Network Working Group B. Black
Internet-Draft Microsoft
Intended status: Informational J. Bos
Expires: January 1, 2015 NXP Semiconductors
C. Costello
P. Longa
M. Naehrig
Microsoft Research
June 30, 2014
Elliptic Curve Cryptography (ECC) Nothing Up My Sleeve (NUMS) Curves and
Curve Generation
draft-black-numscurves-00
Abstract
This memo describes a family of deterministically generated Nothing
Up My Sleeve (NUMS) elliptic curves over prime fields offering high
practical security in cryptographic applications, including Transport
Layer Security (TLS) and X.509 certificates. The domain parameters
are defined for both classical Weierstrass curves, for compatibility
with existing applications, and modern twisted Edwards curves,
allowing further efficiency improvements for a given security level.
Status of This Memo
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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. Requirements . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1. Technical Requirements . . . . . . . . . . . . . . . . . 4
3.2. Security Requirements . . . . . . . . . . . . . . . . . . 4
4. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5. Curve Parameters . . . . . . . . . . . . . . . . . . . . . . 5
5.1. Parameters for 256-bit Curves . . . . . . . . . . . . . . 5
5.2. Parameters for 384-bit Curves . . . . . . . . . . . . . . 6
5.3. Parameters for 512-bit Curves . . . . . . . . . . . . . . 7
6. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 8
7. Security Considerations . . . . . . . . . . . . . . . . . . . 8
8. Intellectual Property Rights . . . . . . . . . . . . . . . . 8
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 8
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 8
10.1. Normative References . . . . . . . . . . . . . . . . . . 8
10.2. Informative References . . . . . . . . . . . . . . . . . 9
Appendix A. Parameter Generation . . . . . . . . . . . . . . . . 10
A.1. Prime Generation . . . . . . . . . . . . . . . . . . . . 10
A.2. Deterministic Curve Parameter Generation . . . . . . . . 11
A.2.1. Weierstrass Curves . . . . . . . . . . . . . . . . . 11
A.2.2. Twisted Edwards Curves . . . . . . . . . . . . . . . 11
Appendix B. Generators . . . . . . . . . . . . . . . . . . . . . 12
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 set of elliptic curves for cryptography,
defined in [MSR] which have been specifically chosen to achieve
extremely high performance, security, and attack resistance. These
curves are deterministically generated based on algorithms defined in
this document and without any hidden parameters or reliance on
randomness, hence they are called Nothing Up My Sleeve (NUMS) curves.
The domain parameters are defined for both classical Weierstrass
curves, for compatibility with existing applications while delivering
better performance and stronger security, and modern twisted Edwards
curves, allowing even further efficiency improvements for a given
security level.
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 RFC specifies elliptic curve domain parameters over prime fields
GF(p) with p having a length of 256, 384, and 512 bits, in both
Weierstrass and twisted Edwards form. These parameters were
generated in a transparent and deterministic way and have been shown
to resist current cryptanalytic approaches. Furthermore, this
document identifies the security and implementation requirements for
the parameters, and describes the methods used for the deterministic
generation of the parameters.
This document addresses neither the cryptographic algorithms to be
used with the specified parameters nor their application in other
standards. However, it is consistent with the following RFCs that
specify the usage of ECC in protocols and applications:
o [RFC3279] for X.509 certificates
o [RFC4050] for XML signatures
o [RFC4492] for TLS
o [RFC4754] for IKE
o [RFC5480] for CRLs
o [RFC5753] for cryptographic message syntax (CMS)
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3. Requirements
3.1. Technical Requirements
1. Applicability to multiple cryptographic algorithms without
transformation, in particular key exchange (e.g. ECDHE) and
digital signature algorithms (e.g., ECDSA, Schnorr).
2. Multiple security levels using the same curve generation
algorithm with only a security parameter change. The curve
generation algorithm must be extensible to any security level.
3. Ability to use pre-computation for increased performance. In
particular, speed-up in key generation is important when a curve
is used with ephemeral key exchange algorithm, such as ECDHE.
4. The bit length of prime and order of curves for a given security
level MUST be divisible by 8. Specifically, constructions such
as NIST P-521 are to be avoided as they introduce
interoperability and implementation problems.
3.2. Security Requirements
For each curve type (twisted Edwards or Weierstrass) at a specific
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 Appendix A.
2. The curve SHALL NOT restrict the scalars to a small subset.
Using full-set scalars prevents implementation pitfalls that
might otherwise go unnoticed.
3. The curve selection SHALL include prime order curves with
cofactor 1 only. Composite order curves require changes in
protocols and in implementations. Additionally, implementations
for composite order curves must thwart subgroup attacks.
4. 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].
5. MOV Degree: the embedding degree k MUST be greater than (r - 1) /
100, as in [EBP].
6. 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:
s: Denotes the bit length, here s in {256,384,512}.
p: Denotes the prime number defining the base field.
c: A positive integer used in the representation of the prime
p = 2^s - c.
GF(p): The finite field with p elements.
b: An element in the finite field GF(p), different from -2,2.
Eb: The elliptic curve Eb/GF(p):
y^2 = x^3 - 3x + b
in short Weierstrass form, defined over GF(p) by the
parameter b.
rb: The order rb = #Eb(GF(p)) of the group of GF(p)-rational
points on Eb.
tb: The trace of Frobenius tb = p + 1 - rb of Eb.
rb': The order rb' = #E'b(GF(p)) = p + 1 + tb of the group of
GF(p)-rational points on the quadratic twist Eb':
y^2 = x^3 - 3x - b.
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
twisted Edwards form, defined over GF(p) by the parameter d.
rd: The subgroup order such that 4 * rd = #Ed(GF(p)) is the
order of the group of GF(p)-rational points on Ed.
td: The trace of Frobenius td = p + 1 - 4 * rd of Ed.
rd': The subgroup order such that 4 * rd' = #Ed'(GF(p)) = p + 1 + tb
is the order of the group of GF(p)-rational points on the
quadratic twist Ed':
-x^2 = y^2 = 1 + (1 / d) * x^2 * y^2.
P: A generator point defined over GF(p) either of prime order
rb in the Weierstrass curve Eb, or of prime order rd on the
twisted Edwards curve Ed.
X(P): The x-coordinate of the elliptic curve point P.
Y(P): The y-coordinate of the elliptic curve point P.
5. Curve Parameters
5.1. Parameters for 256-bit Curves
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p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFF43
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFF40
b = 0x25581
r = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE43C8275EA265C60E43C8275E
A265C60
X(P) = 0x01
Y(P) = 0x696F1853C1E466D7FC82C96CCEEEDD6BD02C2F9375894EC10BF46306C
2B56C77
h = 0x01
Curve-Id: numsp256d1
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFF43
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFF42
d = 0x3BEE
r = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFBE6AA55AD0A6BC64E5B84E6F1
122B4AD
X(P) = 0x0D
Y(P) = 0x7D0AB41E2A1276DBA3D330B39FA046BFBE2A6D63824D303F707F6FB53
31CADBA
h = 0x04
Curve-Id: numsp256t1
5.2. Parameters for 384-bit Curves
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC3
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC0
b = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF77BB
r = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD61EAF1EE
B5D6881BEDA9D3D4C37E27A604D81F67B0E61B9
X(P) = 0x02
Y(P) = 0x3C9F82CB4B87B4DC71E763E0663E5DBD8034ED422F04F82673330DC58
D15FFA2B4A3D0BAD5D30F865BCBBF503EA66F43
h = 0x01
Curve-Id: numsp384d1
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p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC3
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC2
d = 0x5158A
r = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFECD7D11ED
5A259A25A13A0458E39F4E451D6D71F70426E25
X(P) = 0x08
Y(P) = 0x749CDABA136CE9B65BD4471794AA619DAA5C7B4C930BFF8EBD798A8AE
753C6D72F003860FEBABAD534A4ACF5FA7F5BEE
h = 0x04
Curve-Id: numsp384t1
5.3. Parameters for 512-bit Curves
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFDC7
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFDC4
b = 0x1D99B
r = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFF5B3CA4FB94E7831B4FC258ED97D0BDC63B568B36607CD243CE
153F390433555D
X(P) = 0x02
Y(P) = 0x1C282EB23327F9711952C250EA61AD53FCC13031CF6DD336E0B932843
3AFBDD8CC5A1C1F0C716FDC724DDE537C2B0ADB00BB3D08DC83755B20
5CC30D7F83CF28
h = 0x01
Curve-Id: numsp512d1
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p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFDC7
a = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFDC6
d = 0x9BAA8
r = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFA7E50809EFDABBB9A624784F449545F0DCEA5FF0CB800F894E
78D1CB0B5F0189
X(P) = 0x20
Y(P) = 0x7D67E841DC4C467B605091D80869212F9CEB124BF726973F9FF048779
E1D614E62AE2ECE5057B5DAD96B7A897C1D72799261134638750F4F0C
B91027543B1C5E
h = 0x04
Curve-Id: numsp512t1
6. Acknowledgements
7. Security Considerations
In addition to the discussion in the requirements, [MSR], [SC], and
the other reference documents on EC security, users SHOULD match
curves with cryptographic functions of similar strength.
8. Intellectual Property Rights
The authors have no knowledge about any intellectual property rights
that cover the usage of the domain parameters defined herein.
However, readers should be aware that implementations based on these
domain parameters may require use of inventions covered by patent
rights.
9. IANA Considerations
This memo includes no request to IANA.
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
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10.2. 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,
.
[MSR] Bos, J., Costello, C., Longa, P., and M. Naehrig,
"Selecting Elliptic Curves for Cryptography: An Efficiency
and Security Analysis", February 2014,
.
[NIST] National Institute of Standards, "Recommended Elliptic
Curves for Federal Government Use", July 1999,
.
[RFC2629] Rose, M., "Writing I-Ds and RFCs using XML", RFC 2629,
June 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.
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[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.
[S] Semaev, I., "Evaluation of discrete logarithms on some
elliptic curves", 1998.
[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.
Appendix A. Parameter Generation
This section describes the generation of the curve parameters, namely
the base field prime p, the curve parameters b and d for the
Weierstrass and twisted Edwards curves, respectively, and a generator
point P of the prime order subgroup of the elliptic curve.
A.1. Prime Generation
For a given bitlength s in {256, 384, 512}, a prime p is selected as
a pseudo-Mersenne prime of the form p = 2^s - c for a positive
integer c. Each prime is determined by the smallest positive integer
c such that p = 2^s - c is prime and p = 3 mod 4.
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Input: a bit length s in {256, 384, 512}
Output: a prime p = 2^s - c with p = 3 mod 4
1. Set c = 1
2. while (p = 2^s - c is not prime) do
c = c + 4
end while
3. Output p
GenerateP
A.2. Deterministic Curve Parameter Generation
A.2.1. Weierstrass Curves
For a given bitlength s in {256, 384, 512} and a corresponding prime
p = 2^s - c selected according to Section A.1, the elliptic curve Eb
in short Weierstrass form is determined by the element b from GF(p),
different from -2,2 with smallest absolute value (when represented as
an integer in the interval [-(p - 1) / 2, (p - 1) / 2]) such that
both group orders rb and rb' are prime, and the group order rb < p,
i.e. tb > 1. In addition, care must be taken to ensure the MOV
degree and CM discriminant requirements from Section 3.2 are met.
Input: a prime p = 2^s - c with p = 3 mod 4
Output: the parameter b defining the curve Eb
1. Set b = 1
2. while (rb is not prime or rb' is not prime) do
b = b + 1
end while
3. if p + 1 < rb then
b = -b
end if
4. Output b
GenerateCurveWeierstrass
A.2.2. Twisted Edwards Curves
For a given bitlength s in {256, 384, 512} and a corresponding prime
p = 2^s - c selected according to Section A.1, the elliptic curve Ed
in twisted Edwards form is determined by the element d from GF(p),
different from -1,0 with smallest value (when represented as a
positive integer) such that both subgroup orders rd and rd' are
prime, and the group order 4 * rd < p, i.e. td > 1. In addition,
care must be taken to ensure the MOV degree and CM discriminant
requirements from Section 3.2 are met.
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Input: a prime p = 2^s - c with p = 3 mod 4
Output: the parameter d defining the curve Ed
1. Set d = 1
2. while (rd is not prime or rd' is not prime or 4*rd > p) do
d = d + 1;
end while
3. Output d
GenerateCurveTEdwards
Appendix B. Generators
The generator points on all six curves are selected as the points of
order rb and rd, respectively, with the smallest value for x(P) when
represented as a positive integer.
Input: a prime p, and a Weierstrass curve parameter b
Output: a generator point P = (x(P), y(P)) of order rb
1. Set x = 1
2. while ((x^3 - 3 * x + b) is not a quadratic residue modulo p) do
x = x + 1
end while
3. Compute an integer s, 0 < s < p, such that
s^2 = x^3 - 3 * x + b mod p
4. Set y = min(s, p - s)
5. Output P = (x, y)
GenerateGenWeierstrass
Input: a prime p and a twisted Edwards curve parameter d
Output: a generator point P = (x(P), y(P)) of order rd
1. Set x = 1
2. while ((d * x^2 = 1 mod p)
or ((1 + x^2) * (1 - d * x^2) is not a quadratic residue
modulo p)) do x = x + 1
end while
3. Compute an integer s, 0 < s < p, such that
s^2 * (1 - d * x^2) = 1 + x^2 mod p
4. Set y = min(s, p - s)
5. Output P = (x, y)
GenerateGenTEdwards
Authors' Addresses
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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
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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