Elliptic-Curve Algorithm Integration in the Secure Shell Transport Layer
draft-green-secsh-ecc-09

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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.

Table of Contents

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

1.  Introduction

This document adds 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] (National Institute of Standards and Technology, “Recommendation for Key Management - Part 1: General (Revised),” March 2007.), 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.

Implementation of this specification requires familiarity with both SSH [RFC4251] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Protocol Architecture,” January 2006.) [RFC4253] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Transport Layer Protocol,” January 2006.) [RFC4250] (Lehtinen, S. and C. Lonvick, “The Secure Shell (SSH) Protocol Assigned Numbers,” January 2006.) and ECC [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.) (additional information on ECC available in [HMV04] (Hankerson, D., Menezes, A., and S. Vanstone, “Guide to Elliptic Curve Cryptography,” 2004.), [IEEE1363] (Institute of Electrical and Electronics Engineers, “Standard Specifications for Public Key Cryptography,” 2000.), [ANSI‑X9.62] (American National Standards Institute, “Public Key Cryptography For The Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA),” 1998.), and [ANSI‑X9.63] (American National Standards Institute, “Public Key Cryptography For The Financial Services Industry: Key Agreement and Key Transport Using Elliptic Curve Cryptography,” January 1999.)).

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] (Bradner, S., “Key words for use in RFCs to Indicate Requirement Levels,” March 1997.).

The data types boolean, uint32, uint64, string, and mpint are to be interpreted in this document as described in [RFC4251] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Protocol Architecture,” January 2006.).

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

3.  ECC Public Key Algorithm

The 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 (Elliptic Curve Domain Parameter Identifiers). Information on the required and recommended sets of elliptic curve domain parameters for use with this algorithm can be found in Section 10 (Named Elliptic Curve Domain Parameters).

Q is the public key encoded from an elliptic curve point into an octet string as defined in Section 2.3.3 of [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.); point compression MAY be used.

The algorithm for ECC key generation can be found in Section 3.2 of [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.). 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.). The message hashing algorithm MUST be from the SHA2 family of hash functions [FIPS‑180‑3] (National Institute of Standards and Technology, “Secure Hash Standard,” October 2008.) and is chosen according to the curve size as specified in Section 6.2.1 (Elliptic Curve Digital Signature Algorithm).

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 (Elliptic Curve Domain Parameter Identifiers). Information on the required and recommended sets of elliptic curve domain parameters for use with this algorithm can be found in Section 10 (Named Elliptic Curve Domain Parameters).

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] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Transport Layer Protocol,” January 2006.) 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.). The algorithm for key pair generation can be found in Section 3.2 of [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.).

The family of key exchange method names defined for use with this key exchange can be found in Section 6.3 (ECDH Key Exchange Method Names (ecdh-sha2-*)). 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 (Named Elliptic Curve Domain Parameters).

All elliptic curve public keys MUST be validated after they are received. An example of a validation algorithm can be found in A.16.10 of [IEEE1363] (Institute of Electrical and Electronics Engineers, “Standard Specifications for Public Key Cryptography,” 2000.). 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.); 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.).

Specification of the message numbers SSH_MSG_KEX_ECDH_INIT and SSH_MSG_KEX_ECDH_REPLY are found in Section 7 (Key Exchange Messages).

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 is allowed to 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] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Protocol Architecture,” January 2006.).

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] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Transport Layer Protocol,” January 2006.). 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 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 (ECMQV Key Exchange and Verification Method Name (ecmqv-sha2)).

In general the ECMQV key exchange is performed using the ephemeral and long term key pair of both the client and server, 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 does not adversely affect security (it is analogous to the one-pass protocol in Section 6.1 of [LMQSV98] (Law, L., Menezes, A., Qu, M., Solinas, J., and S. Vanstone, “An Efficient Protocol for Authenticated Key Agreement,” August 1998.)).

A full description of the ECMQV primitive can be found in Section 3.4 of [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.). The algorithm for key pair generation can be found in Section 3.2 of [SEC1] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.).

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] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Transport Layer Protocol,” January 2006.). 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 A.16.10 of [IEEE1363] (Institute of Electrical and Electronics Engineers, “Standard Specifications for Public Key Cryptography,” 2000.). 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.); 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.).

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 is allowed to 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 (Key Exchange Messages).

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   Q_S, server's ephemeral public key octet
   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 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] (International Telecommunications Union, “Abstract Syntax Notation One (ASN.1): Specification of basic notation,” July 2002.) 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 identifer (or the method name and the elliptic curve domain parameter identifier) does not exceed the maximum specified by the SSH Protocol Architecture [RFC4251] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Protocol Architecture,” January 2006.), 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 (Named Elliptic Curve Domain Parameters).

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 ECC Public Key Algorithm is specified by a family of public key format identifiers. Each identifer is the concatenation of the string "ecdsa-sha2-" with the elliptic curve domain parameter identifier as defined in Section 6.1 (Elliptic Curve Domain Parameter Identifiers). A list of the required and recommended curves and their OIDs can be found in Section 10 (Named Elliptic Curve Domain Parameters).

For example: The method name for ECDH key exchange with ephemeral keys generated on the nistp256 curve would be "ecdsa-sha2-nistp256".

6.2.1.  Elliptic Curve Digital Signature Algorithm

The Elliptic Curve Digital Signature Algorithm (ECDSA) is specified for use with the ECC Public Key Algorithm.

The hashing algorithm defined by this family of method names is the SHA2 family of hashing algorithms [FIPS‑180‑3] (National Institute of Standards and Technology, “Secure Hash Standard,” October 2008.). 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:

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

The Elliptic Curve Diffie-Hellman 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 (Elliptic Curve Domain Parameter Identifiers). A list of the required and recommended curves and their OIDs can be found in Section 10 (Named Elliptic Curve Domain Parameters).

For example: The method name for ECDH key exchange with ephemeral keys generated on the sect409k1 curve would be "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] (National Institute of Standards and Technology, “Secure Hash Standard,” October 2008.). 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 (Elliptic Curve Digital Signature Algorithm).

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 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] (National Institute of Standards and Technology, “Secure Hash Standard,” October 2008.). 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 (Elliptic Curve Digital Signature Algorithm).

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] (Krawczyk, H., Bellare, M., and R. Canetti, “HMAC: Keyed-Hashing for Message Authentication,” February 1997.).

7.  Key Exchange Messages

The message numbers 30-49 are key exchange-specific and in a private namespace defined in [RFC4250] (Lehtinen, S. and C. Lonvick, “The Secure Shell (SSH) Protocol Assigned Numbers,” January 2006.) that may be redefined by any key exchange method [RFC4253] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Transport Layer Protocol,” January 2006.) 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 (Named Elliptic Curve Domain Parameters) 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 (Named Elliptic Curve Domain Parameters) do not have this problem. System administrators should be cautious when enabling curves other than the ones specified in Section 10 (Named Elliptic Curve Domain Parameters) 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.)) 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 (Named Elliptic Curve Domain Parameters) 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 (Introduction) 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 documents 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] (Standards for Efficient Cryptography Group, “Elliptic Curve Cryptography,” September 2000.).

Additionally, the key exchange methods defined in this document rely on the SHA2 family of hash functions, defined in [FIPS‑180‑3] (National Institute of Standards and Technology, “Secure Hash Standard,” October 2008.). 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 (Elliptic Curve Digital Signature Algorithm) 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] (National Institute of Standards and Technology, “Recommendation for Key Management - Part 1: General (Revised),” March 2007.); the discussion in [RFC3766] (Orman, H. and P. Hoffman, “Determining Strengths For Public Keys Used For Exchanging Symmetric Keys,” April 2004.) 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 (Introduction).

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-seed 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] (International Telecommunications Union, “Abstract Syntax Notation One (ASN.1): Specification of basic notation,” July 2002.).

10.1.  Required Curves

Every SSH ECC implementation MUST support the named curves below. These curves are defined in [SEC2] (Standards for Efficient Cryptography Group, “Recommended Elliptic Curve Domain Parameters,” September 2000.); the NIST curves were originally defined in [NIST‑CURVES] (National Institute of Standards and Technology, “Recommended Elliptic Curves for Federal Government Use,” July 1999.). These curves SHOULD always be enabled unless specifically disabled by local security policy.

* 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 (Elliptic Curve Domain Parameter Identifiers).)

10.2.  Recommended Curves

It is RECOMMENDED that SSH ECC implementations also support the following curves. These curves are defined in [SEC2] (Standards for Efficient Cryptography Group, “Recommended Elliptic Curve Domain Parameters,” September 2000.).

* 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 (Elliptic Curve Domain Parameter Identifiers).)

11.  IANA Considerations

Consistent with Section 8 of [RFC4251] (Ylonen, T. and C. Lonvick, “The Secure Shell (SSH) Protocol Architecture,” January 2006.) and Section 4.6 of [RFC4250] (Lehtinen, S. and C. Lonvick, “The Secure Shell (SSH) Protocol Assigned Numbers,” January 2006.), 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 (ECC Public Key Algorithm).

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 (ECDH Key Exchange).

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 (ECMQV Key Exchange).

This document creates no new registries.

12.  References

12.1. Normative References

12.2. Informative References

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