Network Working Group S. Josefsson
Internet-Draft SJD AB
Updates: 4492, 5246 (if approved) M. Pegourie-Gonnard
Intended status: Informational Independent / PolarSSL
Expires: October 15, 2014 April 13, 2014

Curve25519 for ephemeral key exchange in Transport Layer Security (TLS)


This document specifies the use of Curve25519 for ephemeral key exchange in the Transport Layer Security (TLS) protocol, as well as its DTLS variant. It updates RFC 5246 (TLS 1.2) and RFC 4492 (Elliptic Curve Cryptography for TLS).

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.

[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 Curve25519, as well as the minor changes in key selection and representation that are required to accommodate for Curve25519's slightly different nature.

This document only describes usage of Curve25519 for ephemeral key exchange (ECDHE). It does not define its use for signature, since the primitive considered here is a Diffie-Hellman function; the related signature scheme, Ed25519, is outside the scope of this document. The use of Curve25519 with long-term keys embedded in X.509 certificates is also out of scope here.

1.1. Requirements Terminology

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].

2. Data Structures and Computations

2.1. Cryptographic computations

All cryptographic computations are done using the Curve25519 function defined in [Curve25519]. In this memo, this function is considered as a black box that takes as input a (secret key, public key) pair and outputs a public key. Public keys are defined as strings of 32 bytes. Secret keys are defined as 255 bits numbers such as the high-order bit (bit 254) is set, and the three lowest-order bits are unset. In addition, a common public key, denoted by G, is shared by all users.

An ECDHE key exchange using Curve25519 goes as follows. Each party picks a secret key d uniformly at random and computes the corresponding public key x = Curve25519(d, G). Parties exchange their public keys (see Section 2.3) and compute a shared secret as x_S = Curve25519(d, x_peer). This shared secret is used directly as the premaster secret, which is always exactly 32 bytes when ECDHE with Curve25519 is used.

A complete description of the Curve25519 function, as well as a few implementation notes, are provided in Appendix A.

2.2. Curve negotiation and new NamedCurve value

Curve negotiation uses the mechanisms introduced by [RFC4492], without modification except the following restriction: in the ECParameters structure, only the named_curve case can be used with Curve25519. Accordingly, arbitrary_explicit_prime_curves in the Supported Curves extension does not imply support for Curve25519, even though the Curve25519 function happens to be defined using an elliptic curve over a prime field.

The reason for this restriction is that explicit_prime is only suited to the so-called Short Weierstrass representation of elliptic curves, while Curve25519 uses a different representation for performance and security reasons.

This document adds a new NamedCurve value for Curve25519 as follows.

      enum {
      } NamedCurve;

Curve25519 is suitable for use with DTLS [RFC6347].

Since Curve25519 is not designed to be used in signatures, clients who offer ECDHE_ECDSA ciphersuites and advertise support for Curve25519 in the elliptic_curves ClientHello extension SHOULD also advertise support for at least one curve suitable for ECDSA. Servers MUST NOT select an ECDHE_ECDSA ciphersuite if there are no common curves suitable for ECDSA.

2.3. Public Key representation and new ECPointFormat value

This section defines a new point format suitable to encode Curve25519 public keys, as well as an identifier to negotiate this new format in TLS, and includes guidance on their use.

The curves defined in [RFC4492] define a public key as a point on the curve. In order to exchange public keys, the points are serialized as a string of bytes using one of the formats defined in [SEC1]. These encodings begin with a leading byte identifying the format, followed by a string of bytes, whose length is uniquely determined by the leading byte and curve used.

Since Curve25519 public keys already are string of bytes, no serialization is needed. However, a leading byte with value 0x41 is prepended to the public key to identify the format. The goal, besides consistency with the SEC1 formats, is to allow using other formats with Curve25519 in the future if needed.

In order to negotiate this format in TLS, a new ECPointFormat is defined as follows.

      enum {
      } ECPointFormat;

This format is currently the only format defined for use with Curve25519. Clients offering Curve25519 in the Supported Elliptic Curves extension MUST also offer montgomery_x_le in the Supported Point Format extension. Servers selecting Curve25519 for key exchange MUST include montgomery_x_le in their Supported Point Format extension. Servers willing to use Curve25519 MUST NOT assume that the client supports the montgomery_x_le format if the client did not advertise it explicitly.

When included in a ServerKeyExchange or ClientKeyExchange message, the public key is wrapped in an ECPoint structure as defined in [RFC4492], whose payload is as described above. For example, a public key with value 2A ... 2A appears on the wire as follows (including the length byte of ECPoint.point).

        21 41 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A
        2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A

2.4. Public key validation

With the curves defined by [RFC4492], each party must validate the public key sent by its peer before performing cryptographic computations with it. Failing to do so allows attackers to gain information about the private key, to the point that they may recover the entire private key in a few requests, if that key is not really ephemeral.

Curve25519 was designed in a way that the result of Curve25519(x, d) will never reveal information about d, provided is was chosen as prescribed, for any value of x.

Let's define legitimate values of x as the values that can be obtained as x = Curve25519(G, d') for some d, and call the other values illegitimate. The definition of the Curve25519 function shows that legitimate values all share the following property: the high-order bit of the last byte is not set.

Since there are some implementation of the Curve25519 function that impose this restriction on their input and others that don't, implementations of Curve25519 in TLS SHOULD reject public keys when the high-order bit of the last byte is set (in other words, when the value of the leftmost byte is greater than 0x7F) in order to prevent implementation fingerprinting.

Other than this recommended check, implementations do not need to ensure that the public keys they receive are legitimate: this is not necessary for security with Curve25519.

3. IANA Considerations

IANA is requested to assign numbers for Curve25519 listed in Section 2.2 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

IANA is also requested to assign numbers for Curve25519 listed in Section 2.3 to the Transport Layer Security (TLS) Parameters registry EC Point Format [IANA-TLS] as follows.

Value Description DTLS-OK Reference
TBD2 montgomery_x_le Y This doc

4. Security Considerations

The security considerations of [RFC5246] and most of the security considerations of [RFC4492] apply accordingly.

Curve25519 is designed to facilitate the production of high-performance constant-time implementations of the Curve25519 function. Implementors are encouraged to use a constant-time implementation of the Curve25519 function. This point is of crucial importance if the implementation chooses to reuse its supposedly ephemeral key pair for many key exchanges, which some implementations do in order to improve performance.

Curve25519 is believed to be at least as secure as the secp256r1 curve defined in [RFC4492], also know as NIST P-256. While the NIST curves are advertised as being chosen verifiably at random, there is no explanation for the seeds used to generate them. In contrast, the process used to pick Curve25519 is fully documented and rigid enough so that independent verification has been done. This is widely seen as a security advantage for Curve25519, since it prevents the generating party from maliciously manipulating the parameters.

Another family of curves available in TLS, generated in a fully verifiable way, is the Brainpool curves [RFC7027]. Specifically, brainpoolP256 is expected to provide a level of security comparable to Curve25519 and NIST P-256. However, due to the use of pseudo-random prime, it is significantly slower than NIST P-256, which is itself slower than Curve25519.

See [SafeCurves] for more comparisons between curves.

5. Acknowledgements

Several people provided comments and suggestions that helped improve this document: Kurt Roeckx, Andrey Jivsov, Robert Ransom, Rich Salz, David McGrew.

6. References

6.1. Normative References

[Curve25519] Bernstein, J., "Curve25519: New Diffie-Hellman Speed Records ", LNCS 3958, pp. 207-228, February 2006.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997.
[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.J. and T. Lange, "SafeCurves: choosing safe curves for elliptic-curve cryptography.", 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.
[EFD] Bernstein, D.J. and T. Lange, "Explicit-Formulas Database: XZ coordinates for Montgomery curves", January 2014.
[NaCl] Bernstein, D.J., "Cryptography in NaCL", March 2013.

Appendix A. The curve25519 function

A.1. Formulas

This section completes Section 2.1 by defining the Curve25519 function and the common public key G. It is meant as an alternative, self-contained specification for the Curve25519 function, possibly easier to follow than the original paper for most implementors.

A.1.1. Field Arithmetic

Throughout this section, P denotes the integer 2^255-19 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFED. The letters X and Z, and their numbered variants such as x1, z2, etc. denote integers modulo P, that is integers between 0 and P-1 and every operation between them is implictly done modulo P. For addition, subtraction and multiplication this means doing the operation in the usual way and then replacing the result with the remainder of its division by P. For division, "X / Z" means mutliplying (mod P) X by the modular inverse of Z mod P.

A convenient way to define the modular inverse of Z mod P is as Z^(P-2) mod P, that is Z to the power of 2^255-21 mod P. It is also a practical way of computing it, using a square-and-multiply method.

The four operations +, -, *, / modulo P are known as the field operations. Techniques for efficient implementation of the field operations are outside the scope of this document.

A.1.2. Conversion to and from internal format

For the purpose of this section, we will define a Curve25519 point as a pair (X, Z) were X and Z are integers mod P (as defined above). Though public keys were defined to be strings of 32 bytes, internally they are represented as curve points. This subsection describes the conversion process as two functions: PubkeyToPoint and PointToPubkey.

    Input: a public key b_0, ..., b_31
    Output: a Curve25519 point (X, Z)
        1. Set X = b_0 + 256 * b_1 + ... + 256^31 * b_31 mod P
        2. Set Z = 1
        3. Output (X, Z)

    Input: a Curve25519 point (X, Z)
    Output: a public key b_0, ..., b_31
        1. Set x1 = X / Z mod P
        2. Set b_0, ... b_31 such that
            x1 = b_0 + 256 * b_1 + ... + 256^31 * b_31 mod P
        3. Output b_0, ..., b_31

A.1.3. Scalar Multiplication

We first introduce the DoubleAndAdd function, defined as follows (formulas taken from [EFD]).

    Input: two points (X2, Z2), (X3, Z3), and an integer mod P: X1
    Output: two points (X4, Z4), (X5, Z5)
    Constant: the integer mod P: a24 = 121666 = 0x01DB42
    Variables: A, AA, B, BB, E, C, D, DA, CB are integers mod P
        1. Do the following computations mod P:
            A  = X2 + Z2
            AA = A2
            B  = X2 - Z2
            BB = B2
            E  = AA - BB
            C  = X3 + Z3
            D  = X3 - Z3
            DA = D * A
            CB = C * B
            X5 = (DA + CB)^2
            Z5 = X1 * (DA - CB)^2
            X4 = AA * BB
            Z4 = E * (BB + a24 * E)
        2. Output (X4, Z4) and (X5, Z5)

This may be taken as the abstract definition of an arbitrary-looking function. However, let's mention "the true meaning" of this function, without justification, in order to help the reader make more sense of it. It is possible to define operations "+" and "-" between Curve25519 points. Then, assuming (X2, Z2) - (X3, Z3) = (X1, 1), the DoubleAndAdd function returns points such that (X4, Z4) = (X2, Z2) + (X2, Z2) and (X5, Z5) = (X2, Z2) + (X3, Z3).

Taking the "+" operation as granted, we can define multiplication of a Curve25519 point by a positive integer as N * (X, Z) = (X, Z) + ... + (X, Z), with N point additions. It is possible to compute this operation, known as scalar multiplication, using an algorithm called the Montgomery ladder, as follows.

    Input: a Curve25519 point: (X, 1) and a 255-bits integer: N
    Output: a point (X1, Z1)
    Variable: a point (X2, Z2)
        0. View N as a sequence of bits b_254, ..., b_0,
            with b_254 the most significant bit
            and b_0 the least significant bit.
        1. Set X1 = 1 and Z1 = 0
        2. Set X2 = X and Z2 = 1
        3. For i from 254 downwards to 0, do:
            If b_i == 0, then:
                Set (X2, Z2) and (X1, Z1) to the output of
                DoubleAndAdd((X2, Z2), (X1, Z1), X)
                Set (X1, Z1) and (X2, Z2) to the output of
                DoubleAndAdd((X1, Z1), (X2, Z2), X)
        4. Output (X1, Z1)

A.1.4. Conclusion

We are now ready to define the Curve25519 function itself.

    Input: a public key P and a secret key S
    Output: a public key Q
    Variables: two Curve25519 points (X, Z) and (X1, Z1)
        1. Set (X, Z) = PubkeyToPoint(P)
        2. Set (X1, Z1) = ScalarMult((X, Z), S)
        3. Set Q = PointToPubkey((X1, Z1))
        4. Output Q

The common public key G mentioned in the first paragraph of Section 2.1 is defined as G = PointToPubkey((9, 1).

A.2. Test vectors

The following test vectors are taken from [NaCl]. Compared to this reference, the private key strings have been applied the ClampC function of the reference and converted to integers in order to fit the description given in [Curve25519] and the present memo.

The secret key of party A is denoted by S_a, it public key by P_a, and similarly for party B. The shared secret is SS.

          S_a = 0x6A2CB91DA5FB77B12A99C0EB872F4CDF

          P_a = 85 20 F0 09 89 30 A7 54 74 8B 7D DC B4 3E F7 5A
                0D BF 3A 0D 26 38 1A F4 EB A4 A9 8E AA 9B 4E 6A

          S_b = 0x6BE088FF278B2F1CFDB6182629B13B6F

          P_b = DE 9E DB 7D 7B 7D C1 B4 D3 5B 61 C2 EC E4 35 37
                3F 83 43 C8 5B 78 67 4D AD FC 7E 14 6F 88 2B 4F

           SS = 4A 5D 9D 5B A4 CE 2D E1 72 8E 3B F4 80 35 0F 25
                E0 7E 21 C9 47 D1 9E 33 76 F0 9B 3C 1E 16 17 42

A.3. Side-channel considerations

Curve25519 was specifically designed so that correct, fast, constant-time implementations are easier to produce. In particular, using a Montgomery ladder as described in the previous section ensures that, for any valid value of the secret key, the same sequence of field operations are performed, which eliminates a major source of side-channel leakage.

However, merely using Curve25519 with a Montgomery ladder does not prevent all side-channels by itself, and some point are the responsibility of implementors:

  1. In step 3 of SclarMult, avoid branches depending on b_i, as well as memory access patterns depending on b_i, for example by using safe conditional swaps on the inputs and outputs of DoubleAndAdd.
  2. Avoid data-dependant branches and memory access patterns in the implementation of field operations.

Techniques for implementing the field operations in constant time and with high performance are out of scope of this document. Let's mention however that, provided constant-time multiplication is available, division can be computed in constant time using exponentiation as described in Appendix A.1.1.

If using constant-time implementations of the field operations is not convenient, an option to reduce the information leaked this way is to replace step 2 of the SclarMult function with:

        2a. Pick Z uniformly randomly between 1 and P-1 included
        2b. Set X2 = X * Z and Z2 = Z

This method is known as randomizing projective coordinates. However, it is no guaranteed to avoid all side-channel leaks related to field operations.

Side-channel attacks are an active reseach domain that still sees new significant results, so implementors of the Curve25519 function are advised to follow recent security research closely.

Authors' Addresses

Simon Josefsson SJD AB EMail:
Manuel Pegourie-Gonnard Independent / PolarSSL EMail:

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