Secure Shell Working Group J. Green
Internet-Draft Certicom
Expires: April 19, 2007 D. Stebila
U Waterloo
October 16, 2006
Elliptic-Curve Algorithm Integration in the Secure Shell Transport Layer
draft-green-secsh-ecc-01
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Copyright (C) The Internet Society (2006).
Abstract
This document describes algorithms based on Elliptic Curve
Cryptography (ECC) for use within the Secure Shell (SSH) transport
protocol. In particular, it specifies: Elliptic Curve Diffie-Hellman
(ECDH) key agreement, Elliptic Curve Menezes-Qu-Vanstone (ECMQV) key
agreement and Elliptic Curve Digital Signature Algorithm (ECDSA) for
use in the SSH Transport Layer protocol.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. ECC Public Key Algorithm . . . . . . . . . . . . . . . . . . . 5
3.1. Key/Signature Encoding . . . . . . . . . . . . . . . . . . 6
4. ECDH Key Exchange . . . . . . . . . . . . . . . . . . . . . . 7
4.1. Description . . . . . . . . . . . . . . . . . . . . . . . 7
4.2. Implementation . . . . . . . . . . . . . . . . . . . . . . 8
5. ECMQV Key Exchange and Verification . . . . . . . . . . . . . 10
5.1. Description . . . . . . . . . . . . . . . . . . . . . . . 10
5.2. Implementation . . . . . . . . . . . . . . . . . . . . . . 11
6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 14
6.1. ECC Public Key Algorithm Identifiers . . . . . . . . . . . 14
6.2. ECDH Key Exchange Method Names . . . . . . . . . . . . . . 14
6.3. ECMQV Key Exchange and Verification Method Names . . . . . 15
7. Key Exchange Messages . . . . . . . . . . . . . . . . . . . . 16
7.1. ECDH Message Numbers . . . . . . . . . . . . . . . . . . . 16
7.2. ECMQV Message Numbers . . . . . . . . . . . . . . . . . . 16
8. Security Considerations . . . . . . . . . . . . . . . . . . . 17
Appendix A. Named Elliptic Curve Domain Parameters . . . . . . . 18
Appendix A.1. Required and Recommended Curves . . . . . . . . . . 18
Appendix A.2. SEC Equivalent NIST Curves and OIDs . . . . . . . . 18
9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 19
10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1. Normative References . . . . . . . . . . . . . . . . . . . 20
10.2. Informative References . . . . . . . . . . . . . . . . . . 21
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 22
Intellectual Property and Copyright Statements . . . . . . . . . . 23
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1. Introduction
Due to its inclusion in NSA's Suit B and its small key sizes elliptic
curve cryptography (ECC) is becoming a widely utilized and attractive
public-key cryptosystem.
In the interest of adding Suit B algorithms to SSH this document adds
three ECC Suit B algorithms to the Secure Shell arsenal: Elliptic
Curve Menezes-Qu-Vanstone (ECMQV), Elliptic Curve Diffie-Hellman
(ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) as well
as utilizing the SHA2 family of secure hash algorithms.
Compared to cryptosystems such as RSA, DSA, and DH, ECC variations on
these schemes offer equivalent security with smaller key sizes. This
is illustrated in the following table, based on Section 5.6.1 of NIST
800-57 [9], which gives approximate comparable key sizes for
symmetric- and asymmetric-key cryptosystems based on the best known
algorithms for attacking them. L is field size and N is sub-field
size.
+-----------+-----------------------------+-------+---------+
| Symmetric | Discrete Log (eg. DSA, DH) | RSA | ECC |
+-----------+-----------------------------+-------+---------+
| 80 | L = 1024 N = 160 | 1024 | 160-223 |
| | | | |
| 112 | L = 2048 N = 256 | 2048 | 224-255 |
| | | | |
| 128 | L = 3072 N = 256 | 3072 | 256-383 |
| | | | |
| 192 | L = 7680 N = 384 | 7680 | 384-511 |
| | | | |
| 256 | L = 15360 N = 512 | 15360 | 512+ |
+-----------+-----------------------------+-------+---------+
Figure 1: Comparable key sizes (in bits).
Smaller key sizes result in power, bandwidth, and computational
savings that make ECC especially attractive for constrained
environments.
Implementation of this specification requires familiarity with both
SSH [2] [3] [4] and ECC [6] [10] [11].
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2. Notation
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 [1].
The data types boolean, uint32, uint64, string, and mpint are to be
interpreted in this document as described in RFC 4251 [2].
The size of a set of elliptic curve domain parameters on a prime
curve is defined as the number of bits in the binary representation
of the field order, commonly denoted p. Size on a characteristric-2
curve is defined as the number of bits in the binary representation
of the field, commonly denoted m.
Protocol fields and possible values to fill them in are defined in
this set of documents. As an example SSH_MSG_KEX_ECDH_INIT is
defined as follows.
byte SSH_MSG_KEX_ECDH_INIT
string pK, public key octet string.
Throughout these documents, when the fields are referenced, they will
appear within single quotes. When values to fill in those fields are
referenced they will appear in double quotes.
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3. ECC Public Key Algorithm
The ECC public key algorithm defines ECC public keys, private keys
and signatures for use within the SSH protocol. When ECC is chosen
as the public key algorithm through key exchange negotiations the
server's long term ECC key pair, which are called host keys
throughout this draft, is used for identification and verification.
When asked to sign communications or verify signatures by the key
exchange method the elliptic curve digital signature algorithm
(ECDSA) is used. The algorithmic details of ECDSA can be found in
Section 4 of SEC 1 [6] and in [13]. This document is concerned with
the SSH implementation details; specification of the algorithm is
left to other standards documents.
The message hashing algorithm to be used with ECDSA is the same one
specified to generate the exchange hash in the key exchange method
chosen during algorithm negotiation. If the chosen key exchange
method doesn't specify a hashing function then SHA-256 [5] will be
used.
The algorithm for ECC key generation can be found in section 3.2 of
SEC 1 [6]. Given some elliptic curve domain parameters, an ECC key
pair can be generated containing an integer d that makes up the
secret key and an elliptic curve point Q which makes up the public
key.
The elliptic curve domain parameters to generate EC keys can be
produced at random, but this is a costly operation and may result in
the use of domain parameters of which the security hasn't been
tested. Because of this, standards bodies have produced named sets
of elliptic curve domain parameters or named curves. This public key
algorithm only allows named curves, specified by their ASN.1 OIDs, to
be used. Details of the curves that are required and recommended are
outlined in Appendix A.
The family of public key algorithm identifiers for ECC is specified
in Section 6.1. For the remainder of this section [identifier] will
represent the ECC Public Key Algorithm identifier determined in
Section 6.1.
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3.1. Key/Signature Encoding
The ecc public key has the following encoding:
string [identifier]
string Q
Where Q is the elliptic curve point that forms the public key. Here
Q is encoded from an elliptic curve point into an octet string as
defined in Section 2.3.3 of SEC1 [6].
The ECDSA signature has the following encoding:
string [identifier]
mpint r
mpint s
Where the integers R and S are the output of the ECDSA algorithm.
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4. ECDH Key Exchange
4.1. Description
The Elliptic Curve Diffie-Hellman (ECDH) key exchange algorithm
generates a shared secret from an ephemeral elliptic curve private
key contributed by the local entity and an ephemeral elliptic curve
public key contributed by the remote entity. When this algorithm is
applied on both the client and the server both will derive the same
shared key.
The family of key exchange method names defined for use with this key
exchange can be found in Section 6.2.
The rest of this section is an informal overview of the ECDH key
exchange mechanism. A formal description can be found in
Section 4.2.
In the following: C is the client, S is the server; NC is the named
curve specified in the algorithm name; cPK is the client's ephemeral
public key; sPK is the server's ephemeral public key; K_S is the
server's public host key; H is the exchange hash; s is the signature
on H; and K is the shared secret.
1. C generates an ephemeral elliptic curve key pair on NC and sends
the public-key value 'cPK'.
2. S SHOULD validate 'cPK' using, for example, algorithm A.16.10 of
[10]. S generates an ephemeral public-key / private-key pair on
NC. S uses 'cPK' and S's ephemeral private key value to compute
the shared secret K using ECDH. S computes H and the signature
's' on H using its private host key. S sends "K_S || sPK || s".
3. C SHOULD validate 'sPK', for example, algorithm A.16.10 of [10].
C verifies that 'K_S' really is the public host key for S (e.g.,
using certificates or a local database). C is also allowed to
accept the key without verification; however, doing so will
render the protocol insecure against active attacks. C uses
'sPK' and C's ephemeral private key to compute the shared secret
and verifies the signature 's' on H.
The size of the named curve specified in the algorithm name SHOULD be
greater than 160 bits and preferably at least 224 bits. The size of
the curve used SHOULD be in line with the bulk encryption algorithm
chosen during algorithm negotiation.
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4.2. Implementation
This document is concerned with describing the implementation of ECDH
in SSH, not the specification of the algorithm itself. The algorithm
used for shared key generation is ECDH, the full specification of
which can be found in Section 3.3.2 of SEC1 [6].
The algorithm for generation of EC key pairs can be found in Section
3.2 of SEC1 [6]. This algorithm is necessary to generate the
ephemeral EC key pairs needed for ECDH.
The elliptic curve points (ephemeral public keys) that must be
transmitted are encoded into octet strings before they are
transmitted. The transformation between elliptic curve points and
octet strings is specified in SEC1 Section 2.3 [6].
The hash algorithm HASH for computing the exchange hash is determined
by the size of the named curve specified in the method name. The
method for choosing a hash function can be seen in the table below
where b is the size of the curve [5]:
+----------------+----------------+
| Curve Size | Hash Algorithm |
+----------------+----------------+
| b <= 256 | SHA-256 |
| | |
| 256 < b <= 384 | SHA-384 |
| | |
| 384 < b | SHA-512 |
+----------------+----------------+
Specification of the message numbers SSH_MSG_KEX_ECDH_INIT and
SSH_MSG_KEX_ECDH_REPLY are found in Section 7.
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The ECDH key exchange algorithm is implemented with the following
messages. The public key algorithm for signing is negotiated with
the KEXINIT messages.
The client sends:
byte SSH_MSG_KEX_ECDH_INIT
string cPK, the octet string formed from the
client's ephemeral public key.
The server responds with:
byte SSH_MSG_KEX_ECDH_REPLY
string K_S, server public host key and/or certificates
string sPK, the octet string formed from the server's
ephemeral public key.
string s, the signature of H
The exchange hash H is computed as the HASH of the concatenation of
the following.
string V_C, the client's version string (CR and NL excluded)
string V_S, the server's version string (CR and NL excluded)
string I_C, the payload of the client's SSH_MSG_KEXINIT
string I_S, the payload of the server's SSH_MSG_KEXINIT
string K_S, the server's public host key
string cPK, the client's ephemeral public key octet string.
string sPK, the server's ephemeral public key octet string.
mpint K, the shared secret
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5. ECMQV Key Exchange and Verification
5.1. Description
The Elliptic Curve Menezes-Qu-Vanstone (ECMQV) key exchange algorithm
generates a shared secret from two elliptic curve key pairs owned by
one entity and two elliptic curve public keys owned by another
entity. Both entities' roles are analogous in the algorithm. Using
it's key pairs and the other entity's public keys, both entities'
will derive the same secret.
The family of key exchange method names defined for use with this key
exchange can be found in Section 6.3.
The following is an informal discussion of ECMQV key exchange and
verification for a formal description see Section 5.2.
The client doesn't necessarily have a long term key under the
existing SSH protocol. Instead of generating two ephemeral key pairs
the client generates one ephemeral key pair and uses it for all the
keys it is required to supply. This is illustrated below considering
the ECMQV algorithm in terms of a black box:
Local Keys |-----|-|-----| Remote Keys
ECMQV Block: _______
Key Pair ----| |---- Public Key
| ECMQV |
Key Pair ----|_______|---- Public Key
|
shared secret
Server Configuration: _______
Server Host Key ----| |----
Pair | ECMQV | |-- Client Ephemeral
Server Ephemeral Key ----|_______|---- Public Key
Pair |
shared secret
Client Configuration: _______
----| |---- Server Public Host Key
Client Ephemeral --| | ECMQV |
Key Pair ----|_______|---- Server Ephemeral Public
| Key
shared secret
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In the following: C is the client; S is the server; NC is the curve
from the method name; K_S is the server's public host key; sPK is the
server's ephemeral public key; cPK is the client's ephemeral public
key; H is the exchange hash; T is the HMAC tag of H; and K is the
shared secret.
1. C uses the domain parameters associated with NC to generate an
ephemeral ECC public/private key pair. C sends 'cPK'.
2. S SHOULD validate 'cPK' using, for example, algorithm A.16.10 of
[10]. S generates an ephemeral public/private key pair using NC.
S generates K using its private host key, its private ephemeral
key and 'cPK'. S computes H and the message authentication code
(MAC) tag 'T' on H using K as the key. S then sends "K_S || sPK
|| T".
3. C SHOULD validate 'sPK' and 'K_S' as above. C verifies that
'K_S' really is the public host key for S (e.g., using
certificates or a local database). C is also allowed to accept
the key without verification; however, doing so will render the
protocol insecure against active attacks. C generates K using
its private ephemeral key, 'sPK' and 'K_S'. C independently
computes H and verifies the tag 'T' is valid. We use a MAC tag
with K as a key for verification of communication with S because
the private host key was used to create K, eliminating the need
for a costly signing algorithm.
5.2. Implementation
This document is concerned with describing the implementation of
ECMQV in SSH, not the specification of the algorithm itself. A full
description of ECMQV can be found in Section 3.4 of SEC1 [6].
An implementation of SSH supporting ECMQV MUST support the ECC Public
Key Algorithm (Section 3). If during the key exchange algorithm
negotiation ECMQV is chosen as the key exchange algorithm then the
ECC Public Key Algorithm MUST be selected as the server host key
algorithm. This is due to the fact that ECMQV uses the servers ECC
host keys within the key exchange.
The server's host keys are used in shared key generation; therefore,
the named curve used to generate them is the curve that must be used
by the ECMQV algorithm. Care should be taken that the server's host
key is sufficiently strong to ensure that the ECMQV algorithm isn't
the weakest point in the SSH encryption suite.
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The elliptic curve points (public keys) that must be transmitted are
encoded into octet strings before they are transmitted. The
transformation between elliptic curve points and octet strings is
specified in SEC1 Section 2.3 [6].
The hash algorithm HASH for computing the exchange hash is determined
by the strength of the named curve used by ECMQV. The method for
choosing a hash function can be seen in the table below where b is
the size of the curve [5]:
+----------------+----------------+
| Curve Size | Hash Algorithm |
+----------------+----------------+
| b <= 256 | SHA-256 |
| | |
| 256 < b <= 384 | SHA-384 |
| | |
| 384 < b | SHA-512 |
+----------------+----------------+
The hash function chosen above is also the hash function that is used
to implement the HMAC used for server identification and
verification. The algorithm for implementing HMAC with any of the
above hash functions can be found in [5].
The specification of the message numbers SSH_MSG_ECMQV_INIT and
SSH_MSG_ECMQV_REPLY can be found in Section 7.
The algorithm for generation of EC key pairs can be found in SEC1
Section 3.2 [6]. This algorithm is necessary to generate the
ephemeral EC key pairs needed for ECMQV.
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This key exchange algorithm is implemented with the following
messages.
The client sends:
byte SSH_MSG_ECMQV_INIT
string cPK, The octet string formed from the client's
ephemeral public key.
The server sends:
byte SSH_MSG_ECMQV_REPLY
string K_S, Server public host key octet string
string sPK, Server ephemeral public key octet string
string T, The HMAC tag computed on H using the shared secret
The hash H is formed by applying the algorithm HASH on a
concatenation of the following:
string V_C, the client's version string (CR and NL excluded)
string V_S, the server's version string (CR and NL excluded)
string I_C, the payload of the client's SSH_MSG_KEXINIT
string I_S, the payload of the server's SSH_MSG_KEXINIT
string cPK, client's ephemeral public key octet
string K_S, server's public host key octet
string sPK, server's ephemeral public key octet
mpint K, the shared secret
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6. IANA Considerations
This document defines two new families of key exchange method names
and one new family of public key algorithm name in the SSH name
registry. These additions to the SSH name space will have to be
approved the IANA. The specification of these families is found
below.
6.1. ECC Public Key Algorithm Identifiers
The ECC Public Key Algorithm specifies a family of identifiers. The
general format for this family of identifiers is the string "secg-
ecc-" concatenated with the ASN.1 OID, in dotted decimal format, of
the named curve domain parameters that are associated with the
server's ECC host keys [8].
There are two other identifiers in the ECC Public Key family that are
only to be used only by the client during algorithm negotiation in
SSH_MSG_KEXINIT messages. These identifiers can: not be used at all,
used on their own, or used in concert with normal "secg-ecc-[oid]"
identifiers to specify additional supported named curves that the are
not included in the special identifiers.
"secg-ecc-standard" tells the server that the clients local security
policy has not disabled any of the required curves specified in
Appendix A and is analogous to sending "secg-ecc-1.3.132.0.37,secg-
ecc-1.3.132.0.34,secg-ecc-1.3.132.0.16,secg-ecc-1.2.840.10045.3.1.7".
"secg-ecc-extended" tells the server that the client supports all of
the curves specified in Appendix A and is analogous to sending "secg-
ecc-1.3.132.0.38,secg-ecc-1.3.132.0.35,secg-ecc-1.3.132.0.37,secg-
ecc-1.3.132.0.36,secg-ecc-1.3.132.0.34,secg-ecc-1.3.132.0.16,secg-
ecc-1.2.840.10045.3.1.7,secg-ecc-1.3.132.0.27,secg-ecc-
1.3.132.0.26,secg-ecc-1.3.132.0.33,secg-ecc-1.2.840.10045.3.1.1,secg-
ecc-1.3.132.0.1".
6.2. ECDH Key Exchange Method Names
The Elliptic Curve Diffie-Hellman key exchange is defined by a family
of method names. Each method name consists of the string "ecdh-
sha2-" concatenated with the ASN.1 OID of the named curve, in dotted
decimal notation, to be used for ephemeral key generation within the
key exchange algorithm [8]. Information on the named curves that are
required and recommended can be found in Appendix A.
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6.3. ECMQV Key Exchange and Verification Method Names
The Elliptic Curve Menezes-Qu-Vanstone key exchange is defined by a
family of method names that specify what named curve will be used
within the key exchange. Each method name consists of the string
"ecmqv-sha2-" concatenated with the ASN.1 OID of the named curve to
be used in dotted decimal notation. Information on the named curves
that are required and recommended can be found in Appendix A [8].
When the server forms its 'server_host_key_algorithms' name-list it
MUST include only the algorithm from the "ecmqv-sha2-*" family that
corresponds to the named curve used to produce its ECC host keys. If
the server doesn't have an ECC host key then the server MUST NOT put
any members of the "ecmqv-sha2-*" family of algorithms in the
'server_host_key_algorithms' name-list.
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7. Key Exchange Messages
The message numbers 30-49 are key exchange-specific and in a private
namespace defined in RFC4250 [4] that may be redefined by any key
exchange method [3] without being granted IANA permission.
The following message numbers have been defined in this document:
7.1. ECDH Message Numbers
#define SSH_MSG_KEX_ECDH_INIT 30
#define SSH_MSG_KEX_ECDH_REPLY 31
7.2. ECMQV Message Numbers
#define SSH_MSG_ECMQV_INIT 30
#define SSH_MSG_ECMQV_REPLY 31
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8. Security Considerations
The Elliptic Curve Diffie-Hellman key agreement algorithm is defined
in [6], [10] and [11]. The appropriate security considerations of
those documents apply.
The Elliptic Curve Menezes-Qu-Vanstone key agreement algorithm is
defined in [6]. The security considerations raised in that document
also apply. A more detailed discussion of security considerations
can be found in The Guide to Elliptic Curve Cryptography section 4.7
[14].
The methods defined in Section 4 rely on the SHA family of hashing
functions as defined in [15]. The appropriate security
considerations of that document apply.
Additionally a good general discussion of the security considerations
that must be taken into account when creating an ECC implementation
can be found in The Guide to Elliptic Curve Cryptography section 5
[14].
Since ECDH and ECMQV allow for elliptic curves of arbitrary sizes and
thus arbitrary security strength, it is important that the size of
elliptic curve be chosen to match the security strength of other
elements of the SSH handshake. In particular, host key sizes,
hashing algorithms and bulk encryption algorithms must be chosen
appropriately. Information regarding estimated equivalence of key
sizes is available in [9].
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Appendix A. Named Elliptic Curve Domain Parameters
Implementations may support any ASN.1 object identifier (OID) in the
ASN.1 object tree that defines a set of elliptic curve domain
parameters [8].
Appendix A.1. Required and Recommended Curves
Every SSH ECC implementation MUST support the named curves below,
these curves are defined in SEC2 [7]. These curves should always be
enabled unless specifically disabled by local security policy.
secp256r1 sect283k1 secp384r1 sect409k1
It is RECOMMENDED that SSH ECC implementations also support the
following curves.
sect163k1 secp192r1 sect233k1 secp224r1
sect233r1 sect409r1 secp521r1 sect571k1
Appendix A.2. SEC Equivalent NIST Curves and OIDs
+-----------+----------+---------------------+
| SEC | NIST[12] | OID[7] |
+-----------+----------+---------------------+
| sect163k1 | nistk163 | 1.3.132.0.1 |
| | | |
| secp192r1 | nistp192 | 1.2.840.10045.3.1.1 |
| | | |
| secp224r1 | nistp224 | 1.3.132.0.33 |
| | | |
| sect233k1 | nistk233 | 1.3.132.0.26 |
| | | |
| sect233r1 | nistb233 | 1.3.132.0.27 |
| | | |
| secp256r1 | nistp256 | 1.2.840.10045.3.1.7 |
| | | |
| sect283k1 | nistk283 | 1.3.132.0.16 |
| | | |
| secp384r1 | nistp384 | 1.3.132.0.34 |
| | | |
| sect409k1 | nistk409 | 1.3.132.0.36 |
| | | |
| sect409r1 | nistb409 | 1.3.132.0.37 |
| | | |
| secp521r1 | nistp521 | 1.3.132.0.35 |
| | | |
| sect571k1 | nistk571 | 1.3.132.0.38 |
+-----------+----------+---------------------+
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9. Acknowledgements
Douglas Stebila wishes to thank Sheueling Chang and Vipul Gupta of
Sun Microsystems. The work on draft-stebila-secsh-ecdh-01, of which
this work is a derivative document, was largely performed during a
student internship at Sun Microsystems Laboratories from the
University of Waterloo.
Jon Green would like to thank Robert Lambert of Certicom for all the
help and knowledge he provided. This document was written during an
internship at Certicom from Queens University.
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10. References
10.1. Normative References
[1] Bradner, S., "Key Words for Use in RFCs to Indicate Requirement
Levels", RFC 2119, March 1997.
[2] Ylonen, T. and C. Lonvick, Ed., "The Secure Shell Protocol
Architecture", RFC 4251, January 2006.
[3] Ylonen, T. and C. Lonvick, Ed., "The Secure Shell Transport
Layer Protocol", RFC 4253, January 2006.
[4] Lehtinen, S. and C. Lonvick, Ed., "The Secure Shell Protocol
Assigned Numbers", RFC 4250, January 2006.
[5] Eastlake, 3rd, D. and T. Hansen, "US Secure Hash Algorithms (SHA
and HMAC-SHA)", RFC 4634, July 2006.
[6] Standards for Efficient Cryptography Group, "Elliptic Curve
Cryptography", SEC 1 v1.0, September 2000.
[7] Standards for Efficient Cryptography Group, "Recommended
Elliptic Curve Domain Parameters", SEC 2 v1.0, September 2000.
[8] International Telecommunication Union, "Abstract Syntax Notation
One (ASN.1): Specification of basic notation", X.680 ,
July 2002.
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10.2. Informative References
[9] National Institute of Standards and Technology, "Recommendation
for Key Management - Part 1", NIST Special Publication 800-57.
[10] Institute of Electrical and Electronics Engineers, "Standard
Specifications for Public Key Cryptography", IEEE 1363, 2000.
[11] American National Standards Institute, "Public Key Cryptography
For The Financial Services Industry: Key Agreement and key
Transport Using Elliptic Curve Cryptography", ANSI X9.63,
November 2001.
[12] National Institute of Standards and Technology, "Recommended
Elliptic Curves for Federal Government Use", August 1999.
[13] American National Standards Institute, "Public Key Cryptography
For The Financial Services Industry The Elliptic Curve Digital
Signature Algorithm", ANSI X9.62, 1998.
[14] Hankerson, Menezes, and Vanstone, "Guide to Elliptic Curve
Cryptography", 2004, <urn:isbn:038795273X>.
[15] National Institute of Standards and Technology, "Secure Hash
Standard", FIPS 180-2, August 2002.
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Authors' Addresses
Jon Green
Certicom
5520 Explorer Drive
4th Floor
Mississauga, ON L4W 5L1
Canada
Email: jgreen@certicom.com
Douglas Stebila
Department of Combinatorics and Optimization
University of Waterloo
Waterloo, ON N2L 3G1
Canada
Email: douglas@stebila.ca
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