federal Security Service of the Russian Federation with participation
of the Open Joint-Stock company "Information Technologies and
Communication Systems" (InfoTeCS JSC). GOST R 34.12-2015 was
approved and introduced by Decree #749 of the Federal Agency on
Technical Regulating and Metrology on 19.06.2015.
Terms and concepts in the standard comply with the following
international standards:
o ISO/IEC 10116 [ISO-IEC10116],
o series of standards ISO/IEC 18033 [ISO-IEC18033-1],
[ISO-IEC18033-3].
3. Definitions and Notations
The following terms and their corresponding definitions are used in
the standard.
3.1. Definitions
Definitions
encryption algorithm: process which transforms plaintext into
ciphertext (Clause 2.19 of [ISO-IEC18033-1]),
decryption algorithm: process which transforms ciphertext into
plaintext (Clause 2.14 of [ISO-IEC18033-1]),
basic block cipher: block cipher which for a given key provides a
single invertible mapping of the set of fixed-length plaintext
blocks into ciphertext blocks of the same length,
block: string of bits of a defined length (Clause 2.6 of
[ISO-IEC18033-1]),
block cipher: symmetric encipherment system with the property that
the encryption algorithm operates on a block of plaintext, i.e. a
string of bits of a defined length, to yield a block of ciphertext
(Clause 2.7 of [ISO-IEC18033-1]),
Note: In GOST R 34.12-2015, it is established that the terms
"block cipher" and "block encryption algorithm" are synonyms.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 3]
Internet-Draft Abbreviated Title June 2019
encryption: reversible transformation of data by a cryptographic
algorithm to produce ciphertext, i.e., to hide the information
content of the data (Clause 2.18 of [ISO-IEC18033-1]),
round key: sequence of symbols which is calculated from the key
and controls a transformation for one round of a block cipher,
key: sequence of symbols that controls the operation of a
cryptographic transformation (e.g., encipherment, decipherment)
(Clause 2.21 of [ISO-IEC18033-1]),
Note: In GOST R 34.12-2015, the key must be a binary sequence.
plaintext: unencrypted information (Clause 3.11 of
[ISO-IEC10116]),
key schedule: calculation of round keys from the key,
decryption: reversal of a corresponding encipherment (Clause 2.13
of [ISO-IEC18033-1]),
symmetric cryptographic technique: cryptographic technique that
uses the same secret key for both the originator's and the
recipient's transformation (Clause 2.32 of [ISO-IEC18033-1]),
cipher: alternative term for encipherment system (Clause 2.20 of
[ISO-IEC18033-1]),
ciphertext: data which has been transformed to hide its
information content (Clause 3.3 of [ISO-IEC10116]).
3.2. Notations
The following notations are used in the standard:
V* the set of all binary vector-strings of a finite length
(hereinafter referred to as the strings) including the empty
string,
V_s the set of all binary strings of length s, where s is a
non-negative integer; substrings and string components are
enumerated from right to left starting from zero,
U[*]W direct (Cartesian) product of two set U and W,
|A| the number of components (the length) of a string A
belonging to V* (if A is an empty string, then |A| = 0),
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 4]
Internet-Draft Abbreviated Title June 2019
A||B concatenation of strings A and B both belonging to V*,
i.e., a string from V_(|A|+|B|), where the left substring from
V_|A| is equal to A and the right substring from V_|B| is equal to
B,
A<<<_11 cyclic rotation of string A belonging to V_32 by 11
components in the direction of components having greater indices
Z_(2^n) ring of residues modulo 2^n,
(xor) exclusive-or of the two binary strings of the same length,
[+] addition in the ring Z_(2^32)
Vec_s: Z_(2^s) -> V_s bijective mapping which maps an element from
ring Z_(2^s) into its binary representation, i.e., for an element
z of the ring Z_(2^s), represented by the residue z_0 + (2*z_1) +
... + (2^(s-1)*z_(s-1)), where z_i in {0, 1}, i = 0, ..., n-1, the
equality Vec_s(z) = z_(s-1)||...||z_1||z_0 holds,
Int_s: V_s -> Z_(2^s) the mapping inverse to the mapping Vec_s,
i.e., Int_s = Vec_s^(-1),
PS composition of mappings, where the mapping S applies first,
P^s composition of mappings P^(s-1) and P, where P^1=P,
4. Description of Kuznechik cipher
This section corresponds to "Kuznechik" cipher and is described in
[RFC7801]
5. Parameter Values
5.1. Nonlinear Bijection
The bijective nonlinear mapping is a set of substitutions:
Pi_i = Vec_4 Pi'_i Int_4: V_4 -> V_4,
where
Pi'_i: Z_(2^4) -> Z_(2^4), i = 0, 1, ..., 7.
The values of the substitution Pi' are specified below as arrays
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 5]
Internet-Draft Abbreviated Title June 2019
Pi'_i = (Pi'_i(0), Pi'_i(1), ... , Pi'_i(15)), i = 0, 1, ..., 7:
Pi'_0 = (12, 4, 6, 2, 10, 5, 11, 9, 14, 8, 13, 7, 0, 3, 15, 1);
Pi'_1 = (6, 8, 2, 3, 9, 10, 5, 12, 1, 14, 4, 7, 11, 13, 0, 15);
Pi'_2 = (11, 3, 5, 8, 2, 15, 10, 13, 14, 1, 7, 4, 12, 9, 6, 0);
Pi'_3 = (12, 8, 2, 1, 13, 4, 15, 6, 7, 0, 10, 5, 3, 14, 9, 11);
Pi'_4 = (7, 15, 5, 10, 8, 1, 6, 13, 0, 9, 3, 14, 11, 4, 2, 12);
Pi'_5 = (5, 13, 15, 6, 9, 2, 12, 10, 11, 7, 8, 1, 4, 3, 14, 0);
Pi'_6 = (8, 14, 2, 5, 6, 9, 1, 12, 15, 4, 11, 0, 13, 10, 3, 7);
Pi'_7 = (1, 7, 14, 13, 0, 5, 8, 3, 4, 15, 10, 6, 9, 12, 11, 2);
5.2. Transformations
The following transformations are applicable for encryption and
decryption algorithms:
t: V_32 -> V_32 t(a) = t(a_7||...||a_0) = Pi_7(a_7)||...||Pi_0(a_0),
where a=a_7||...||a_0 belongs to V_32, a_i belongs to V_4, i=0, 1,
..., 7;
g[k]: V_32 -> V_32 g[k](a) = (t(Vec_32(Int_32(a) [+] Int_32(k))))
<<<_11, where k, a belong to V_32;
G[k]: V_32[*]V_32 -> V_32[*]V_32 G[k](a_1, a_0) = (a_0, g[k](a_0)
(xor) a_1), where k, a_0, a_1 belong to V_32;
G^*[k]: V_32[*]V_32 -> V_64 G^*[k](a_1, a_0) = (g[k](a_0) (xor)
a_1) || a_0, where k, a_0, a_1 belong to V_32.
5.3. Key schedule
Round keys K_i belonging to V_32, i=1, 2, ..., 32 are derived from
key K=k_255||...||k_0 belonging to V_256, k_i belongs to V_1, i=0, 1,
..., 255, as follows:
K_1=k_255||...||k_224;
K_2=k_223||...||k_192;
K_3=k_191||...||k_160;
K_4=k_159||...||k_128;
K_5=k_127||...||k_96;
K_6=k_95||...||k_64;
K_7=k_63||...||k_32;
K_8=k_31||...||k_0;
K_(i+8)=K_i, i = 1, 2, ..., 8;
K_(i+16)=K_i, i = 1, 2, ..., 8;
K_(i+24)=K_(9-i), i = 1, 2, ..., 8.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 6]
Internet-Draft Abbreviated Title June 2019
6. Basic encryption algorithm
6.1. Encryption
Depending on the values of round keys K_1,...,K_32, the encryption
algorithm is a substitution E_(K_1,...,K_32) defined as follows:
E_(K_1,...,K_32)(a)=G^*[K_32]G[K_31]...G[K_2]G[K_1](a_1, a_0),
where a=(a_1, a_0) belongs to V_64, and a_0, a_1 belong to V_32.
6.2. Decryption
Depending on the values of round keys K_1,...,K_32, the decryption
algorithm is a substitution D_(K_1,...,K_32) defined as follows:
D_(K_1,...,K_32)(a)=G^*[K_1]G[K_2]...G[K_31]G[K_32](a_1, a_0),
where a=(a_1, a_0) belongs to V_64, and a_0, a_1 belong to V_32.
7. IANA Considerations
This memo includes no request to IANA.
8. Security Considerations
This entire document is about security considerations.
9. References
9.1. Normative References
[GOSTR3412-2015]
Federal Agency on Technical Regulating and Metrology,
"Information technology. Cryptographic data security.
Block ciphers. GOST R 34.12-2015", 2015.
[RFC5830] Dolmatov, V., Ed., "GOST 28147-89: Encryption, Decryption,
and Message Authentication Code (MAC) Algorithms",
RFC 5830, DOI 10.17487/RFC5830, March 2010,
.
[RFC7801] Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher
"Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016,
.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 7]
Internet-Draft Abbreviated Title June 2019
[RFC7836] Smyshlyaev, S., Ed., Alekseev, E., Oshkin, I., Popov, V.,
Leontiev, S., Podobaev, V., and D. Belyavsky, "Guidelines
on the Cryptographic Algorithms to Accompany the Usage of
Standards GOST R 34.10-2012 and GOST R 34.11-2012",
RFC 7836, DOI 10.17487/RFC7836, March 2016,
.
9.2. Informative References
[GOST28147-89]
Government Committee of the USSR for Standards,
""Cryptographic Protection for Data Processing System",
GOST 28147-89, Gosudarstvennyi Standard of USSR", 1989.
[ISO-IEC10116]
ISO-IEC, "Information technology - Security techniques -
Modes of operation for an n-bit block cipher, ISO-IEC
10116", 2006.
[ISO-IEC18033-1]
ISO-IEC, "Information technology - Security techniques -
Encryption algorithms - Part 1: General, ISO-IEC 18033-1",
2013.
[ISO-IEC18033-3]
ISO-IEC, "Information technology - Security techniques -
Encryption algorithms - Part 3: Block ciphers, ISO-IEC
18033-3", 2010.
Appendix A. Test Examples
This section is for information only and is not a normative part of
the standard.
A.1. Transformation t
t(fdb97531) = 2a196f34,
t(2a196f34) = ebd9f03a,
t(ebd9f03a) = b039bb3d,
t(b039bb3d) = 68695433.
A.2. Transformation g
g[87654321](fedcba98) = fdcbc20c,
g[fdcbc20c](87654321) = 7e791a4b,
g[7e791a4b](fdcbc20c) = c76549ec,
g[c76549ec](7e791a4b) = 9791c849.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 8]
Internet-Draft Abbreviated Title June 2019
A.3. Key schedule
With key set to
K = ffeeddccbbaa99887766554433221100f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff,
following round keys are generated:
K_1 = ffeeddcc,
K_2 = bbaa9988,
K_3 = 77665544,
K_4 = 33221100,
K_5 = f0f1f2f3,
K_6 = f4f5f6f7,
K_7 = f8f9fafb,
K_8 = fcfdfeff,
K_9 = ffeeddcc,
K_10 = bbaa9988,
K_11 = 77665544,
K_12 = 33221100,
K_13 = f0f1f2f3,
K_14 = f4f5f6f7,
K_15 = f8f9fafb,
K_16 = fcfdfeff,
K_17 = ffeeddcc,
K_18 = bbaa9988,
K_19 = 77665544,
K_20 = 33221100,
K_21 = f0f1f2f3,
K_22 = f4f5f6f7,
K_23 = f8f9fafb,
K_24 = fcfdfeff,
K_25 = fcfdfeff,
K_26 = f8f9fafb,
K_27 = f4f5f6f7,
K_28 = f0f1f2f3,
K_29 = 33221100,
K_30 = 77665544,
K_31 = bbaa9988,
K_32 = ffeeddcc.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 9]
Internet-Draft Abbreviated Title June 2019
A.4. Test Encryption
In this test example, encryption is performed on the round keys
specified in clause A.3. Let the plaintext be
a = fedcba9876543210,
then
(a_1, a_0) = (fedcba98, 76543210),
G[K_1](a_1, a_0) = (76543210, 28da3b14),
G[K_2]G[K_1](a_1, a_0) = (28da3b14, b14337a5),
G[K_3]...G[K_1](a_1, a_0) = (b14337a5, 633a7c68),
G[K_4]...G[K_1](a_1, a_0) = (633a7c68, ea89c02c),
G[K_5]...G[K_1](a_1, a_0) = (ea89c02c, 11fe726d),
G[K_6]...G[K_1](a_1, a_0) = (11fe726d, ad0310a4),
G[K_7]...G[K_1](a_1, a_0) = (ad0310a4, 37d97f25),
G[K_8]...G[K_1](a_1, a_0) = (37d97f25, 46324615),
G[K_9]...G[K_1](a_1, a_0) = (46324615, ce995f2a),
G[K_10]...G[K_1](a_1, a_0) = (ce995f2a, 93c1f449),
G[K_11]...G[K_1](a_1, a_0) = (93c1f449, 4811c7ad),
G[K_12]...G[K_1](a_1, a_0) = (4811c7ad, c4b3edca),
G[K_13]...G[K_1](a_1, a_0) = (c4b3edca, 44ca5ce1),
G[K_14]...G[K_1](a_1, a_0) = (44ca5ce1, fef51b68),
G[K_15]...G[K_1](a_1, a_0) = (fef51b68, 2098cd86)
G[K_16]...G[K_1](a_1, a_0) = (2098cd86, 4f15b0bb),
G[K_17]...G[K_1](a_1, a_0) = (4f15b0bb, e32805bc),
G[K_18]...G[K_1](a_1, a_0) = (e32805bc, e7116722),
G[K_19]...G[K_1](a_1, a_0) = (e7116722, 89cadf21),
G[K_20]...G[K_1](a_1, a_0) = (89cadf21, bac8444d),
G[K_21]...G[K_1](a_1, a_0) = (bac8444d, 11263a21),
G[K_22]...G[K_1](a_1, a_0) = (11263a21, 625434c3),
G[K_23]...G[K_1](a_1, a_0) = (625434c3, 8025c0a5),
G[K_24]...G[K_1](a_1, a_0) = (8025c0a5, b0d66514),
G[K_25]...G[K_1](a_1, a_0) = (b0d66514, 47b1d5f4),
G[K_26]...G[K_1](a_1, a_0) = (47b1d5f4, c78e6d50),
G[K_27]...G[K_1](a_1, a_0) = (c78e6d50, 80251e99),
G[K_28]...G[K_1](a_1, a_0) = (80251e99, 2b96eca6),
G[K_29]...G[K_1](a_1, a_0) = (2b96eca6, 05ef4401),
G[K_30]...G[K_1](a_1, a_0) = (05ef4401, 239a4577),
G[K_31]...G[K_1](a_1, a_0) = (239a4577, c2d8ca3d).
Then the ciphertext is
b = G^*[K_32]G[K_31]...G[K_1](a_1, a_0) = 4ee901e5c2d8ca3d.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 10]
Internet-Draft Abbreviated Title June 2019
A.5. Test Decryption
In this test example, decryption is performed on the round keys
specified in clause A.3. Let the ciphertext be
b = 4ee901e5c2d8ca3d,
then
(b_1, b_0) = (4ee901e5, c2d8ca3d),
G[K_32](b_1, b_0) = (c2d8ca3d, 239a4577),
G[K_31]G[K_32](b_1, b_0) = (239a4577, 05ef4401),
G[K_30]...G[K_32](b_1, b_0) = (05ef4401, 2b96eca6),
G[K_29]...G[K_32](b_1, b_0) = (2b96eca6, 80251e99),
G[K_28]...G[K_32](b_1, b_0) = (80251e99, c78e6d50),
G[K_27]...G[K_32](b_1, b_0) = (c78e6d50, 47b1d5f4),
G[K_26]...G[K_32](b_1, b_0) = (47b1d5f4, b0d66514),
G[K_25]...G[K_32](b_1, b_0) = (b0d66514, 8025c0a5),
G[K_24]...G[K_32](b_1, b_0) = (8025c0a5, 625434c3),
G[K_23]...G[K_32](b_1, b_0) = (625434c3, 11263a21),
G[K_22]...G[K_32](b_1, b_0) = (11263a21, bac8444d),
G[K_21]...G[K_32](b_1, b_0) = (bac8444d, 89cadf21),
G[K_20]...G[K_32](b_1, b_0) = (89cadf21, e7116722),
G[K_19]...G[K_32](b_1, b_0) = (e7116722, e32805bc),
G[K_18]...G[K_32](b_1, b_0) = (e32805bc, 4f15b0bb),
G[K_17]...G[K_32](b_1, b_0) = (4f15b0bb, 2098cd86),
G[K_16]...G[K_32](b_1, b_0) = (2098cd86, fef51b68),
G[K_15]...G[K_32](b_1, b_0) = (fef51b68, 44ca5ce1),
G[K_14]...G[K_32](b_1, b_0) = (44ca5ce1, c4b3edca),
G[K_13]...G[K_32](b_1, b_0) = (c4b3edca, 4811c7ad),
G[K_12]...G[K_32](b_1, b_0) = (4811c7ad, 93c1f449),
G[K_11]...G[K_32](b_1, b_0) = (93c1f449, ce995f2a),
G[K_10]...G[K_32](b_1, b_0) = (ce995f2a, 46324615),
G[K_9]...G[K_32](b_1, b_0) = (46324615, 37d97f25),
G[K_8]...G[K_32](b_1, b_0) = (37d97f25, ad0310a4),
G[K_7]...G[K_32](b_1, b_0) = (ad0310a4, 11fe726d),
G[K_6]...G[K_32](b_1, b_0) = (11fe726d, ea89c02c),
G[K_5]...G[K_32](b_1, b_0) = (ea89c02c, 633a7c68),
G[K_4]...G[K_32](b_1, b_0) = (633a7c68, b14337a5),
G[K_3]...G[K_32](b_1, b_0) = (b14337a5, 28da3b14),
G[K_2]...G[K_32](b_1, b_0) = (28da3b14, 76543210).
Then the plaintext is
a = G^*[K_1]G[K_2]...G[K_32](b_1, b_0) = fedcba9876543210.
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 11]
Internet-Draft Abbreviated Title June 2019
Appendix B. Background
Algoritmically Magma is a variation of block cipher defined in
[RFC5830] ([GOST28147-89]) with the following clarifications and
minor modifications:
1. S-BOX set is fixed at id-tc26-gost-28147-param-Z (See Appendix C
of [RFC7836]);
2. key is parsed as a single big-endian integer (compared to little-
endian approach used in [GOST28147-89]), which results in
different subkey values being used;
3. data are also parsed as single big-endian integer (instead of
being parsed as little-endian integer).
Authors' Addresses
Vasily Dolmatov (editor)
JSC "NPK Kryptonite"
Spartakovskaya sq., 14, bld 2, JSC "NPK Kryptonite"
Moscow 105082
Russian Federation
Email: vdolmatov@gmail.com
Dmitry Eremin-Solenikov
Auriga, Inc
Torfyanaya Doroga, 7F, office 1410
Saint-Petersburg 197374
Russia
Email: dbaryshkov@gmail.com
Dolmatov & Eremin-SoleniExpires December 27, 2019 [Page 12]