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Internet Engineering Task Force (IETF) A. Jivsov
Request for Comments: 6637 Symantec Corporation
Category: Standards Track June 2012
ISSN: 2070-1721
Elliptic Curve Cryptography (ECC) in OpenPGP
Abstract
This document defines an Elliptic Curve Cryptography extension to the
OpenPGP public key format and specifies three Elliptic Curves that
enjoy broad support by other standards, including standards published
by the US National Institute of Standards and Technology. The
document specifies the conventions for interoperability between
compliant OpenPGP implementations that make use of this extension and
these Elliptic Curves.
Status of This Memo
This is an Internet Standards Track document.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Further information on
Internet Standards is available in Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc6637.
Copyright Notice
Copyright (c) 2012 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
(http://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 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.
Jivsov Standards Track [Page 1]
RFC 6637 ECC in OpenPGP June 2012
Table of Contents
1. Introduction ....................................................3
2. Conventions used in This Document ...............................3
3. Elliptic Curve Cryptography .....................................3
4. Supported ECC Curves ............................................3
5. Supported Public Key Algorithms .................................4
6. Conversion Primitives ...........................................4
7. Key Derivation Function .........................................5
8. EC DH Algorithm (ECDH) ..........................................5
9. Encoding of Public and Private Keys .............................8
10. Message Encoding with Public Keys ..............................9
11. ECC Curve OID .................................................10
12. Compatibility Profiles ........................................10
12.1. OpenPGP ECC Profile ......................................10
12.2. Suite-B Profile ..........................................11
12.2.1. Security Strength at 192 Bits .....................11
12.2.2. Security Strength at 128 Bits .....................11
13. Security Considerations .......................................12
14. IANA Considerations ...........................................14
15. References ....................................................14
15.1. Normative References .....................................14
15.2. Informative References ...................................15
16. Contributors ..................................................15
17. Acknowledgment ................................................15
Jivsov Standards Track [Page 2]
RFC 6637 ECC in OpenPGP June 2012
1. Introduction
The OpenPGP protocol [RFC4880] supports RSA and DSA (Digital
Signature Algorithm) public key formats. This document defines the
extension to incorporate support for public keys that are based on
Elliptic Curve Cryptography (ECC).
2. Conventions Used in This Document
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 [RFC2119]. Any
implementation that adheres to the format and methods specified in
this document is called a compliant application. Compliant
applications are a subset of the broader set of OpenPGP applications
described in [RFC4880]. Any [RFC2119] keyword within this document
applies to compliant applications only.
3. Elliptic Curve Cryptography
This document establishes the minimum set of Elliptic Curve
Cryptography (ECC) public key parameters and cryptographic methods
that will likely satisfy the widest range of platforms and
applications and facilitate interoperability. It adds a more
efficient method for applications to balance the overall level of
security with any AES algorithm specified in [RFC4880] than by simply
increasing the size of RSA keys. This document defines a path to
expand ECC support in the future.
The National Security Agency (NSA) of the United States specifies ECC
for use in its [SuiteB] set of algorithms. This document includes
algorithms required by Suite B that are not present in [RFC4880].
[KOBLITZ] provides a thorough introduction to ECC.
4. Supported ECC Curves
This document references three named prime field curves, defined in
[FIPS-186-3] as "Curve P-256", "Curve P-384", and "Curve P-521".
The named curves are referenced as a sequence of bytes in this
document, called throughout, curve OID. Section 11 describes in
detail how this sequence of bytes is formed.
Jivsov Standards Track [Page 3]
RFC 6637 ECC in OpenPGP June 2012
5. Supported Public Key Algorithms
The supported public key algorithms are the Elliptic Curve Digital
Signature Algorithm (ECDSA) [FIPS-186-3] and the Elliptic Curve
Diffie-Hellman (ECDH). A compatible specification of ECDSA is given
in [RFC6090] as "KT-I Signatures" and in [SEC1]; ECDH is defined in
Section 8 of this document.
The following public key algorithm IDs are added to expand Section
9.1 of [RFC4880], "Public-Key Algorithms":
ID Description of Algorithm
-- --------------------------
18 ECDH public key algorithm
19 ECDSA public key algorithm
Compliant applications MUST support ECDSA and ECDH.
6. Conversion Primitives
This document only defines the uncompressed point format. The point
is encoded in the Multiprecision Integer (MPI) format [RFC4880]. The
content of the MPI is the following:
B = 04 || x || y
where x and y are coordinates of the point P = (x, y), each encoded
in the big-endian format and zero-padded to the adjusted underlying
field size. The adjusted underlying field size is the underlying
field size that is rounded up to the nearest 8-bit boundary.
Therefore, the exact size of the MPI payload is 515 bits for "Curve
P-256", 771 for "Curve P-384", and 1059 for "Curve P-521".
Even though the zero point, also called the point at infinity, may
occur as a result of arithmetic operations on points of an elliptic
curve, it SHALL NOT appear in data structures defined in this
document.
This encoding is compatible with the definition given in [SEC1].
If other conversion methods are defined in the future, a compliant
application MUST NOT use a new format when in doubt that any
recipient can support it. Consider, for example, that while both the
public key and the per-recipient ECDH data structure, respectively
defined in Sections 9 and 10, contain an encoded point field, the
format changes to the field in Section 10 only affect a given
recipient of a given message.
Jivsov Standards Track [Page 4]
RFC 6637 ECC in OpenPGP June 2012
7. Key Derivation Function
A key derivation function (KDF) is necessary to implement the EC
encryption. The Concatenation Key Derivation Function (Approved
Alternative 1) [NIST-SP800-56A] with the KDF hash function that is
SHA2-256 [FIPS-180-3] or stronger is REQUIRED. See Section 12 for
the details regarding the choice of the hash function.
For convenience, the synopsis of the encoding method is given below
with significant simplifications attributable to the restricted
choice of hash functions in this document. However, [NIST-SP800-56A]
is the normative source of the definition.
// Implements KDF( X, oBits, Param );
// Input: point X = (x,y)
// oBits - the desired size of output
// hBits - the size of output of hash function Hash
// Param - octets representing the parameters
// Assumes that oBits <= hBits
// Convert the point X to the octet string, see section 6:
// ZB' = 04 || x || y
// and extract the x portion from ZB'
ZB = x;
MB = Hash ( 00 || 00 || 00 || 01 || ZB || Param );
return oBits leftmost bits of MB.
Note that ZB in the KDF description above is the compact
representation of X, defined in Section 4.2 of [RFC6090].
8. EC DH Algorithm (ECDH)
The method is a combination of an ECC Diffie-Hellman method to
establish a shared secret, a key derivation method to process the
shared secret into a derived key, and a key wrapping method that uses
the derived key to protect a session key used to encrypt a message.
The One-Pass Diffie-Hellman method C(1, 1, ECC CDH) [NIST-SP800-56A]
MUST be implemented with the following restrictions: the ECC CDH
primitive employed by this method is modified to always assume the
cofactor as 1, the KDF specified in Section 7 is used, and the KDF
parameters specified below are used.
Jivsov Standards Track [Page 5]
RFC 6637 ECC in OpenPGP June 2012
The KDF parameters are encoded as a concatenation of the following 5
variable-length and fixed-length fields, compatible with the
definition of the OtherInfo bitstring [NIST-SP800-56A]:
o a variable-length field containing a curve OID, formatted as
follows:
- a one-octet size of the following field
- the octets representing a curve OID, defined in Section 11
o a one-octet public key algorithm ID defined in Section 5
o a variable-length field containing KDF parameters, identical to
the corresponding field in the ECDH public key, formatted as
follows:
- a one-octet size of the following fields; values 0 and 0xff
are reserved for future extensions
- a one-octet value 01, reserved for future extensions
- a one-octet hash function ID used with the KDF
- a one-octet algorithm ID for the symmetric algorithm used to
wrap the symmetric key for message encryption; see Section 8
for details
o 20 octets representing the UTF-8 encoding of the string
"Anonymous Sender ", which is the octet sequence
41 6E 6F 6E 79 6D 6F 75 73 20 53 65 6E 64 65 72 20 20 20 20
o 20 octets representing a recipient encryption subkey or a master
key fingerprint, identifying the key material that is needed for
the decryption
The size of the KDF parameters sequence, defined above, is either 54
or 51 for the three curves defined in this document.
The key wrapping method is described in [RFC3394]. KDF produces a
symmetric key that is used as a key-encryption key (KEK) as specified
in [RFC3394]. Refer to Section 13 for the details regarding the
choice of the KEK algorithm, which SHOULD be one of three AES
algorithms. Key wrapping and unwrapping is performed with the
default initial value of [RFC3394].
Jivsov Standards Track [Page 6]
RFC 6637 ECC in OpenPGP June 2012
The input to the key wrapping method is the value "m" derived from
the session key, as described in Section 5.1 of [RFC4880], "Public-
Key Encrypted Session Key Packets (Tag 1)", except that the PKCS #1.5
(Public-Key Cryptography Standards version 1.5) padding step is
omitted. The result is padded using the method described in [PKCS5]
to the 8-byte granularity. For example, the following AES-256
session key, in which 32 octets are denoted from k0 to k31, is
composed to form the following 40 octet sequence:
09 k0 k1 ... k31 c0 c1 05 05 05 05 05
The octets c0 and c1 above denote the checksum. This encoding allows
the sender to obfuscate the size of the symmetric encryption key used
to encrypt the data. For example, assuming that an AES algorithm is
used for the session key, the sender MAY use 21, 13, and 5 bytes of
padding for AES-128, AES-192, and AES-256, respectively, to provide
the same number of octets, 40 total, as an input to the key wrapping
method.
The output of the method consists of two fields. The first field is
the MPI containing the ephemeral key used to establish the shared
secret. The second field is composed of the following two fields:
o a one-octet encoding the size in octets of the result of the key
wrapping method; the value 255 is reserved for future extensions
o up to 254 octets representing the result of the key wrapping
method, applied to the 8-byte padded session key, as described
above
Note that for session key sizes 128, 192, and 256 bits, the size of
the result of the key wrapping method is, respectively, 32, 40, and
48 octets, unless the size obfuscation is used.
For convenience, the synopsis of the encoding method is given below;
however, this section, [NIST-SP800-56A], and [RFC3394] are the
normative sources of the definition.
Jivsov Standards Track [Page 7]
RFC 6637 ECC in OpenPGP June 2012
Obtain the authenticated recipient public key R
Generate an ephemeral key pair {v, V=vG}
Compute the shared point S = vR;
m = symm_alg_ID || session key || checksum || pkcs5_padding;
curve_OID_len = (byte)len(curve_OID);
Param = curve_OID_len || curve_OID || public_key_alg_ID || 03
|| 01 || KDF_hash_ID || KEK_alg_ID for AESKeyWrap || "Anonymous
Sender " || recipient_fingerprint;
Z_len = the key size for the KEK_alg_ID used with AESKeyWrap
Compute Z = KDF( S, Z_len, Param );
Compute C = AESKeyWrap( Z, m ) as per [RFC3394]
VB = convert point V to the octet string
Output (MPI(VB) || len(C) || C).
The decryption is the inverse of the method given. Note that the
recipient obtains the shared secret by calculating
S = rV = rvG, where (r,R) is the recipient's key pair.
Consistent with Section 5.13 of [RFC4880], "Sym. Encrypted Integrity
Protected Data Packet (Tag 18)", a Modification Detection Code (MDC)
MUST be used anytime the symmetric key is protected by ECDH.
9. Encoding of Public and Private Keys
The following algorithm-specific packets are added to Section 5.5.2
of [RFC4880], "Public-Key Packet Formats", to support ECDH and ECDSA.
This algorithm-specific portion is:
Algorithm-Specific Fields for ECDSA keys:
o a variable-length field containing a curve OID, formatted
as follows:
- a one-octet size of the following field; values 0 and
0xFF are reserved for future extensions
- octets representing a curve OID, defined in Section 11
o MPI of an EC point representing a public key
Jivsov Standards Track [Page 8]
RFC 6637 ECC in OpenPGP June 2012
Algorithm-Specific Fields for ECDH keys:
o a variable-length field containing a curve OID, formatted
as follows:
- a one-octet size of the following field; values 0 and
0xFF are reserved for future extensions
- the octets representing a curve OID, defined in
Section 11
- MPI of an EC point representing a public key
o a variable-length field containing KDF parameters,
formatted as follows:
- a one-octet size of the following fields; values 0 and
0xff are reserved for future extensions
- a one-octet value 01, reserved for future extensions
- a one-octet hash function ID used with a KDF
- a one-octet algorithm ID for the symmetric algorithm
used to wrap the symmetric key used for the message
encryption; see Section 8 for details
Observe that an ECDH public key is composed of the same sequence of
fields that define an ECDSA key, plus the KDF parameters field.
The following algorithm-specific packets are added to Section 5.5.3.
of [RFC4880], "Secret-Key Packet Formats", to support ECDH and ECDSA.
Algorithm-Specific Fields for ECDH or ECDSA secret keys:
o an MPI of an integer representing the secret key, which is a
scalar of the public EC point
10. Message Encoding with Public Keys
Section 5.2.2 of [RFC4880], "Version 3 Signature Packet Format"
defines signature formats. No changes in the format are needed for
ECDSA.
Section 5.1 of [RFC4880], "Public-Key Encrypted Session Key Packets
(Tag 1)" is extended to support ECDH. The following two fields are
the result of applying the KDF, as described in Section 8.
Jivsov Standards Track [Page 9]
RFC 6637 ECC in OpenPGP June 2012
Algorithm-Specific Fields for ECDH:
o an MPI of an EC point representing an ephemeral public key
o a one-octet size, followed by a symmetric key encoded using the
method described in Section 8
11. ECC Curve OID
The parameter curve OID is an array of octets that define a named
curve. The table below specifies the exact sequence of bytes for
each named curve referenced in this document:
ASN.1 Object OID Curve OID bytes in Curve name in
Identifier len hexadecimal [FIPS-186-3]
representation
1.2.840.10045.3.1.7 8 2A 86 48 CE 3D 03 01 07 NIST curve P-256
1.3.132.0.34 5 2B 81 04 00 22 NIST curve P-384
1.3.132.0.35 5 2B 81 04 00 23 NIST curve P-521
The sequence of octets in the third column is the result of applying
the Distinguished Encoding Rules (DER) to the ASN.1 Object Identifier
with subsequent truncation. The truncation removes the two fields of
encoded Object Identifier. The first omitted field is one octet
representing the Object Identifier tag, and the second omitted field
is the length of the Object Identifier body. For example, the
complete ASN.1 DER encoding for the NIST P-256 curve OID is "06 08 2A
86 48 CE 3D 03 01 07", from which the first entry in the table above
is constructed by omitting the first two octets. Only the truncated
sequence of octets is the valid representation of a curve OID.
12. Compatibility Profiles
12.1. OpenPGP ECC Profile
A compliant application MUST implement NIST curve P-256, MAY
implement NIST curve P-384, and SHOULD implement NIST curve P-521, as
defined in Section 11. A compliant application MUST implement
SHA2-256 and SHOULD implement SHA2-384 and SHA2-512. A compliant
application MUST implement AES-128 and SHOULD implement AES-256.
Jivsov Standards Track [Page 10]
RFC 6637 ECC in OpenPGP June 2012
A compliant application SHOULD follow Section 13 regarding the choice
of the following algorithms for each curve:
o the KDF hash algorithm
o the KEK algorithm
o the message digest algorithm and the hash algorithm used in the
key certifications
o the symmetric algorithm used for message encryption.
It is recommended that the chosen symmetric algorithm for message
encryption be no less secure than the KEK algorithm.
12.2. Suite-B Profile
A subset of algorithms allowed by this document can be used to
achieve [SuiteB] compatibility. The references to [SuiteB] in this
document are informative. This document is primarily concerned with
format specification, leaving additional security restrictions
unspecified, such as matching the assigned security level of
information to authorized recipients or interoperability concerns
arising from fewer allowed algorithms in [SuiteB] than allowed by
[RFC4880].
12.2.1. Security Strength at 192 Bits
To achieve the security strength of 192 bits, [SuiteB] requires NIST
curve P-384, AES-256, and SHA2-384. The symmetric algorithm
restriction means that the algorithm of KEK used for key wrapping in
Section 8 and an [RFC4880] session key used for message encryption
must be AES-256. The hash algorithm restriction means that the hash
algorithms of KDF and the [RFC4880] message digest calculation must
be SHA-384.
12.2.2. Security Strength at 128 Bits
The set of algorithms in Section 12.2.1 is extended to allow NIST
curve P-256, AES-128, and SHA2-256.
Jivsov Standards Track [Page 11]
RFC 6637 ECC in OpenPGP June 2012
13. Security Considerations
Refer to [FIPS-186-3], B.4.1, for the method to generate a uniformly
distributed ECC private key.
The curves proposed in this document correspond to the symmetric key
sizes 128 bits, 192 bits, and 256 bits, as described in the table
below. This allows a compliant application to offer balanced public
key security, which is compatible with the symmetric key strength for
each AES algorithm allowed by [RFC4880].
The following table defines the hash and the symmetric encryption
algorithm that SHOULD be used with a given curve for ECDSA or ECDH.
A stronger hash algorithm or a symmetric key algorithm MAY be used
for a given ECC curve. However, note that the increase in the
strength of the hash algorithm or the symmetric key algorithm may not
increase the overall security offered by the given ECC key.
Curve name ECC RSA Hash size Symmetric
strength strength, key size
informative
NIST curve P-256 256 3072 256 128
NIST curve P-384 384 7680 384 192
NIST curve P-521 521 15360 512 256
Requirement levels indicated elsewhere in this document lead to the
following combinations of algorithms in the OpenPGP profile: MUST
implement NIST curve P-256 / SHA2-256 / AES-128, SHOULD implement
NIST curve P-521 / SHA2-512 / AES-256, MAY implement NIST curve P-384
/ SHA2-384 / AES-256, among other allowed combinations.
Consistent with the table above, the following table defines the KDF
hash algorithm and the AES KEK encryption algorithm that SHOULD be
used with a given curve for ECDH. A stronger KDF hash algorithm or
AES KEK algorithm MAY be used for a given ECC curve.
Curve name Recommended KDF Recommended KEK
hash algorithm encryption algorithm
NIST curve P-256 SHA2-256 AES-128
NIST curve P-384 SHA2-384 AES-192
NIST curve P-521 SHA2-512 AES-256
Jivsov Standards Track [Page 12]
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This document explicitly discourages the use of algorithms other than
AES as a KEK algorithm because backward compatibility of the ECDH
format is not a concern. The KEK algorithm is only used within the
scope of a Public-Key Encrypted Session Key Packet, which represents
an ECDH key recipient of a message. Compare this with the algorithm
used for the session key of the message, which MAY be different from
a KEK algorithm.
Compliant applications SHOULD implement, advertise through key
preferences, and use in compliance with [RFC4880], the strongest
algorithms specified in this document.
Note that the [RFC4880] symmetric algorithm preference list may make
it impossible to use the balanced strength of symmetric key
algorithms for a corresponding public key. For example, the presence
of the symmetric key algorithm IDs and their order in the key
preference list affects the algorithm choices available to the
encoding side, which in turn may make the adherence to the table
above infeasible. Therefore, compliance with this specification is a
concern throughout the life of the key, starting immediately after
the key generation when the key preferences are first added to a key.
It is generally advisable to position a symmetric algorithm ID of
strength matching the public key at the head of the key preference
list.
Encryption to multiple recipients often results in an unordered
intersection subset. For example, if the first recipient's set is
{A, B} and the second's is {B, A}, the intersection is an unordered
set of two algorithms, A and B. In this case, a compliant
application SHOULD choose the stronger encryption algorithm.
Resource constraints, such as limited computational power, is a
likely reason why an application might prefer to use the weakest
algorithm. On the other side of the spectrum are applications that
can implement every algorithm defined in this document. Most
applications are expected to fall into either of two categories. A
compliant application in the second, or strongest, category SHOULD
prefer AES-256 to AES-192.
SHA-1 MUST NOT be used with the ECDSA or the KDF in the ECDH method.
MDC MUST be used when a symmetric encryption key is protected by
ECDH. None of the ECC methods described in this document are allowed
with deprecated V3 keys. A compliant application MUST only use
iterated and salted S2K to protect private keys, as defined in
Section 3.7.1.3 of [RFC4880], "Iterated and Salted S2K".
Jivsov Standards Track [Page 13]
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Side channel attacks are a concern when a compliant application's use
of the OpenPGP format can be modeled by a decryption or signing
oracle model, for example, when an application is a network service
performing decryption to unauthenticated remote users. ECC scalar
multiplication operations used in ECDSA and ECDH are vulnerable to
side channel attacks. Countermeasures can often be taken at the
higher protocol level, such as limiting the number of allowed
failures or time-blinding of the operations associated with each
network interface. Mitigations at the scalar multiplication level
seek to eliminate any measurable distinction between the ECC point
addition and doubling operations.
14. IANA Considerations
Per this document, IANA has assigned an algorithm number from the
"Public Key Algorithms" range (or the "name space" in the terminology
of [RFC5226]) of the "Pretty Good Privacy (PGP)" registry, created by
[RFC4880]. Two ID numbers have been assigned, as defined in Section
5. The first one, value 19, is already designated for ECDSA and is
currently unused, while the other one, value 18, is new.
15. References
15.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC4880] Callas, J., Donnerhacke, L., Finney, H., Shaw, D.,
and R. Thayer, "OpenPGP Message Format", RFC 4880,
November 2007.
[SuiteB] National Security Agency, "NSA Suite B
Cryptography", March 11, 2010,
http://www.nsa.gov/ia/programs/suiteb_cryptography/.
[FIPS-186-3] National Institute of Standards and Technology, U.S.
Department of Commerce, "Digital Signature
Standard", FIPS 186-3, June 2009.
[NIST-SP800-56A] Barker, E., Johnson, D., and M. Smid,
"Recommendation for Pair-Wise Key Establishment
Schemes Using Discrete Logarithm Cryptography", NIST
Special Publication 800-56A Revision 1, March 2007.
[FIPS-180-3] National Institute of Standards and Technology, U.S.
Department of Commerce, "Secure Hash Standard
(SHS)", FIPS 180-3, October 2008.
Jivsov Standards Track [Page 14]
RFC 6637 ECC in OpenPGP June 2012
[RFC3394] Schaad, J. and R. Housley, "Advanced Encryption
Standard (AES) Key Wrap Algorithm", RFC 3394,
September 2002.
[PKCS5] RSA Laboratories, "PKCS #5 v2.0: Password-Based
Cryptography Standard", March 25, 1999.
[RFC5226] Narten, T. and H. Alvestrand, "Guidelines for
Writing an IANA Considerations Section in RFCs", BCP
26, RFC 5226, May 2008.
15.2. Informative References
[KOBLITZ] N. Koblitz, "A course in number theory and
cryptography", Chapter VI. Elliptic Curves, ISBN:
0-387-96576-9, Springer-Verlag, 1987
[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental
Elliptic Curve Cryptography Algorithms", RFC 6090,
February 2011.
[SEC1] Standards for Efficient Cryptography Group, "SEC 1:
Elliptic Curve Cryptography", September 2000.
16. Contributors
Hal Finney provided important criticism on compliance with
[NIST-SP800-56A] and [SuiteB], and pointed out a few other mistakes.
17. Acknowledgment
The author would like to acknowledge the help of many individuals who
kindly voiced their opinions on the IETF OpenPGP Working Group
mailing list, in particular, the help of Jon Callas, David Crick, Ian
G, Werner Koch, and Marko Kreen.
Author's Address
Andrey Jivsov
Symantec Corporation
EMail: Andrey_Jivsov@symantec.com
Jivsov Standards Track [Page 15]