Network Working Group S. Josefsson
Internet-Draft SJD AB
Updates: 4492 (if approved) M. Pegourie-Gonnard
Intended status: Informational Independant / PolarSSL
Expires: July 15, 2014 January 11, 2014
Additional Elliptic Curves for Transport Layer Security (TLS) Elliptic
Curve Diffie-Hellman Key Agreement
draft-josefsson-tls-curve25519-03
Abstract
This document specifies the use of additional elliptic curves
(Curve25519, E382, M383, Curve3617, M511, E521) for key exchange in
the Transport Layer Security (TLS) protocol.
Status of this Memo
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1. Introduction
In [Curve25519], a new elliptic curve function for use in
cryptographic applications was specified. Curve25519 is a Diffie-
Hellman function designed with performance and security in mind. In
[I-D.ladd-safecurves] additional curves are describe with similar
properties, but of different strengths. The complete list of curves
is Curve25519, E382, M383, Curve3617, M511, and E521. In this
document we refer to this list of curves informally as "additional
curves".
[RFC4492] defines the usage of elliptic curves for authentication and
key agreement in TLS 1.0 and TLS 1.1, and these mechanisms are also
applicable to TLS 1.2 [RFC5246]. The use of ECC curves for key
exchange requires the definition and assignment of additional
NamedCurve IDs. This document specify that value for the additional
curves, as well as the minor changes in key selection and
representation that are required to accomodate for the curves
slightly different nature.
This document only describes usage of additional curves for ephemeral
key exchange (ECDHE), not for use with long-term keys embedded in
PKIX certificates (ECDH_ECDSA and ECDH_ECDSA). This is because
Curve25519 is not directly suitable for authentication with ECDSA,
and thus not applicable for signing of e.g. PKIX certificates. See
draft-josefsson-eddsa-ed25519 for a parallel effort.
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2. Data Structures and Computations
[RFC4492] and related standards [SEC1] define an elliptic curve over
GF(p) as the set of solutions of the equation y^2 = x^3 + a x + b
(commonly referred to as a short Weierstrass equation) with both x
and y in GF(p), plus the point at infinity. For each curve, a point
G is fixed, generating a subgroup of large (prime) order N.
The Diffie-Hellman key exchange is then defined as follows: each
party generates a random number 1 <= d < N (the private key),
computes Q = d G (the public key). The parties exchange their public
keys and compute the shared secret as Z = d Q_peer.
[RFC4492] defines formats for representing public keys during the
exchange, and extensions for negotiating the format used by each
party and the underlying curve used by both parties.
While retaining the same overall structure for the Diffie-Hellman key
exchange, the additional curves makes some changes to the way the
curve equation is presented, private keys are selected and public
keys exchanged, in order to ease secure and efficient
implementations.
The following subsections describe the differences between using
additional curves and the curves defined by RFC 4492 for key exchange
in TLS.
2.1. Group negotiation and new NamedCurve type
Curve negotiation is the same for the additional curves as for other
curves, but is restricted to using the named_curve type in the
ServerKeyEchange message: the explicit_prime type is only suited to
curves in short Weierstrass form. This document adds a new
NamedCurve value for the additional curves as follows.
enum {
Curve25519(TBD1),
E382(TBD2),
M383(TBD3),
Curve3617(TBD4),
M511(TBD5),
E521(TBD6)
} NamedCurve;
The curves are suitable for use with DTLS [RFC6347].
Since these curves are not designed to be used in signatures, clients
who offer ECDHE_ECDSA ciphersuites and advertise support for any of
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these curves in the elliptic_curves ClientHello extension SHOULD also
advertise support for at least one other curve, suitable for ECDSA.
Servers MUST NOT select an ECDHE_ECDSA ciphersuite if the only common
curve is one of these curves.
2.2. Curve25519 Private key generation
Private keys MUST be selected as specified in [Curve25519]. That is,
private keys are 255-bits numbers with the following properties: the
most significant bit (bit 254) is set, and the three least
significants bits are cleared; the remaining bits (3 to 253 included)
are chosen uniformly at random.
2.3. Curve25519 Public key representation
For ECDHE, public keys need to be transmitted in the
ServerKeyExchange and ClientKeyExchange messages, both of which
encode it as follows.
struct {
opaque point <1..2^8-1>;
} ECPoint;
For short Weierstrass curves, the contents of ECPoint.point are
defined by X9.62. For Curve25519, the ECpoint structure is the same,
but the contents of ECPoint.point are encoded and interpreted as
follows: each peer's public key is a number between 0 and 2^255 - 20
included, and ECPoint.point contains the 32 bytes string representing
this number in big endian convention. (The receiving party can
accept any 32 bytes string, interpreted as a 256 bits number, as
public key: by design, no validation is needed.)
Note that ECPoint.point differs from the definition of public keys in
[Curve25519] in two ways: (1) the byte-ordering is big-endian, wich
is more uniform with how big integers are represented in TLS, and (2)
there is an additional length byte (so ECpoint.point is actually 33
bytes), again for uniformity (and extensibility).
Since only one point format can be used with Curve25519, which is
distinct from the formats used by short Weierstrass curves, the
contents of the "Supported Point Formats" extension is irrelevant for
this curve. Peers do not need to advertise support for the above
point format in any way (nor check that the orther party supports it)
when planning to use Curve25519 for key agreement: support for
Curve25519 implies support for the above point format, which is tied
to it.
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2.4. Curve25519 Shared secret computation
As in the standard Elliptic Curve Diffie-Hellman protocol [SEC1],
each party computes the shared secret by multiplying the peer's
public value (seen as a point on the curve) by its own private value,
except that in the case of Curve25519, only the x coordinate is
computed. This is merely an internal detail since [RFC4492]
specifies that only the x coordinate is used as the premaster secret
anyway.
Again, in line with [RFC4492] and as a departure from the convention
chosen in [Curve25519], the x coordinate is converted to a bytes
string using big endian order. As in [RFC4492], leading zeros are
preserved, so the premaster secret is always a 32 bytes string with
Curve25519.
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3. Acknowledgement
This document was inspired by the content and structure of [RFC7027].
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4. IANA Considerations
IANA is requested to assign numbers for the additional curves listed
in Section 2.1 to the Transport Layer Security (TLS) Parameters
registry EC Named Curve [IANA-TLS] as follows.
+-------+-------------+---------+-----------+
| Value | Description | DTLS-OK | Reference |
+-------+-------------+---------+-----------+
| TBD1 | Curve25519 | Y | This doc |
| | | | |
| TBD2 | E382 | Y | This doc |
| | | | |
| TBD3 | M383 | Y | This doc |
| | | | |
| TBD4 | Curve3617 | Y | This doc |
| | | | |
| TBD5 | M511 | Y | This doc |
| | | | |
| TBD6 | E521 | Y | This doc |
+-------+-------------+---------+-----------+
Table 1
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5. Security Considerations
The security considerations of [RFC5246] and most of the security
considerations of [RFC4492] apply accordingly.
Curve25519 was specifically designed so that secure, fast
implementations are easier to produce. In particular, no validation
of public keys is required, and point multiplication (using only the
x coordinate) can be efficiently [EFD] computed with a Montgomery
ladder using a constant number of operations (since the actual bit
length of the private key is fixed), which avoids a number of side-
channel attacks. However, in the fight against side-channel leaks,
implementors should also pay attention to the following points:
1. In the Montgomery ladder, avoid branches depending on secret data
(the individual bits of the secret key);
2. In the same place, avoid memory access patterns dependant on
secret data;
3. Either avoid data-dependant branches and memory access patterns
in the underlying field arithmetic (that is, the bignum
arithmetic, including the mod 2^255-19 operation) or randomize
projective (that is, x/z) coordinates by multiplying both x and z
with the same 256-bit value, chosen at random.
Some of the curves defined in [RFC4492], namely all whose name ends
with r1 or r2, have been advertised as pseudo-randomly chosen, but
the lack of verifiability of the seeds has raised concerns that the
those curves might be weaker than expected aginst some attackers.
The Koblitz curves (those whose name end with k1) of [RFC4492] do not
suffer from this problem, but are char2 curves and there seems to be
a consensus that curves over prime fields are a safer bet against
future progress in discrete log computation. The Brainpool curves
[RFC7027] are prime curves generated in a fully verifiable pseudo-
random way to avoid manipulation concerns, but do not perform as well
due to the use of pseudo-random primes. Curve22519 is also chosen in
a fully verifiable way, but offers better performances (better than
the curves form [RFC4492]) while facilitating secure implementations
as mentioned above.
See also [SafeCurves] and [I-D.ladd-safecurves] for more security
discussions.
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6. References
6.1. Normative References
[Curve25519]
Bernstein, J., "Curve25519: new Diffie-Hellman speed
records",
WWW http://cr.yp.to/ecdh/curve25519-20060209.pdf,
February 2006.
[RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B.
Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites
for Transport Layer Security (TLS)", RFC 4492, May 2006.
[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.2", RFC 5246, August 2008.
[RFC6347] Rescorla, E. and N. Modadugu, "Datagram Transport Layer
Security Version 1.2", RFC 6347, January 2012.
6.2. Informative References
[IANA-TLS]
Internet Assigned Numbers Authority, "Transport Layer
Security (TLS) Parameters", .
[SafeCurves]
Bernstein, D. and T. Lange, "SafeCurves: choosing safe
curves for elliptic-curve cryptography.", January 2014,
.
[I-D.ladd-safecurves]
Ladd, W., "Additional Elliptic Curves for IETF protocols",
draft-ladd-safecurves-02 (work in progress), January 2014.
[EFD] Bernstein, D. and T. Lange, "Explicit-Formulas Database:
XZ coordinates for Montgomery curves", January 2014, .
[RFC7027] Merkle, J. and M. Lochter, "Elliptic Curve Cryptography
(ECC) Brainpool Curves for Transport Layer Security
(TLS)", RFC 7027, October 2013.
[SEC1] Certicom Research, "Standards for Efficient Cryptography
(SEC) 1", September 2000.
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Appendix A. Test vectors
This section provides some test vectors for example Diffie-Hellman
key exchanges using Curve25519. The following notations are used:
d_A the secret key of party A
x_A the public key of party A
d_B the secret key of party B
x_B the public key of party B
x_S the shared secret that results from completion of the Diffie-
Hellman computation, i.e., the hex representation of the pre-
master secret.
The field elements x_A, x_B, and x_S are represented as hexadecimal
values using the FieldElement-to-OctetString conversion method
specified in [SEC1].
d_A = 5AC99F33632E5A768DE7E81BF854C27C46E3FBF2ABBACD29EC4AFF51
7369C660
d_B = 47DC3D214174820E1154B49BC6CDB2ABD45EE95817055D255AA35831
B70D3260
x_A = 057E23EA9F1CBE8A27168F6E696A791DE61DD3AF7ACD4EEACC6E7BA5
14FDA863
x_B = 6EB89DA91989AE37C7EAC7618D9E5C4951DBA1D73C285AE1CD26A855
020EEF04
x_S = 61450CD98E36016B58776A897A9F0AEF738B99F09468B8D6B8511184
D53494AB
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Authors' Addresses
Simon Josefsson
SJD AB
Email: simon@josefsson.org
Manuel Pegourie-Gonnard
Independant / PolarSSL
Email: mpg@elzevir.fr
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