ECC in SSH: Elliptic Curve Algorithm Integration in the Secure Shell Transport Layer

Network Working Group                                         D. Stebila
Request for Comments: 5656           Queensland University of Technology
Category: Standards Track                                       J. Green
                                                      Queen's University
                                                           December 2009


Elliptic Curve Algorithm Integration in the Secure Shell Transport Layer

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.

Status of This Memo

   This document specifies an Internet standards track protocol for the
   Internet community, and requests discussion and suggestions for
   improvements.  Please refer to the current edition of the "Internet
   Official Protocol Standards" (STD 1) for the standardization state
   and status of this protocol.  Distribution of this memo is unlimited.

Copyright Notice

   Copyright (c) 2009 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 BSD License.

   This document may contain material from IETF Documents or IETF
   Contributions published or made publicly available before November
   10, 2008.  The person(s) controlling the copyright in some of this
   material may not have granted the IETF Trust the right to allow
   modifications of such material outside the IETF Standards Process.
   Without obtaining an adequate license from the person(s) controlling
   the copyright in such materials, this document may not be modified
   outside the IETF Standards Process, and derivative works of it may
   not be created outside the IETF Standards Process, except to format
   it for publication as an RFC or to translate it into languages other
   than English.

Table of Contents

   1. Introduction ....................................................3
   2. Notation ........................................................4
   3. SSH ECC Public Key Algorithm ....................................4
      3.1. Key Format .................................................4
           3.1.1. Signature Algorithm .................................5
           3.1.2. Signature Encoding ..................................5
   4. ECDH Key Exchange ...............................................5
   5. ECMQV Key Exchange ..............................................8
   6. Method Names ...................................................10
      6.1. Elliptic Curve Domain Parameter Identifiers ...............10
      6.2. ECC Public Key Algorithm (ecdsa-sha2-*) ...................11
           6.2.1. Elliptic Curve Digital Signature Algorithm .........11
      6.3. ECDH Key Exchange Method Names (ecdh-sha2-*) ..............12
      6.4. ECMQV Key Exchange and Verification Method Name
           (ecmqv-sha2) ..............................................12
   7. Key Exchange Messages ..........................................13
      7.1. ECDH Message Numbers ......................................13
      7.2. ECMQV Message Numbers .....................................13
   8. Manageability Considerations ...................................13
      8.1. Control of Function through Configuration and Policy ......13
      8.2. Impact on Network Operation ...............................14
   9. Security Considerations ........................................14
   10. Named Elliptic Curve Domain Parameters ........................16
      10.1. Required Curves ..........................................16
      10.2. Recommended Curves .......................................17
   11. IANA Considerations ...........................................17
   12. References ....................................................18
      12.1. Normative References .....................................18
      12.2. Informative References ...................................19
   Appendix A.  Acknowledgements .....................................20

1.  Introduction

   This document adds the following elliptic curve cryptography
   algorithms to the Secure Shell arsenal: Elliptic Curve Diffie-Hellman
   (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA), as
   well as utilizing the SHA2 family of secure hash algorithms.
   Additionally, support is provided for Elliptic Curve Menezes-Qu-
   Vanstone (ECMQV).

   Due to its small key sizes and its inclusion in the National Security
   Agency's Suite B, Elliptic Curve Cryptography (ECC) is becoming a
   widely utilized and attractive public-key cryptosystem.

   Compared to cryptosystems such as RSA, the Digital Signature
   Algorithm (DSA), and Diffie-Hellman (DH) key exchange, 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 [NIST-800-57], which gives approximate comparable key
   sizes for symmetric- and asymmetric-key cryptosystems based on the
   best known algorithms for attacking them.  L is the field size and N
   is the sub-field size.

      +-----------+------------------------------+-------+---------+
      | Symmetric | Discrete Log (e.g., 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+  |
      +-----------+------------------------------+-------+---------+

   Implementation of this specification requires familiarity with both
   SSH [RFC4251] [RFC4253] [RFC4250] and ECC [SEC1] (additional
   information on ECC available in [HMV04], [ANSI-X9.62], and
   [ANSI-X9.63]).

   This document is concerned with SSH implementation details;
   specification of the underlying cryptographic algorithms is left to
   other standards documents.

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 [RFC2119].

   The data types boolean, byte, uint32, uint64, string, and mpint are
   to be interpreted in this document as described in [RFC4251].

   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 by p.  Size on a
   characteristic-2 curve is defined as the number of bits in the binary
   representation of the field, commonly denoted by m.  A set of
   elliptic curve domain parameters defines a group of order n generated
   by a base point P.

3.  SSH ECC Public Key Algorithm

   The SSH ECC public key algorithm is defined by its key format,
   corresponding signature algorithm ECDSA, signature encoding, and
   algorithm identifiers.

   This section defines the family of "ecdsa-sha2-*" public key formats
   and corresponding signature formats.  Every compliant SSH ECC
   implementation MUST implement this public key format.

3.1.  Key Format

   The "ecdsa-sha2-*" key formats all have the following encoding:

      string   "ecdsa-sha2-[identifier]"
      byte[n]  ecc_key_blob

   The ecc_key_blob value has the following specific encoding:

      string   [identifier]
      string   Q

   The string [identifier] is the identifier of the elliptic curve
   domain parameters.  The format of this string is specified in
   Section 6.1.  Information on the REQUIRED and RECOMMENDED sets of
   elliptic curve domain parameters for use with this algorithm can be
   found in Section 10.

   Q is the public key encoded from an elliptic curve point into an
   octet string as defined in Section 2.3.3 of [SEC1]; point compression
   MAY be used.

   The algorithm for ECC key generation can be found in Section 3.2 of
   [SEC1].  Given some elliptic curve domain parameters, an ECC key pair
   can be generated containing a private key (an integer d), and a
   public key (an elliptic curve point Q).

3.1.1.  Signature Algorithm

   Signing and verifying is done using the Elliptic Curve Digital
   Signature Algorithm (ECDSA).  ECDSA is specified in [SEC1].  The
   message hashing algorithm MUST be from the SHA2 family of hash
   functions [FIPS-180-3] and is chosen according to the curve size as
   specified in Section 6.2.1.

3.1.2.  Signature Encoding

   Signatures are encoded as follows:

      string   "ecdsa-sha2-[identifier]"
      string   ecdsa_signature_blob

   The string [identifier] is the identifier of the elliptic curve
   domain parameters.  The format of this string is specified in
   Section 6.1.  Information on the REQUIRED and RECOMMENDED sets of
   elliptic curve domain parameters for use with this algorithm can be
   found in Section 10.

   The ecdsa_signature_blob value has the following specific encoding:

      mpint    r
      mpint    s

   The integers r and s are the output of the ECDSA algorithm.

   The width of the integer fields is determined by the curve being
   used.  Note that the integers r and s are integers modulo the order
   of the cryptographic subgroup, which may be larger than the size of
   the finite field.

4.  ECDH Key Exchange

   The Elliptic Curve Diffie-Hellman (ECDH) key exchange method
   generates a shared secret from an ephemeral local elliptic curve
   private key and ephemeral remote elliptic curve public key.  This key
   exchange method provides explicit server authentication as defined in
   [RFC4253] using a signature on the exchange hash.  Every compliant
   SSH ECC implementation MUST implement ECDH key exchange.

   The primitive used for shared key generation is ECDH with cofactor
   multiplication, the full specification of which can be found in
   Section 3.3.2 of [SEC1].  The algorithm for key pair generation can
   be found in Section 3.2.1 of [SEC1].

   The family of key exchange method names defined for use with this key
   exchange can be found in Section 6.3.  Algorithm negotiation chooses
   the public key algorithm to be used for signing and the method name
   of the key exchange.  The method name of the key exchange chosen
   determines the elliptic curve domain parameters and hash function to
   be used in the remainder of this section.

   Information on the REQUIRED and RECOMMENDED elliptic curve domain
   parameters for use with this method can be found in Section 10.

   All elliptic curve public keys MUST be validated after they are
   received.  An example of a validation algorithm can be found in
   Section 3.2.2 of [SEC1].  If a key fails validation, the key exchange
   MUST fail.

   The elliptic curve public keys (points) 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 Sections 2.3.3 and 2.3.4 of [SEC1]; point compression
   MAY be used.  The output of shared key generation is a field element
   xp.  The SSH framework requires that the shared key be an integer.
   The conversion between a field element and an integer is specified in
   Section 2.3.9 of [SEC1].

   Specification of the message numbers SSH_MSG_KEX_ECDH_INIT and
   SSH_MSG_KEX_ECDH_REPLY is found in Section 7.

   The following is an overview of the key exchange process:

      Client                                                Server
      ------                                                ------
      Generate ephemeral key pair.
      SSH_MSG_KEX_ECDH_INIT  -------------->

                                      Verify received key is valid.
                                       Generate ephemeral key pair.
                                             Compute shared secret.
                                   Generate and sign exchange hash.
                             <------------- SSH_MSG_KEX_ECDH_REPLY

      Verify received key is valid.
      *Verify host key belongs to server.
      Compute shared secret.
      Generate exchange hash.
      Verify server's signature.

      *  It is RECOMMENDED that the client verify that the host key sent
         is the server's host key (for example, using a local database).
         The client MAY accept the host key without verification, but
         doing so will render the protocol insecure against active
         attacks; see the discussion in Section 4.1 of [RFC4251].

   This is implemented using the following messages.

   The client sends:

      byte     SSH_MSG_KEX_ECDH_INIT
      string   Q_C, client's ephemeral public key octet string

   The server responds with:

      byte     SSH_MSG_KEX_ECDH_REPLY
      string   K_S, server's public host key
      string   Q_S, server's ephemeral public key octet string
      string   the signature on the exchange hash
   The exchange hash H is computed as the hash of the concatenation of
   the following.

      string   V_C, client's identification string (CR and LF excluded)
      string   V_S, server's identification string (CR and LF excluded)
      string   I_C, payload of the client's SSH_MSG_KEXINIT
      string   I_S, payload of the server's SSH_MSG_KEXINIT
      string   K_S, server's public host key
      string   Q_C, client's ephemeral public key octet string
      string   Q_S, server's ephemeral public key octet string
      mpint    K,   shared secret

5.  ECMQV Key Exchange

   The Elliptic Curve Menezes-Qu-Vanstone (ECMQV) key exchange algorithm
   generates a shared secret from two local elliptic curve key pairs and
   two remote public keys.  This key exchange method provides implicit
   server authentication as defined in [RFC4253].  The ECMQV key
   exchange method is OPTIONAL.

   The key exchange method name defined for use with this key exchange
   is "ecmqv-sha2".  This method name gives a hashing algorithm that is
   to be used for the Hashed Message Authentication Code (HMAC) below.
   Future RFCs may define new method names specifying new hash
   algorithms for use with ECMQV.  More information about the method
   name and HMAC can be found in Section 6.4.

   In general, the ECMQV key exchange is performed using the ephemeral
   and long-term key pair of both the client and server, which is a
   total of 4 keys.  Within the framework of SSH, the client does not
   have a long-term key pair that needs to be authenticated.  Therefore,
   we generate an ephemeral key and use that as both the clients keys.
   This is more efficient than using two different ephemeral keys, and
   it does not adversely affect security (it is analogous to the one-
   pass protocol in Section 6.1 of [LMQSV98]).

   A full description of the ECMQV primitive can be found in Section 3.4
   of [SEC1].  The algorithm for key pair generation can be found in
   Section 3.2.1 of [SEC1].

   During algorithm negotiation with the SSH_MSG_KEXINIT messages, the
   ECMQV key exchange method can only be chosen if a public key
   algorithm supporting ECC host keys can also be chosen.  This is due
   to the use of implicit server authentication in this key exchange
   method.  This case is handled the same way that key exchange methods
   requiring encryption/signature capable public key algorithms are
   handled in Section 7.1 of [RFC4253].  If ECMQV key exchange is
   chosen, then the public key algorithm supporting ECC host keys MUST
   also be chosen.

   ECMQV requires that all the keys used to generate a shared secret are
   generated over the same elliptic curve domain parameters.  Since the
   host key is used in the generation of the shared secret, allowing for
   implicit server authentication, the domain parameters associated with
   the host key are used throughout this section.

   All elliptic curve public keys MUST be validated after they are
   received.  An example of a validation algorithm can be found in
   Section 3.2.2 of [SEC1].  If a key fails validation, the key exchange
   MUST fail.

   The elliptic curve ephemeral public keys (points) 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 Sections 2.3.3 and 2.3.4 of [SEC1];
   point compression MAY be used.  The output of shared key generation
   is a field element xp.  The SSH framework requires that the shared
   key be an integer.  The conversion between a field element and an
   integer is specified in Section 2.3.9 of [SEC1].

   The following is an overview of the key exchange process:

      Client                                                Server
      ------                                                ------
      Generate ephemeral key pair.
      SSH_MSG_KEX_ECMQV_INIT ------------->

                                      Verify received key is valid.
                                       Generate ephemeral key pair.
                                             Compute shared secret.
                                Generate exchange hash and compute
                              HMAC over it using the shared secret.
                            <------------- SSH_MSG_KEX_ECMQV_REPLY

      Verify received keys are valid.
      *Verify host key belongs to server.
      Compute shared secret.
      Verify HMAC.

      *  It is RECOMMENDED that the client verify that the host key sent
         is the server's host key (for example, using a local database).
         The client MAY accept the host key without verification, but
         doing so will render the protocol insecure against active
         attacks.

   The specification of the message numbers SSH_MSG_ECMQV_INIT and
   SSH_MSG_ECMQV_REPLY can be found in Section 7.

   This key exchange algorithm is implemented with the following
   messages.

   The client sends:

      byte     SSH_MSG_ECMQV_INIT
      string   Q_C, client's ephemeral public key octet string

   The server sends:

      byte     SSH_MSG_ECMQV_REPLY
      string   K_S, server's public host key
      string   Q_S, server's ephemeral public key octet string
      string   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, client's identification string (CR and LF excluded)
      string   V_S, server's identification string (CR and LF excluded)
      string   I_C, payload of the client's SSH_MSG_KEXINIT
      string   I_S, payload of the server's SSH_MSG_KEXINIT
      string   K_S, server's public host key
      string   Q_C, client's ephemeral public key octet string
      string   Q_S, server's ephemeral public key octet string
      mpint    K,   shared secret

6.  Method Names

   This document defines a new family of key exchange method names, a
   new key exchange method name, and a new family of public key
   algorithm names in the SSH name registry.

6.1.  Elliptic Curve Domain Parameter Identifiers

   This section specifies identifiers encoding named elliptic curve
   domain parameters.  These identifiers are used in this document to
   identify the curve used in the SSH ECC public key format, the ECDSA
   signature blob, and the ECDH method name.

   For the REQUIRED elliptic curves nistp256, nistp384, and nistp521,
   the elliptic curve domain parameter identifiers are the strings
   "nistp256", "nistp384", and "nistp521".

   For all other elliptic curves, including all other NIST curves and
   all other RECOMMENDED curves, the elliptic curve domain parameter
   identifier is the ASCII period-separated decimal representation of
   the Abstract Syntax Notation One (ASN.1) [ASN1] Object Identifier
   (OID) of the named curve domain parameters that are associated with
   the server's ECC host keys.  This identifier is defined provided that
   the concatenation of the public key format identifier and the
   elliptic curve domain parameter identifier (or the method name and
   the elliptic curve domain parameter identifier) does not exceed the
   maximum specified by the SSH protocol architecture [RFC4251], namely
   64 characters; otherwise, the identifier for that curve is undefined,
   and the curve is not supported by this specification.

   A list of the REQUIRED and RECOMMENDED curves and their OIDs can be
   found in Section 10.

   Note that implementations MUST use the string identifiers for the
   three REQUIRED NIST curves, even when an OID exists for that curve.

6.2.  ECC Public Key Algorithm (ecdsa-sha2-*)

   The SSH ECC public key algorithm is specified by a family of public
   key format identifiers.  Each identifier is the concatenation of the
   string "ecdsa-sha2-" with the elliptic curve domain parameter
   identifier as defined in Section 6.1.  A list of the required and
   recommended curves and their OIDs can be found in Section 10.

   For example, the method name for ECDH key exchange with ephemeral
   keys generated on the nistp256 curve is "ecdsa-sha2-nistp256".

6.2.1.  Elliptic Curve Digital Signature Algorithm

   The Elliptic Curve Digital Signature Algorithm (ECDSA) is specified
   for use with the SSH ECC public key algorithm.

   The hashing algorithm defined by this family of method names is the
   SHA2 family of hashing algorithms [FIPS-180-3].  The algorithm from
   the SHA2 family that will be used is chosen based on the size of the
   named curve specified in the public key:

                    +----------------+----------------+
                    |   Curve Size   | Hash Algorithm |
                    +----------------+----------------+
                    |    b <= 256    |     SHA-256    |
                    |                |                |
                    | 256 < b <= 384 |     SHA-384    |
                    |                |                |
                    |     384 < b    |     SHA-512    |
                    +----------------+----------------+

6.3.  ECDH Key Exchange Method Names (ecdh-sha2-*)

   The Elliptic Curve Diffie-Hellman (ECDH) key exchange is defined by a
   family of method names.  Each method name is the concatenation of the
   string "ecdh-sha2-" with the elliptic curve domain parameter
   identifier as defined in Section 6.1.  A list of the required and
   recommended curves and their OIDs can be found in Section 10.

   For example, the method name for ECDH key exchange with ephemeral
   keys generated on the sect409k1 curve is "ecdh-sha2-1.3.132.0.36".

   The hashing algorithm defined by this family of method names is the
   SHA2 family of hashing algorithms [FIPS-180-3].  The hashing
   algorithm is defined in the method name to allow room for other
   algorithms to be defined in future documents.  The algorithm from the
   SHA2 family that will be used is chosen based on the size of the
   named curve specified in the method name according to the table in
   Section 6.2.1.

   The concatenation of any so encoded ASN.1 OID specifying a set of
   elliptic curve domain parameters with "ecdh-sha2-" is implicitly
   registered under this specification.

6.4.  ECMQV Key Exchange and Verification Method Name (ecmqv-sha2)

   The Elliptic Curve Menezes-Qu-Vanstone (ECMQV) key exchange is
   defined by the method name "ecmqv-sha2".  Unlike the ECDH key
   exchange method, ECMQV relies on a public key algorithm that uses ECC
   keys: it does not need a family of method names because the curve
   information can be gained from the public key algorithm.

   The hashing and message authentication code algorithms are defined by
   the method name to allow room for other algorithms to be defined for
   use with ECMQV in future documents.

   The hashing algorithm defined by this method name is the SHA2 family
   of hashing algorithms [FIPS-180-3].  The algorithm from the SHA2
   family that will be used is chosen based on the size of the named
   curve specified for use with ECMQV by the chosen public key algorithm
   according to the table in Section 6.2.1.

   The keyed-hash message authentication code that is used to identify
   the server and verify communications is based on the hash chosen
   above.  The information on implementing the HMAC based on the chosen
   hash algorithm can be found in [RFC2104].

7.  Key Exchange Messages

   The message numbers 30-49 are key-exchange-specific and in a private
   namespace defined in [RFC4250] that may be redefined by any key
   exchange method [RFC4253] without requiring an IANA registration
   process.

   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

8.  Manageability Considerations

   As this document only provides new public key algorithms and key
   exchange methods within the existing Secure Shell protocol
   architecture, there are few manageability considerations beyond those
   that apply for existing Secure Shell implementations.  Additional
   manageability considerations are listed below.

8.1.  Control of Function through Configuration and Policy

   Section 10 specifies REQUIRED and RECOMMENDED elliptic curve domain
   parameters to be used with the public key algorithms and key exchange
   methods defined in this document.  Implementers SHOULD allow system
   administrators to disable some curves, including REQUIRED or
   RECOMMENDED curves, to meet local security policy.

8.2.  Impact on Network Operation

   As this document provides new functionality within the Secure Shell
   protocol architecture, the only impact on network operations is the
   impact on existing Secure Shell implementations.  The Secure Shell
   protocol provides negotiation mechanisms for public key algorithms
   and key exchange methods: any implementations that do not recognize
   the algorithms and methods defined in this document will ignore them
   in the negotiation and use the next mutually supported algorithm or
   method, causing no negative impact on backward compatibility.

   The use of elliptic curve cryptography should not place a significant
   computational burden on an implementing server.  In fact, due to its
   smaller key sizes, elliptic curve cryptography can be implemented
   more efficiently for the same security level than RSA, finite field
   Diffie-Hellman, or DSA.

9.  Security Considerations

   This document provides new public key algorithms and new key
   agreement methods for the Secure Shell protocol.  For the most part,
   the security considerations involved in using the Secure Shell
   protocol apply.  Additionally, implementers should be aware of
   security considerations specific to elliptic curve cryptography.

   For all three classes of functionality added by this document (the
   public key algorithms involving ECDSA, key exchange involving ECDH,
   and authenticated key exchange involving ECMQV), the current best
   known technique for breaking the cryptosystems is by solving the
   elliptic curve discrete logarithm problem (ECDLP).

   The difficulty of breaking the ECDLP depends on the size and quality
   of the elliptic curve parameters.  Certain types of curves can be
   more susceptible to known attacks than others.  For example, curves
   over finite fields GF(2^m), where m is composite, may be susceptible
   to an attack based on the Weil descent.  All of the RECOMMENDED
   curves in Section 10 do not have this problem.  System administrators
   should be cautious when enabling curves other than the ones specified
   in Section 10 and should make a more detailed investigation into the
   security of the curve in question.

   It is believed (see, for example, Section B.2.1 of [SEC1]) that when
   curve parameters are generated at random, the curves will be
   resistant to special attacks, and must rely on the most general
   attacks.  The REQUIRED curves in Section 10 were all generated
   verifiably pseudorandomly.  The runtime of general attacks depends on
   the algorithm used.  At present, the best known algorithm is the
   Pollard-rho method.  (Shor's algorithm for quantum computers can
   solve the ECDLP in polynomial time, but at present large-scale
   quantum computers have not been constructed and significant
   experimental physics and engineering work needs to be done before
   large-scale quantum computers can be constructed.  There is no solid
   estimate as to when this may occur, but it is widely believed to be
   at least 20 years from the present.)

   Based on projections of computation power, it is possible to estimate
   the running time of the best known attacks based on the size of the
   finite field.  The table in Section 1 gives an estimate of the
   equivalence between elliptic curve field size and symmetric key size.
   Roughly speaking, an N-bit elliptic curve offers the same security as
   an N/2-bit symmetric cipher, so a 256-bit elliptic curve (such as the
   REQUIRED nistp256 curve) is suitable for use with 128-bit AES, for
   example.

   Many estimates consider that 2^80-2^90 operations are beyond
   feasible, so that would suggest using elliptic curves of at least
   160-180 bits.  The REQUIRED curves in this document are 256-, 384-,
   and 521-bit curves; implementations SHOULD NOT use curves smaller
   than 160 bits.

   A detailed discussion on the security considerations of elliptic
   curve domain parameters and the ECDH, ECDSA, and ECMQV algorithms can
   be found in Appendix B of [SEC1].

   Additionally, the key exchange methods defined in this document rely
   on the SHA2 family of hash functions, defined in [FIPS-180-3].  The
   appropriate security considerations of that document apply.  Although
   some weaknesses have been discovered in the predecessor, SHA-1, no
   weaknesses in the SHA2 family are known at present.  The SHA2 family
   consists of four variants -- SHA-224, SHA-256, SHA-384, and SHA-521
   -- named after their digest lengths.  In the absence of special
   purpose attacks exploiting the specific structure of the hash
   function, the difficulty of finding collisions, preimages, and second
   preimages for the hash function is related to the digest length.
   This document specifies in Section 6.2.1 which SHA2 variant should be
   used with which elliptic curve size based on this guidance.

   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 [NIST-800-57]; the discussion in [RFC3766] is
   also relevant.  We note in particular that when ECDSA is used as the
   signature algorithm and ECDH is used as the key exchange method, if
   curves of different sizes are used, then it is possible that
   different hash functions from the SHA2 family could be used.

   The REQUIRED and RECOMMENDED curves in this document are at present
   believed to offer security at the levels indicated in this section
   and as specified with the table in Section 1.

   System administrators and implementers should take careful
   consideration of the security issues when enabling curves other than
   the REQUIRED or RECOMMENDED curves in this document.  Not all
   elliptic curves are secure, even if they are over a large field.

   Implementers SHOULD ensure that any ephemeral private keys or random
   values -- including the value k used in ECDSA signature generation
   and the ephemeral private key values in ECDH and ECMQV -- are
   generated from a random number generator or a properly seeded
   pseudorandom number generator, are protected from leakage, are not
   reused outside of the context of the protocol in this document, and
   are erased from memory when no longer needed.

10.  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 [ASN1].

10.1.  Required Curves

   Every SSH ECC implementation MUST support the named curves below.
   These curves are defined in [SEC2]; the NIST curves were originally
   defined in [NIST-CURVES].  These curves SHOULD always be enabled
   unless specifically disabled by local security policy.

              +----------+-----------+---------------------+
              |   NIST*  |    SEC    |         OID         |
              +----------+-----------+---------------------+
              | nistp256 | secp256r1 | 1.2.840.10045.3.1.7 |
              |          |           |                     |
              | nistp384 | secp384r1 |     1.3.132.0.34    |
              |          |           |                     |
              | nistp521 | secp521r1 |     1.3.132.0.35    |
              +----------+-----------+---------------------+

      *  For these three REQUIRED curves, the elliptic curve domain
         parameter identifier is the string in the first column of the
         table, the NIST name of the curve.  (See Section 6.1.)

10.2.  Recommended Curves

   It is RECOMMENDED that SSH ECC implementations also support the
   following curves.  These curves are defined in [SEC2].

              +----------+-----------+---------------------+
              |   NIST   |    SEC    |         OID*        |
              +----------+-----------+---------------------+
              | nistk163 | sect163k1 |     1.3.132.0.1     |
              |          |           |                     |
              | nistp192 | secp192r1 | 1.2.840.10045.3.1.1 |
              |          |           |                     |
              | nistp224 | secp224r1 |     1.3.132.0.33    |
              |          |           |                     |
              | nistk233 | sect233k1 |     1.3.132.0.26    |
              |          |           |                     |
              | nistb233 | sect233r1 |     1.3.132.0.27    |
              |          |           |                     |
              | nistk283 | sect283k1 |     1.3.132.0.16    |
              |          |           |                     |
              | nistk409 | sect409k1 |     1.3.132.0.36    |
              |          |           |                     |
              | nistb409 | sect409r1 |     1.3.132.0.37    |
              |          |           |                     |
              | nistt571 | sect571k1 |     1.3.132.0.38    |
              +----------+-----------+---------------------+

      *  For these RECOMMENDED curves, the elliptic curve domain
         parameter identifier is the string in the third column of the
         table, the ASCII representation of the OID of the curve.  (See
         Section 6.1.)

11.  IANA Considerations

   Consistent with Section 8 of [RFC4251] and Section 4.6 of [RFC4250],
   this document makes the following registrations:

   In the Public Key Algorithm Names registry: The family of SSH public
   key algorithm names beginning with "ecdsa-sha2-" and not containing
   the at-sign ('@'), to name the public key algorithms defined in
   Section 3.

   In the Key Exchange Method Names registry: The family of SSH key
   exchange method names beginning with "ecdh-sha2-" and not containing
   the at-sign ('@'), to name the key exchange methods defined in
   Section 4.

   In the Key Exchange Method Names registry: The SSH key exchange
   method name "ecmqv-sha2" to name the key exchange method defined in
   Section 5.

   This document creates no new registries.

12.  References

12.1.  Normative References

   [ASN1]         International Telecommunications Union, "Abstract
                  Syntax Notation One (ASN.1): Specification of basic
                  notation",  X.680, July 2002.

   [FIPS-180-3]   National Institute of Standards and Technology,
                  "Secure Hash Standard", FIPS 180-3, October 2008.

   [RFC2104]      Krawczyk, H., Bellare, M., and R. Canetti, "HMAC:
                  Keyed-Hashing for Message Authentication", RFC 2104,
                  February 1997.

   [RFC2119]      Bradner, S., "Key words for use in RFCs to Indicate
                  Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC3766]      Orman, H. and P. Hoffman, "Determining Strengths For
                  Public Keys Used For Exchanging Symmetric Keys",
                  BCP 86, RFC 3766, April 2004.

   [RFC4250]      Lehtinen, S. and C. Lonvick, "The Secure Shell (SSH)
                  Protocol Assigned Numbers", RFC 4250, January 2006.

   [RFC4251]      Ylonen, T. and C. Lonvick, "The Secure Shell (SSH)
                  Protocol Architecture", RFC 4251, January 2006.

   [RFC4253]      Ylonen, T. and C. Lonvick, "The Secure Shell (SSH)
                  Transport Layer Protocol", RFC 4253, January 2006.

   [SEC1]         Standards for Efficient Cryptography Group, "Elliptic
                  Curve Cryptography", SEC 1, May 2009,
                  <http://www.secg.org/download/aid-780/sec1-v2.pdf>.

   [SEC2]         Standards for Efficient Cryptography Group,
                  "Recommended Elliptic Curve Domain Parameters", SEC 2,
                  September 2000,
                  <http://www.secg.org/download/aid-386/sec2_final.pdf>.

12.2.  Informative References

   [ANSI-X9.62]   American National Standards Institute, "Public Key
                  Cryptography For The Financial Services Industry: The
                  Elliptic Curve Digital Signature Algorithm (ECDSA)",
                  ANSI X9.62, 1998.

   [ANSI-X9.63]   American National Standards Institute, "Public Key
                  Cryptography For The Financial Services Industry: Key
                  Agreement and Key Transport Using Elliptic Curve
                  Cryptography", ANSI X9.63, January 1999.

   [HMV04]        Hankerson, D., Menezes, A., and S. Vanstone, "Guide to
                  Elliptic Curve Cryptography", Springer ISBN
                  038795273X, 2004.

   [LMQSV98]      Law, L., Menezes, A., Qu, M., Solinas, J., and S.
                  Vanstone, "An Efficient Protocol for Authenticated Key
                  Agreement", University of Waterloo Technical Report
                  CORR 98-05, August 1998, <http://
                  www.cacr.math.uwaterloo.ca/techreports/1998/
                  corr98-05.pdf>.

   [NIST-800-57]  National Institute of Standards and Technology,
                  "Recommendation for Key Management - Part 1: General
                  (Revised)", NIST Special Publication 800-57,
                  March 2007.

   [NIST-CURVES]  National Institute of Standards and Technology,
                  "Recommended Elliptic Curves for Federal Government
                  Use", July 1999.

Appendix A.  Acknowledgements

   The authors acknowledge helpful comments from James Blaisdell, David
   Harrington, Alfred Hoenes, Russ Housley, Jeffrey Hutzelman, Kevin
   Igoe, Rob Lambert, Jan Pechanek, Tim Polk, Sean Turner, Nicolas
   Williams, and members of the ietf-ssh@netbsd.org mailing list.

Authors' Addresses

   Douglas Stebila
   Queensland University of Technology
   Information Security Institute
   Level 7, 126 Margaret St
   Brisbane, Queensland  4000
   Australia

   EMail: douglas@stebila.ca


   Jon Green
   Queen's University
   Parallel Processing Research Laboratory
   Department of Electrical and Computer Engineering
   Room 614, Walter Light Hall
   Kingston, Ontario  K7L 3N6
   Canada

   EMail: jonathan.green@queensu.ca