Network Working Group B. Black
Internet-Draft Microsoft
Intended status: Informational J. Bos
Expires: June 24, 2015 NXP Semiconductors
C. Costello
Microsoft Research
A. Langley
Google Inc
P. Longa
M. Naehrig
Microsoft Research
December 21, 2014
Rigid Parameter Generation for Elliptic Curve Cryptography
draft-black-rpgecc-01
Abstract
This memo describes algorithms for deterministically generating
parameters for elliptic curves over prime fields offering high
practical security in cryptographic applications, including Transport
Layer Security (TLS) and X.509 certificates. The algorithms can
generate domain parameters at any security level for modern (twisted)
Edwards curves.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
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and may be updated, replaced, or obsoleted by other documents at any
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This Internet-Draft will expire on June 24, 2015.
Copyright Notice
Copyright (c) 2014 IETF Trust and the persons identified as the
document authors. All rights reserved.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3
2. Scope and Relation to Other Specifications . . . . . . . . . 3
3. Security Requirements . . . . . . . . . . . . . . . . . . . . 3
4. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5. Parameter Generation . . . . . . . . . . . . . . . . . . . . 4
5.1. Deterministic Curve Parameter Generation . . . . . . . . 4
5.1.1. Edwards Curves . . . . . . . . . . . . . . . . . . . 4
5.1.2. Twisted Edwards Curves . . . . . . . . . . . . . . . 5
6. Generators . . . . . . . . . . . . . . . . . . . . . . . . . 6
7. Isogenies from the (twisted) Edwards to the Montgomery model 6
7.1. Edwards to Montgomery for p = 3 (mod 4) . . . . . . . . . 6
7.2. Twisted Edwards to Montogmery for p = 1 (mod 4) . . . . . 7
8. Recommended Curves . . . . . . . . . . . . . . . . . . . . . 8
9. TLS NamedCurve Types . . . . . . . . . . . . . . . . . . . . 8
10. Use with ECDSA . . . . . . . . . . . . . . . . . . . . . . . 9
10.1. Object Identifiers . . . . . . . . . . . . . . . . . . . 9
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9
12. Security Considerations . . . . . . . . . . . . . . . . . . . 9
13. Intellectual Property Rights . . . . . . . . . . . . . . . . 9
14. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9
15. References . . . . . . . . . . . . . . . . . . . . . . . . . 10
15.1. Normative References . . . . . . . . . . . . . . . . . . 10
15.2. Informative References . . . . . . . . . . . . . . . . . 10
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 12
1. Introduction
Since the initial standardization of elliptic curve cryptography
(ECC) in [SEC1] there has been significant progress related to both
efficiency and security of curves and implementations. Notable
examples are algorithms protected against certain side-channel
attacks, different 'special' prime shapes which allow faster modular
arithmetic, and a larger set of curve models from which to choose.
There is also concern in the community regarding the generation and
potential weaknesses of the curves defined in [NIST].
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This memo describes a deterministic algorithm for generation of
elliptic curves for cryptography. The constraints in the generation
process produce curves that support constant-time, exception-free
scalar multiplications that are resistant to a wide range of side-
channel attacks including timing and cache attacks, thereby offering
high practical security in cryptographic applications. The
deterministic algorithm operates without any hidden parameters,
reliance on randomness or any other processes offering opportunities
for manipulation of the resulting curves. The selection between
curve models is determined by choosing the curve form that supports
the fastest (currently known) complete formulas for each modularity
option of the underlying field prime. Specifically, the Edwards
curve x^2 + y^2 = 1 + dx^2y^2 is used with primes p with p = 3 mod 4,
and the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 is used for
primes p with p = 1 mod 4.
1.1. Requirements Language
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 RFC 2119 [RFC2119].
2. Scope and Relation to Other Specifications
This document specifies a deterministic algorithm for generating
elliptic curve domain parameters over prime fields GF(p), with p
having a length of twice the desired security level in bits, in
(twisted) Edwards form.
3. Security Requirements
For each curve at a specific security level:
1. The domain parameters SHALL be generated in a simple,
deterministic manner, without any secret or random inputs. The
derivation of the curve parameters is defined in Section 5.
2. The trace of Frobenius MUST NOT be in {0, 1} in order to rule out
the attacks described in [Smart], [AS], and [S], as in [EBP].
3. MOV Degree: the embedding degree k MUST be greater than (r - 1) /
100, as in [EBP].
4. CM Discriminant: discriminant D MUST be greater than 2^100, as in
[SC].
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4. Notation
Throughout this document, the following notation is used:
p: Denotes the prime number defining the base field.
GF(p): The finite field with p elements.
d: An element in the finite field GF(p), different from -1,0.
Ed: The elliptic curve Ed/GF(p): x^2 + y^2 = 1 + dx^2y^2 in
Edwards form, defined over GF(p) by the parameter d.
tEd: The elliptic curve tEd/GF(p): -x^2 + y^2 = 1 + dx^2y^2 in
twisted Edwards form, defined over GF(p) by the parameter d.
rd: The largest odd divisor of the number of GF(p)-rational
points on Ed or tEd.
td: The trace of Frobenius of Ed or tEd such that
#Ed(GF(p)) = p + 1 - td or #tEd(GF(p)) = p + 1 - td,
respectively.
rd': The largest odd divisor of the number of GF(p)-rational
points on the non-trivial quadratic twist Ed' or tEd'.
hd: The index (or cofactor) of the subgroup of order rd in the
group of GF(p)-rational points on Ed or tEd.
hd': The index (or cofactor) of the subgroup of order rd' in the
group of GF(p)-rational points on the non-trivial quadratic
twist of Ed or tEd.
P: A generator point defined over GF(p) of prime order rd on Ed
or tEd.
X(P): The x-coordinate of the elliptic curve point P.
Y(P): The y-coordinate of the elliptic curve point P.
5. Parameter Generation
This section describes the generation of the curve parameters, namely
the curve parameter d, and a generator point P of the prime order
subgroup of the elliptic curve. Best practice is to use primes with
p = 3 mod 4. For compatibility with some deployed implementations, a
generation process for primes with p = 1 mod 4 is also provided.
5.1. Deterministic Curve Parameter Generation
5.1.1. Edwards Curves
For a prime p = 3 mod 4, the elliptic curve Ed in Edwards form is
determined by the non-square element d from GF(p), different from
-1,0 with smallest absolute value such that #Ed(GF(p)) = hd * rd,
#Ed'(GF(p)) = hd' * rd', hd = hd' = 4, and both subgroup orders rd
and rd' are prime. In addition, care must be taken to ensure the MOV
degree and CM discriminant requirements from Section 3 are met.
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Input: a prime p, with p = 3 mod 4
Output: the parameter d defining the curve Ed
1. Set d = 0
2. repeat
repeat
if (d > 0) then
d = -d
else
d = -d + 1
end if
until d is not a square in GF(p)
Compute rd, rd', hd, hd' where #Ed(GF(p)) = hd * rd,
#Ed'(GF(p)) = hd' * rd', hd and hd' are powers of 2 and rd, rd'
are odd
until ((hd = hd' = 4) and rd is prime and rd' is prime)
3. Output d
GenerateCurveEdwards
5.1.2. Twisted Edwards Curves
For a prime p = 1 mod 4, the elliptic curve tEd in twisted Edwards
form is determined by the non-square element d from GF(p), different
from -1,0 with smallest absolute value such that #tEd(GF(p)) = hd *
rd, #tEd'(GF(p)) = hd' * rd', hd = 8, hd' = 4 and both subgroup
orders rd and rd' are prime. In addition, care must be taken to
ensure the MOV degree and CM discriminant requirements from Section 3
are met.
Input: a prime p, with p = 1 mod 4
Output: the parameter d defining the curve tEd
1. Set d = 0
2. repeat
repeat
if (d > 0) then
d = -d
else
d = -d + 1
end if
until d is not a square in GF(p)
Compute rd, rd', hd, hd' where #tEd(GF(p)) = hd * rd,
#tEd'(GF(p)) = hd' * rd', hd and hd' are powers of 2 and rd, rd'
are odd
until (hd = 8 and hd' = 4 and rd is prime and rd' is prime)
3. Output d
GenerateCurveTEdwards
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6. Generators
The generator points P = (X(P),Y(P)) for all curves are selected by
taking the smallest positive value x in GF(p) (when represented as an
integer) such that (x, y) is on the curve and such that (X(P),Y(P)) =
8 * (x, y) has large prime order rd.
Input: a prime p and curve parameters non-square d and
a = -1 for twisted Edwards (p = 1 mod 4) or
a = 1 for Edwards (p = 3 mod 4)
Output: a generator point P = (X(P), Y(P)) of order rd
1. Set x = 0 and found_gen = false
2. while (not found_gen) do
x = x + 1
while ((1 - a * x^2) * (1 - d * x^2) is not a quadratic
residue mod p) do
x = x + 1
end while
Compute an integer s, 0 < s < p, such that
s^2 * (1 - d * x^2) = 1 - a * x^2 mod p
Set y = min(s, p - s)
(X(P), Y(P)) = 8 * (x, y)
if ((X(P), Y(P)) has order rd on Ed or tEd, respectively) then
found_gen = true
end if
end while
3. Output (X(P),Y(P))
GenerateGen
7. Isogenies from the (twisted) Edwards to the Montgomery model
For applications requiring Montgomery curves, such as x-only point
format for elliptic curve Diffie-Hellmann (ECDH) key exchange,
isogenies from the generated (twisted) Edwards curves can be produced
as described in the following sections.
7.1. Edwards to Montgomery for p = 3 (mod 4)
For a prime p = 3 mod 4, and a given Edwards curve Ed: x^2 + y^2 = 1
+ d x^2 y^2 over GF(p) with non-square parameter d, let A = -(4d -
2). Then the Montgomery curve
EM: v^2 = u^3 + Au^2 + u
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is isogenous to Ed over GF(p). The following map is a 4-isogeny from
Ed to EM over GF(p):
phi: Ed -> EM, (x,y) -> (u,v), where
u = y^2 / x^2,
v = -y(x^2 + y^2 - 2) / x^3.
The neutral element (0,1) and the point of order two (0,-1) on Ed are
mapped to the point at infinity on EM. The dual isogeny is given by
phi_d: EM -> Ed, (u,v) -> (x,y), where
x = 4v(u - 1)(u + 1) / (u^4 - 2u^2 + 4v^2 + 1),
y = (u^2 + 2v - 1)(u^2 - 2v - 1) / (-u^4 + 2uv^2 + 2Au + 4u^2 + 1).
It holds phi_d(phi((x,y))) = [4](x,y) on Ed and phi(phi_d((u,v))) =
[4](u,v) on EM.
7.2. Twisted Edwards to Montogmery for p = 1 (mod 4)
For a prime p = 1 mod 4, and a given twisted Edwards curve tEd: -x^2
+ y^2 = 1 + d x^2 y^2 over GF(p) with non-square parameter d, let A =
4d + 2. Then the Montgomery curve
EM: v^2 = u^3 + Au^2 + u
is isogenous to tEd over GF(p). Let s in GF(p) be a fixed square
root of -1, i.e. s is a solution to the equation s^2 + 1 = 0 over
GF(p). Then, the following map is a 4-isogeny from tEd to EM over
GF(p):
phi: tEd -> EM, (x,y) -> (u,v), where
u = -y^2 / x^2,
v = -ys(x^2 - y^2 + 2) / x^3.
The neutral element (0,1) and the point of order two (0,-1) on tEd
are mapped to the point at infinity on EM. The dual isogeny is given
by
phi_d: EM -> tEd, (u,v) -> (x,y), where
x = 4sv(u - 1)(u + 1) / (u^4 - 2u^2 + 4v^2 + 1),
y = (u^2 + 2v - 1)(u^2 - 2v - 1) / (-u^4 + 2uv^2 + 2Au + 4u^2 + 1).
It holds phi_d(phi((x,y))) = [4](x,y) on tEd and phi(phi_d((u,v))) =
[4](u,v) on EM.
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8. Recommended Curves
The following figures give parameters for recommended twisted Edwards
and Edwards curves at the 128 and 192 bit security levels generated
using the algorithms defined in previous sections. All integer
values are unsigned.
p = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFED
d = 0x1DB41
r = 0x1000000000000000000000000000000014DEF9DEA2F79CD65812
631A5CF5D3ED
x(P) = 0x5C88197130371C6958E48E7C57393BDEDBA29F9231D24B3D4DA2
242EC821CDF1
y(P) = 0x6FEC03B956EC4A0E51A838029242F8B107C27399CC7840C34B95
5E478A8FB7A5
h = 0x8
p = 2^255 - 19, twisted Edwards
The isogenous Montgomery curve for p = 2^255 - 19 is given by A =
0x76D06.
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC3
d = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD19F
r = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE2471A1
CB46BE1CF61E4555AAB35C87920B9DCC4E6A3897D
x(P) = 0x61B111FB45A9266CC0B6A2129AE55DB5B30BF446E5BE4C005763FFA
8F33163406FF292B16545941350D540E46C206BDE
y(P) = 0x82983E67B9A6EEB08738B1A423B10DD716AD8274F1425F56830F98F
7F645964B0072B0F946EC48DC9D8D03E1F0729392
h = 0x4
p = 2^384 - 317, Edwards
The isogenous Montgomery curve for p = 2^384 - 317 is given by A =
0xB492.
9. TLS NamedCurve Types
As defined in [RFC4492], the name space NamedCurve is used for the
negotiation of elliptic curve groups for key exchange during TLS
session establishment. This document adds new NamedCurve types for
the elliptic curves defined in this document:
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enum {
ietfp255t1(TBD1),
ietfp255x1(TBD2),
ietfp384e1(TBD3),
ietfp384x1(TBD4)
} NamedCurve;
These curves are suitable for use with Datagram TLS [RFC6347].
10. Use with ECDSA
The (twisted) Edwards curves generated by the procedure defined in
this draft are suitable for use in signature algorithms such as
ECDSA. In compliance with [RFC5480], which only supports named
curves, namedCurve OIDs must be defined for the generated curves and
points must be represented as (x,y) in either uncompressed or
compressed format.
10.1. Object Identifiers
The following object identifiers represent the (twisted) Edwards
domain parameter sets defined in this draft:
ietfp255t1 OBJECT IDENTIFIER ::= {[TBDOID] 1}
ietfp384e1 OBJECT IDENTIFIER ::= {[TBDOID] 2}
11. Acknowledgements
The authors would like to thank Tolga Acar, Karen Easterbrook and
Brian LaMacchia for their contributions to the development of this
draft.
12. Security Considerations
TBD
13. Intellectual Property Rights
The authors have no knowledge about any intellectual property rights
that cover either the generation algorithms or the usage of the
domain parameters defined herein.
14. IANA Considerations
IANA is requested to assign numbers for the curves listed in
Section 9 in the "EC Named Curve" [IANA-TLS] registry of the
"Transport Layer Security (TLS) Parameters" registry as follows:
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+-------+-------------+---------+-----------+
| Value | Description | DTLS-OK | Reference |
+-------+-------------+---------+-----------+
| TBD1 | ietfp255t1 | Y | this doc |
| TBD2 | ietfp255x1 | Y | this doc |
| TBD3 | ietfp384e1 | Y | this doc |
| TBD4 | ietfp384x1 | Y | this doc |
+-------+-------------+---------+-----------+
Table 1
15. References
15.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
15.2. Informative References
[AS] Satoh, T. and K. Araki, "Fermat quotients and the
polynomial time discrete log algorithm for anomalous
elliptic curves", 1998.
[EBP] ECC Brainpool, "ECC Brainpool Standard Curves and Curve
Generation", October 2005, .
[ECCP] Bos, J., Halderman, J., Heninger, N., Moore, J., Naehrig,
M., and E. Wustrow, "Elliptic Curve Cryptography in
Practice", December 2013,
.
[FPPR] Faugere, J., Perret, L., Petit, C., and G. Renault, 2012,
.
[IANA-TLS]
IANA, "EC Named Curve Registry", 2014,
.
[MSR] Bos, J., Costello, C., Longa, P., and M. Naehrig,
"Selecting Elliptic Curves for Cryptography: An Efficiency
and Security Analysis", February 2014,
.
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[NIST] National Institute of Standards, "Recommended Elliptic
Curves for Federal Government Use", July 1999,
.
[RFC3279] Bassham, L., Polk, W., and R. Housley, "Algorithms and
Identifiers for the Internet X.509 Public Key
Infrastructure Certificate and Certificate Revocation List
(CRL) Profile", RFC 3279, April 2002.
[RFC3552] Rescorla, E. and B. Korver, "Guidelines for Writing RFC
Text on Security Considerations", BCP 72, RFC 3552, July
2003.
[RFC4050] Blake-Wilson, S., Karlinger, G., Kobayashi, T., and Y.
Wang, "Using the Elliptic Curve Signature Algorithm
(ECDSA) for XML Digital Signatures", RFC 4050, April 2005.
[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.
[RFC4754] Fu, D. and J. Solinas, "IKE and IKEv2 Authentication Using
the Elliptic Curve Digital Signature Algorithm (ECDSA)",
RFC 4754, January 2007.
[RFC5226] Narten, T. and H. Alvestrand, "Guidelines for Writing an
IANA Considerations Section in RFCs", BCP 26, RFC 5226,
May 2008.
[RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
"Elliptic Curve Cryptography Subject Public Key
Information", RFC 5480, March 2009.
[RFC5753] Turner, S. and D. Brown, "Use of Elliptic Curve
Cryptography (ECC) Algorithms in Cryptographic Message
Syntax (CMS)", RFC 5753, January 2010.
[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
Curve Cryptography Algorithms", RFC 6090, February 2011.
[RFC6347] Rescorla, E. and N. Modadugu, "Datagram Transport Layer
Security Version 1.2", RFC 6347, January 2012.
[S] Semaev, I., "Evaluation of discrete logarithms on some
elliptic curves", 1998.
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[SC] Bernstein, D. and T. Lange, "SafeCurves: choosing safe
curves for elliptic-curve cryptography", June 2014,
.
[SEC1] Certicom Research, "SEC 1: Elliptic Curve Cryptography",
September 2000,
.
[Smart] Smart, N., "The discrete logarithm problem on elliptic
curves of trace one", 1999.
[X9.62] ANSI, "Public Key Cryptography for the Financial Services
Industry, The Elliptic Curve Digital Signature Algorithm
(ECDSA)", 2005.
Authors' Addresses
Benjamin Black
Microsoft
One Microsoft Way
Redmond, WA 98115
US
Email: benblack@microsoft.com
Joppe W. Bos
NXP Semiconductors
Interleuvenlaan 80
3001 Leuven
Belgium
Email: joppe.bos@nxp.com
Craig Costello
Microsoft Research
One Microsoft Way
Redmond, WA 98115
US
Email: craigco@microsoft.com
Adam Langley
Google Inc
Email: agl@google.com
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Patrick Longa
Microsoft Research
One Microsoft Way
Redmond, WA 98115
US
Email: plonga@microsoft.com
Michael Naehrig
Microsoft Research
One Microsoft Way
Redmond, WA 98115
US
Email: mnaehrig@microsoft.com
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