Internet DRAFT - draft-nir-ipsecme-curve25519
draft-nir-ipsecme-curve25519
Network Working Group Y. Nir
Internet-Draft Check Point
Intended status: Standards Track S. Josefsson
Expires: January 8, 2016 SJD
July 7, 2015
New Safe Curves for IKEv2 Key Agreement
draft-nir-ipsecme-curve25519-01
Abstract
This document describes the use of Curve25519 and Curve448
("Goldilocks") for ephemeral key exchange in the Internet Key
Exchange (IKEv2) protocol.
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Conventions Used in This Document . . . . . . . . . . . . 2
2. Curve25519 & Curve448 . . . . . . . . . . . . . . . . . . . . 3
3. Use and Negotiation in IKEv2 . . . . . . . . . . . . . . . . 3
3.1. Key Exchange Payload . . . . . . . . . . . . . . . . . . 4
3.2. Recipient Tests . . . . . . . . . . . . . . . . . . . . . 4
4. Security Considerations . . . . . . . . . . . . . . . . . . . 5
5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 5
6. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 5
7. References . . . . . . . . . . . . . . . . . . . . . . . . . 6
7.1. Normative References . . . . . . . . . . . . . . . . . . 6
7.2. Informative References . . . . . . . . . . . . . . . . . 6
Appendix A. The curve25519 function . . . . . . . . . . . . . . 6
A.1. Formulas . . . . . . . . . . . . . . . . . . . . . . . . 7
A.1.1. Field Arithmetic . . . . . . . . . . . . . . . . . . 7
A.1.2. Conversion to and from internal format . . . . . . . 7
A.1.3. Scalar Multiplication . . . . . . . . . . . . . . . . 8
A.1.4. Conclusion . . . . . . . . . . . . . . . . . . . . . 9
A.2. Test vectors . . . . . . . . . . . . . . . . . . . . . . 9
A.3. Side-channel considerations . . . . . . . . . . . . . . . 10
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 11
1. Introduction
[CFRG-Curves] specifies two new elliptic curve functions for use in
cryptographic applications. Curve25519 and Curve448 (also known as
"Goldilocks") are Diffie-Hellman functions designed with performance
and security in mind.
Almost ten years ago [RFC4753] specified the first elliptic curve
Diffie-Hellman groups for the Internet Key Exchange protocol (IKEv2 -
[RFC7296]). These were the so-called NIST curves. The state of the
art has advanced since then. More modern curves allow faster
implementations while making it much easier to write constant-time
implementations free from side-channel attacks. This document
defines such a curve for use in IKE. See [Curve25519] for details
about the speed and security of the Curve25519 function.
1.1. Conventions Used in This Document
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].
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2. Curve25519 & Curve448
All cryptographic computations are done using the Curve25519 and
Curve448 functions defined in [CFRG-Curves]. In this document, these
functions are considered black boxes that take for input a (secret
key, public key) pair and output a public key. Public keys for are
defined as strings of 32 octets. A common public key, denoted below
as G (or "base point" in the curves document) is shared by all users.
Since the functions only use the u-coordinate of the public key, only
the u coordinate of the base points is necessary. For Curve25519
Gu=9 ; for Curve448 Gu=5.
For Curve25519 secret keys are defined as 255-bit numbers such that
the high-order bit (bit 254) is set, and the three lowest-order bits
are unset.
For Curve448 secret keys are defined as 448-bit numbers such that the
high-order bit (bit 447) is set, and the two lowest-order bits are
unset.
An ephemeral Diffie-Hellman key exchange using Curve25519 or Curve448
goes as follows: Each party picks a secret key d uniformly at random
and computes the corresponding public key. "curve_function" is used
below to denote either Curve25519 or Curve448:
x_mine = curve_function(d, G)
Parties exchange their public keys (see Section 3.1) and compute a
shared secret:
SHARED_SECRET = curve_function(d, x_peer).
This shared secret is used directly as the value denoted g^ir in
section 2.14 of RFC 7296. It is always exactly 32 octets when these
functions are used.
A complete description of the Curve25519 function, as well as a few
implementation notes, are provided in Appendix A.
3. Use and Negotiation in IKEv2
The use of Curve25519 and Curve448 in IKEv2 is negotiated using a
Transform Type 4 (Diffie-Hellman group) in the SA payload of either
an IKE_SA_INIT or a CREATE_CHILD_SA exchange.
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3.1. Key Exchange Payload
The diagram for the Key Exchange Payload from section 3.4 of RFC 7296
is copied below for convenience:
1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Next Payload |C| RESERVED | Payload Length |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Diffie-Hellman Group Num | RESERVED |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |
~ Key Exchange Data ~
| |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
o Payload Length - Since the public key is 32 octets, the Payload
Length field always contains 40.
o The Diffie-Hellman Group Num is xx for Curve25519, or yy for
Curve448 (both TBA by IANA).
o The Key Exchange Data is 32 octets encoded as an array of bytes in
little-endian order as described in section 8 of [CFRG-Curves]
3.2. Recipient Tests
This section describes the checks that a recipient of a public key
needs to perform. It is the equivalent of the tests described in
[RFC6989] for other Diffie-Hellman groups. We use "func" to denote
either Curve25519 or Curve448, as the tests are similar to both.
Both functions were designed in a way that the result of func(d, x)
will never reveal information about d, provided it was chosen as
prescribed, for any value of x.
Define legitimate values of x as the values that can be obtained as x
= func(d, G) for some d, and call the other values illegitimate. The
definitions of the functions show 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 these functions that impose
this restriction on their input and others that don't, IKEv2
implementations 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.
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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.
4. Security Considerations
Curve25519 is designed to facilitate the production of high-
performance constant-time implementations. 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. The same is true for Curve448.
Curve25519 is believed to be at least as secure as the 256-bit random
ECP group (group 19) defined in RFC 4753, also known as NIST P-256 or
secp256r1. Curve448 is believed to be more secure than the 384-bit
random ECP group (group 20), also known as NIST P-384 or secp384r1.
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 these curves 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 IKE, generated in a fully
verifiable way, is the Brainpool curves [RFC6954]. Specifically,
brainpoolP256 (group 28) 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.
5. IANA Considerations
IANA is requested to assign two values from the IKEv2 "Transform Type
4 - Diffie-Hellman Group Transform IDs" registry, with names
"Curve25519" and "Curve448" and this document as reference. The
Recipient Tests field should also point to this document.
6. Acknowledgements
Curve25519 was designed by D. J. Bernstein and Tanja Lange.
Curve448 ("Goldilocks") is by Mike Hamburg. The specification of
wire format is Sean Turner, Rich Salz, and Watson Ladd, with Adam
Langley editing the current document. Much of the text in this
document is copied from Simon's draft for the TLS working group.
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7. References
7.1. Normative References
[CFRG-Curves]
Langley, A., Salz, R., and S. Turner, "Elliptic Curves for
Security", draft-irtf-cfrg-curves-02 (work in progress),
March 2015.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC7296] Kivinen, T., Kaufman, C., Hoffman, P., Nir, Y., and P.
Eronen, "Internet Key Exchange Protocol Version 2
(IKEv2)", RFC 7296, October 2014.
7.2. Informative References
[Curve25519]
Bernstein, J., "Curve25519: New Diffie-Hellman Speed
Records", LNCS 3958, February 2006,
<http://dx.doi.org/10.1007/11745853_14>.
[EFD] Bernstein, D. and T. Lange, "Explicit-Formulas Database:
XZ coordinates for Montgomery curves", January 2014,
<http://www.hyperelliptic.org/EFD/g1p/
auto-montgom-xz.html>.
[NaCl] Bernstein, D., "Cryptography in NaCL", March 2013,
<http://cr.yp.to/highspeed/naclcrypto-20090310.pdf>.
[RFC4753] Fu, D. and J. Solinas, "ECP Groups For IKE and IKEv2", RFC
4753, January 2007.
[RFC6954] Merkle, J. and M. Lochter, "Using the Elliptic Curve
Cryptography (ECC) Brainpool Curves for the Internet Key
Exchange Protocol Version 2 (IKEv2)", RFC 6954, July 2013.
[RFC6989] Sheffer, Y. and S. Fluhrer, "Additional Diffie-Hellman
Tests for the Internet Key Exchange Protocol Version 2
(IKEv2)", RFC 6989, July 2013.
Appendix A. The curve25519 function
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A.1. Formulas
This section completes Section 2 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.
PubkeyToPoint:
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)
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PointToPubkey:
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]).
DoubleAndAdd:
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.
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ScalarMult:
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)
else:
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.
Curve25519:
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
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.
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S_a = 0x6A2CB91DA5FB77B12A99C0EB872F4CDF
4566B25172C1163C7DA518730A6D0770
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
E60E80838B7FE1794B8A4A627E08AB58
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
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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
Yoav Nir
Check Point Software Technologies Ltd.
5 Hasolelim st.
Tel Aviv 6789735
Israel
Email: ynir.ietf@gmail.com
Simon Josefsson
SJD AB
Email: simon@josefsson.org
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