lwig R. Struik
Internet-Draft Struik Security Consultancy
Intended status: Informational July 19, 2018
Expires: January 20, 2019
Alternative Elliptic Curve Representations
draft-struik-lwig-curve-representations-02
Abstract
This document specifies how to represent Montgomery curves and
(twisted) Edwards curves as curves in short-Weierstrass form and
illustrates how this can be used to implement elliptic curve
computations using existing implementations that already implement,
e.g., ECDSA and ECDH using NIST prime curves.
Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in RFC
2119 [RFC2119].
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
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on January 20, 2019.
Copyright Notice
Copyright (c) 2018 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
Struik Expires January 20, 2019 [Page 1]
Internet-Draft lwig-curve-representations July 2018
(https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Fostering Code Reuse with New Elliptic Curves . . . . . . . . 3
2. Specification of Wei25519 . . . . . . . . . . . . . . . . . . 3
3. Example Uses . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1. ECDSA-SHA256-25519 . . . . . . . . . . . . . . . . . . . 3
3.2. Other Uses . . . . . . . . . . . . . . . . . . . . . . . 4
4. Security Considerations . . . . . . . . . . . . . . . . . . . 4
5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 4
6. Normative References . . . . . . . . . . . . . . . . . . . . 4
Appendix A. Some (non-Binary) Elliptic Curves . . . . . . . . . 6
A.1. Curves in short-Weierstrass Form . . . . . . . . . . . . 6
A.2. Montgomery Curves . . . . . . . . . . . . . . . . . . . . 6
A.3. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 6
Appendix B. Elliptic Curve Group Operations . . . . . . . . . . 7
B.1. Group Law for Weierstrass Curves . . . . . . . . . . . . 7
B.2. Group Law for Montgomery Curves . . . . . . . . . . . . . 7
B.3. Group Law for Twisted Edwards Curves . . . . . . . . . . 8
Appendix C. Relationship Between Curve Models . . . . . . . . . 8
C.1. Mapping between twisted Edwards Curves and Montgomery
Curves . . . . . . . . . . . . . . . . . . . . . . . . . 8
C.2. Mapping between Montgomery Curves and Weierstrass Curves 9
C.3. Mapping between twisted Edwards Curves and Weierstrass
Curves . . . . . . . . . . . . . . . . . . . . . . . . . 10
Appendix D. Curve25519 and Cousins . . . . . . . . . . . . . . . 10
D.1. Curve Definition and Alternative Representations . . . . 10
D.2. Switching between Alternative Representations . . . . . . 10
D.3. Domain Parameters . . . . . . . . . . . . . . . . . . . . 12
Appendix E. Further Mappings . . . . . . . . . . . . . . . . . . 14
E.1. Isomorphic Mapping between Weierstrass Curves . . . . . . 14
E.2. Isogeneous Mapping between Weierstrass Curves . . . . . . 15
Appendix F. Further Cousins of Curve25519 . . . . . . . . . . . 15
F.1. Further Alternative Representations . . . . . . . . . . . 15
F.2. Further Switching . . . . . . . . . . . . . . . . . . . . 15
F.3. Further Domain Parameters . . . . . . . . . . . . . . . . 16
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 17
Struik Expires January 20, 2019 [Page 2]
Internet-Draft lwig-curve-representations July 2018
1. Fostering Code Reuse with New Elliptic Curves
It is well-known that elliptic curves can be represented using
different curve models. Recently, IETF standardized elliptic curves
that are claimed to have better performance and improved robustness
against "real world" attacks than curves represented in the
traditional "short" Weierstrass model. This draft specifies an
alternative representation of points of Curve25519, a so-called
Montgomery curve, and of points of Edwards25519, a so-called twisted
Edwards curve, which are both specified in [RFC7748], as points of a
specific so-called "short" Weierstrass curve, called Wei25519. The
draft also defines how to efficiently switch between these different
representations.
Use of Wei25519 allows easy definition of signature schemes and key
agreement schemes already specified for traditional NIST prime
curves, thereby allowing easy integration with existing
specifications, such as NIST SP 800-56a [SP-800-56a], FIPS Pub 186-4
[FIPS-186-4], and ANSI X9.62-2005 [ANSI-X9.62] and fostering code
reuse on platforms that already implement some of these schemes using
elliptic curve arithmetic for curves in "short" Weierstrass form (see
Appendix B.1).
2. Specification of Wei25519
For the specification of Wei25519 and its relationship to Curve25519
and Edwards25519, see Appendix D. For further details and background
information on elliptic curves, we refer to the other appendices.
The use of Wei25519 allows reuse of existing generic code that
implements short-Weierstrass curves, such as the NIST curve P256, to
also implement the CFRG curves Curve25519 and Ed25519. The draft
also caters to reuse of existing code where some domain parameters
may have been hardcoded, thereby widening the scope of applicability;
see Appendix F.
3. Example Uses
3.1. ECDSA-SHA256-25519
RFC 8032 [RFC8032] specifies the use of EdDSA, a "full" Schnorr
signature scheme, with instantiation by Edwards25519 and Ed448, two
so-called twisted Edwards curves. These curves can also be used with
the widely implemented signature scheme ECDSA [FIPS-186-4], by
instantiating ECDSA with the curve Wei25519 and hash function SHA-
256, where "under the hood" an implementation may carry out elliptic
curve scalar multiplication routines using the corresponding
representations of a point of the curve Wei25519 in Weierstrass form
Struik Expires January 20, 2019 [Page 3]
Internet-Draft lwig-curve-representations July 2018
as a point of the Montgomery curve Curve25519 or of the twisted
Edwards curve Edwards25519. (The corresponding ECDSA-SHA512-448
scheme arises if one were to specify a curve in short-Weierstrass
form corresponding to Ed448 and use the hash function SHA512.) Note
that, in either case, one can implement these schemes with the same
representation conventions as used with existing NIST specifications,
including bit/byte-ordering, compression functions, and the-like.
This allows implementations of ECDSA with the hash function SHA-256
and with the NIST curve P-256 or with the curve Wei25519 specified in
this draft to use the same implementation (instantiated with,
respectively, the NIST P-256 elliptic curve domain parameters or with
the domain parameters of curve Wei25519 specified in Appendix D).
3.2. Other Uses
Any existing specification of cryptographic schemes using elliptic
curves in Weierstrass form and that allows introduction of a new
elliptic curve (here: Wei25519) is amenable to similar constructs,
thus spawning "offspring" protocols, simply by instantiating these
using the new curve in "short" Weierstrass form, thereby allowing
code and/or specifications reuse and, for implementations that so
desire, carrying out curve computations "under the hood" on
Montgomery curve and twisted Edwards curve cousins hereof (where
these exist). This would simply require definition of a new object
identifier for any such envisioned "offspring" protocol. This could
significantly simplify standardization of schemes and help keeping
the resource and maintenance cost of implementations supporting
algorithm agility [RFC7696] at bay.
4. Security Considerations
The different representations of elliptic curve points discussed in
this draft are all obtained using a publicly known transformation.
Since this transformation is an isomorphism, this transformation maps
elliptic curve points to equivalent mathematical objects.
5. IANA Considerations
There is *currently* no IANA action required for this document. New
object identifiers would be required in case one wishes to specify
one or more of the "offspring" protocols exemplified in Section 3.
6. Normative References
Struik Expires January 20, 2019 [Page 4]
Internet-Draft lwig-curve-representations July 2018
[ANSI-X9.62]
ANSI X9.62-2005, "Public Key Cryptography for the
Financial Services Industry: The Elliptic Curve Digital
Signature Algorithm (ECDSA)", American National Standard
for Financial Services, Accredited Standards Committee X9,
Inc Anapolis, MD, 2005.
[FIPS-186-4]
FIPS 186-4, "Digital Signature Standard (DSS), Federal
Information Processing Standards Publication 186-4", US
Department of Commerce/National Institute of Standards and
Technology Gaithersburg, MD, July 2013.
[GECC] D. Hankerson, A.J. Menezes, S.A. Vanstone, "Guide to
Elliptic Curve Cryptography", New York: Springer-Verlag,
2004.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>.
[RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
(ECC) Brainpool Standard Curves and Curve Generation",
RFC 5639, DOI 10.17487/RFC5639, March 2010,
<https://www.rfc-editor.org/info/rfc5639>.
[RFC7696] Housley, R., "Guidelines for Cryptographic Algorithm
Agility and Selecting Mandatory-to-Implement Algorithms",
BCP 201, RFC 7696, DOI 10.17487/RFC7696, November 2015,
<https://www.rfc-editor.org/info/rfc7696>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, <https://www.rfc-editor.org/info/rfc7748>.
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
<https://www.rfc-editor.org/info/rfc8032>.
[SP-800-56a]
NIST SP 800-56a, "Recommendation for Pair-Wise Key
Establishment Schemes Using Discrete Log Cryptography,
Revision 2", US Department of Commerce/National Institute
of Standards and Technology Gaithersburg, MD, June 2013.
Struik Expires January 20, 2019 [Page 5]
Internet-Draft lwig-curve-representations July 2018
Appendix A. Some (non-Binary) Elliptic Curves
A.1. Curves in short-Weierstrass Form
Let GF(q) denote the finite field with q elements, where q is an odd
prime power and where q is not divisible by three. Let W_{a,b} be
the Weierstrass curve with defining equation y^2 = x^3 + a*x + b,
where a and b are elements of GF(q) and where 4*a^3 + 27*b^2 is
nonzero. The points of W_{a,b} are the ordered pairs (x, y) whose
coordinates are elements of GF(q) and that satisfy the defining
equation (the so-called affine points), together with the special
point O (the so-called "point at infinity").This set forms a group
under addition, via the so-called "chord-and-tangent" rule, where the
point at infinity serves as the identity element. See Appendix B.1
for details of the group operation.
A.2. Montgomery Curves
Let GF(q) denote the finite field with q elements, where q is an odd
prime power. Let M_{A,B} be the Montgomery curve with defining
equation B*v^2 = u^3 + A*u^2 + u, where A and B are elements of GF(q)
with A unequal to (+/-)2 and with B nonzero. The points of M_{A,B}
are the ordered pairs (u, v) whose coordinates are elements of GF(q)
and that satisfy the defining equation (the so-called affine points),
together with the special point O (the so-called "point at
infinity").This set forms a group under addition, via the so-called
"chord-and-tangent" rule, where the point at infinity serves as the
identity element. See Appendix B.2 for details of the group
operation.
A.3. Twisted Edwards Curves
Let GF(q) denote the finite field with q elements, where q is an odd
prime power. Let E_{a,d} be the twisted Edwards curve with defining
equation a*x^2 + y^2 = 1+ d*x^2*y^2, where a and d are distinct
nonzero elements of GF(q). The points of E_{a,d} are the ordered
pairs (x, y) whose coordinates are elements of GF(q) and that satisfy
the defining equation (the so-called affine points). It can be shown
that this set forms a group under addition if a is a square in GF(q),
whereas d is not, where the point (0, 1) serves as the identity
element. (Note that the identity element satisfies the defining
equation.) See Appendix B.3 for details of the group operation. An
Edwards curve is a twisted Edwards curve with a=1.
Struik Expires January 20, 2019 [Page 6]
Internet-Draft lwig-curve-representations July 2018
Appendix B. Elliptic Curve Group Operations
B.1. Group Law for Weierstrass Curves
For each point P of the Weierstrass curve W_{a,b}, the point at
infinity O serves as identity element, i.e., P + O = O + P = P.
For each affine point P:=(x, y) of the Weierstrass curve W_{a,b}, the
point -P is the point (x, -y) and one has P + (-P) = O.
Let P1:=(x1, y1) and P2:=(x2, y2) be distinct affine points of the
Weierstrass curve W_{a,b} and let Q:=P1 + P2, where Q is not the
identity element. Then Q:=(x, y), where
x + x1 + x2 = lambda^2 and y + y1 = lambda*(x1 - x), where lambda
= (y2 - y1)/(x2 - x1).
Let P:= (x1, y1) be an affine point of the Weierstrass curve W_{a,b}
and let Q:=2P, where Q is not the identity element. Then Q:= (x, y),
where
x + 2*x1 = lambda^2 and y + y1 = lambda*(x1 - x), where
lambda=(3*x1^2 + a)/(2*y1).
B.2. Group Law for Montgomery Curves
For each point P of the Montgomery curve M_{A,B}, the point at
infinity O serves as identity element, i.e., P + O = O + P = P.
For each affine point P:=(x, y) of the Montgomery curve M_{A,B}, the
point -P is the point (x, -y) and one has P + (-P) = O.
Let P1:=(x1, y1) and P2:=(x2, y2) be distinct affine points of the
Montgomery curve M_{A,B} and let Q:=P1 + P2, where Q is not the
identity element. Then Q:=(x, y), where
x + x1 + x2 = B*lambda^2 - A and y + y1 = lambda*(x1 - x), where
lambda=(y2 - y1)/(x2 - x1).
Let P:= (x1, y1) be an affine point of the Montgomery curve M_{A,B}
and let Q:=2P, where Q is not the identity element. Then Q:= (x, y),
where
x + 2*x1 = B*lambda^2 - A and y + y1 = lambda*(x1 - x), where
lambda=(3*x1^2 + 2*A*x1+1)/(2*y1).
Alternative and more efficient group laws exist, e.g., when using the
so-called Montgomery ladder. Details are out of scope.
Struik Expires January 20, 2019 [Page 7]
Internet-Draft lwig-curve-representations July 2018
B.3. Group Law for Twisted Edwards Curves
Note: The group laws below hold for twisted Edwards curves E_{a,d}
where a is a square in GF(q), whereas d is not. In this case, the
addition formulae below are defined for each pair of points, without
exceptions. Generalizations of this group law to other twisted
Edwards curves are out of scope.
For each point P of the twisted Edwards curve E_{a,d}, the point
O=(0,1) serves as identity element, i.e., P + O = O + P = P.
For each point P:=(x, y) of the twisted Edwards curve E_{a,d}, the
point -P is the point (-x, y) and one has P + (-P) = O.
Let P1:=(x1, y1) and P2:=(x2, y2) be points of the twisted Edwards
curve E_{a,d} and let Q:=P1 + P2. Then Q:=(x, y), where
x = (x1*y2 + x2*y1)/(1 + d*x1*x2*y1*y2) and y = (y1*y2 -
a*x1*x2)/(1 - d*x1*x2*y1*y2).
Let P:=(x1, y1) be a point of the twisted Edwards curve E_{a,d} and
let Q:=2P. Then Q:=(x, y), where
x = (2*x1*y1)/(1 + d*x1^2*y1^2) and y = (y1^2 - a*x1^2)/(1 -
d*x1^2*y1^2).
Note that one can use the formulae for point addition to implement
point doubling, taking inverses and adding the identity element as
well (i.e., the point addition formulae are uniform and complete
(subject to our Note above)).
Appendix C. Relationship Between Curve Models
The non-binary curves specified in Appendix A are expressed in
different curve models, viz. as curves in short-Weierstrass form, as
Montgomery curves, or as twisted Edwards curves. These curve models
are related, as follows.
C.1. Mapping between twisted Edwards Curves and Montgomery Curves
One can map points of the Montgomery curve M_{A,B} to points of the
twisted Edwards curve E_{a,d}, where a:=(A+2)/B and d:=(A-2)/B and,
conversely, map points of the twisted Edwards curve E_{a,d} to points
of the Montgomery curve M_{A,B}, where A:=2(a+d)/(a-d) and where
B:=4/(a-d). For twisted Edwards curves we consider (i.e., those
where a is a square in GF(q), whereas d is not), this defines a one-
to-one correspondence, which - in fact - is an isomorphism between
Struik Expires January 20, 2019 [Page 8]
Internet-Draft lwig-curve-representations July 2018
M_{A,B} and E_{a,d}, thereby showing that, e.g., the discrete
logarithm problem in either curve model is equally hard.
For the Montgomery curves and twisted Edwards curves we consider, the
mapping from M_{A,B} to E_{a,d} is defined by mapping the point at
infinity O and the point (0, 0) of order two of M_{A,B} to,
respectively, the point (0, 1) and the point (0, -1) of order two of
E_{a,d}, while mapping each other point (u, v) of M_{A,B} to the
point (x, y):=(u/v, (u-1)/(u+1)) of E_{a,d}. The inverse mapping from
E_{a,d} to M_{A,B} is defined by mapping the point (0, 1) and the
point (0, -1) of order two of E_{a,d} to, respectively, the point at
infinity O and the point (0, 0) of order two of M_{A,B}, while each
other point (x, y) of E_{a,d} is mapped to the point (u,
v):=((1+y)/(1-y), (1+y)/((1-y)*x)) of M_{A,B}.
Implementations may take advantage of this mapping to carry out
elliptic curve group operations originally defined for a twisted
Edwards curve on the corresponding Montgomery curve, or vice-versa,
and translating the result back to the original curve, thereby
potentially allowing code reuse.
C.2. Mapping between Montgomery Curves and Weierstrass Curves
One can map points of the Montgomery curve M_{A,B} to points of the
Weierstrass curve W_{a,b}, where a:=(3-A^2)/(3*B^2) and
b:=(2*A^3-9*A)/(27*B^3). This defines a one-to-one correspondence,
which - in fact - is an isomorphism between M_{A,B} and W_{a,b},
thereby showing that, e.g., the discrete logarithm problem in either
curve model is equally hard.
The mapping from M_{A,B} to W_{a,b} is defined by mapping the point
at infinity O of M_{A,B} to the point at infinity O of W_{a,b}, while
mapping each other point (u, v) of M_{A,B} to the point (x, y):=(u/
B+A/(3*B), v/B) of W_{a,b}. Note that not all Weierstrass curves can
be injectively mapped to Montgomery curves, since the latter have a
point of order two and the former may not. In particular, if a
Weierstrass curve has prime order, such as is the case with the so-
called "NIST curves", this inverse mapping is not defined.
This mapping can be used to implement elliptic curve group operations
originally defined for a twisted Edwards curve or for a Montgomery
curve using group operations on the corresponding elliptic curve in
short-Weierstrass form and translating the result back to the
original curve, thereby potentially allowing code reuse. Note that
implementations for elliptic curves with short-Weierstrass form that
hard-code the domain parameter a to a= -3 (which value is known to
allow more efficient implementations) cannot always be used this way,
Struik Expires January 20, 2019 [Page 9]
Internet-Draft lwig-curve-representations July 2018
since the curve W_{a,b} may not always be expressed in terms of a
Weierstrass curve with a=-3 via a coordinate transformation.
C.3. Mapping between twisted Edwards Curves and Weierstrass Curves
One can map points of the twisted Edwards curve E_{a,d} to points of
the Weierstrass curve W_{a,b}, via function composition, where one
uses the isomorphic mapping between twisted Edwards curve and
Montgomery curves of Appendix C.1 and the one between Montgomery and
Weierstrass curves of Appendix C.2. Obviously, one can use function
composition (now using the respective inverses) to realize the
inverse of this mapping.
Appendix D. Curve25519 and Cousins
D.1. Curve Definition and Alternative Representations
The elliptic curve Curve25519 is the Montgomery curve M_{A,B} defined
over the prime field GF(p), with p:=2^{255}-19, where A:=486662 and
B:=1. This curve has order h*n, where h=8 and where n is a prime
number. For this curve, A^2-4 is not a square in GF(p), whereas A+2
is. The quadratic twist of this curve has order h1*n1, where h1=4
and where n1 is a prime number. For this curve, the base point is
the point (Gu,Gv), where Gu=9 and where Gv is an odd integer in the
interval [0, p-1].
This curve has the same group structure as (is "isomorphic" to) the
twisted Edwards curve E_{a,d} defined over GF(p), with as base point
the point (Gx,Gy), where parameters are as specified in Appendix D.3.
This curve is denoted as Edwards25519. For this curve, the parameter
a is a square in GF(p), whereas d is not, so the group laws of
Appendix B.3 apply.
The curve is also isomorphic to the elliptic curve W_{a,b} in short-
Weierstrass form defined over GF(p), with as base point the point
(Gx',Gy'), where parameters are as specified in Appendix D.3. This
curve is denoted as Wei25519.
D.2. Switching between Alternative Representations
Each affine point (u,v) of Curve25519 corresponds to the point
(x,y):=(u + A/3,y) of Wei25519, while the point at infinity of
Curve25519 corresponds to the point at infinity of Wei25519. (Here,
we used the mapping of Appendix C.2.) Under this mapping, the base
point (Gu,Gv) of Curve25519 corresponds to the base point (Gx',Gy')
of Wei25519. The inverse mapping maps the affine point (x,y) of
Wei25519 to (u,v):=(x - A/3,y) of Curve25519, while mapping the point
at infinity of Wei25519 to the point at infinity of Curve25519. Note
Struik Expires January 20, 2019 [Page 10]
Internet-Draft lwig-curve-representations July 2018
that this mapping involves a simple shift of the first coordinate and
can be implemented via integer-only arithmetic as a shift of (p+A)/3
for the isomorphic mapping and a shift of -(p+A)/3 for its inverse,
where delta=(p+A)/3 is the element of GF(p) defined by
delta 19298681539552699237261830834781317975544997444273427339909597
334652188435537
(=0x2aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaad2
451)
The curve Edwards25519 is isomorphic to the curve Curve25519, where
the base point (Gu,Gv) of Curve25519 corresponds to the base point
(Gx,Gy) of Edwards25519 and where the point at infinity and the point
(0,0) of order two of Curve25519 correspond to, respectively, the
point (0, 1) and the point (0, -1) of order two of Edwards25519 and
where each other point (u, v) of Curve25519 corresponds to the point
(c*u/v, (u-1)/(u+1)) of Edwards25519, where c is the element of GF(p)
defined by
c sqrt(-(A+2))
51042569399160536130206135233146329284152202253034631822681833788
666877215207
(=0x70d9120b 9f5ff944 2d84f723 fc03b081 3a5e2c2e b482e57d
3391fb55 00ba81e7)
(Here, we used the mapping of Appendix C.1.) The inverse mapping
from Edwards25519 to Curve25519 is defined by mapping the point (0,
1) and the point (0, -1) of order two of Edwards25519 to,
respectively, the point at infinity and the point (0,0) of order two
of Curve25519 and having each other point (x, y) of Edwards25519
correspond to the point ((1 + y)/(1 - y), c*(1 + y)/((1-y)*x)).
The curve Edwards25519 is isomorphic to the Weierstrass curve
Wei25519, where the base point (Gx,Gy) of Edwards25519 corresponds to
the base point (Gx',Gy') of Wei25519 and where the identity element
(0,1) and the point (0,-1) of order two of Edwards25519 correspond
to, respectively, the point at infinity O and the point (A/3, 0) of
order two of Wei25519 and where each other point (x, y) of
Edwards25519 corresponds to the point (x', y'):=((1+y)/(1-y)+A/3,
c*(1+y)/((1-y)*x)) of Wei25519, where c was defined before. (Here,
we used the mapping of Appendix C.3.) The inverse mapping from
Wei25519 to Edwards25519 is defined by mapping the point at infinity
O and the point (A/3, 0) of order two of Wei25519 to, respectively,
the identity element (0,1) and the point (0,-1) of order two of
Struik Expires January 20, 2019 [Page 11]
Internet-Draft lwig-curve-representations July 2018
Edwards25519 and having each other point (x, y) of Wei25519
correspond to the point (c*(3*x-A)/(3*y), (3*x-A-3)/(3*x-A+3)).
Note that these mappings can be easily realized in projective
coordinates, using a few field multiplications only, thus allowing
switching between alternative representations with negligible
relative incremental cost.
D.3. Domain Parameters
The parameters of the Montgomery curve and the corresponding
isomorphic curves in twisted Edwards curve and short-Weierstrass form
are as indicated below. Here, the domain parameters of the
Montgomery curve Curve25519 and of the twisted Edwards curve
Edwards25519 are as specified in RFC 7748; the domain parameters of
Wei25519 are "new".
General parameters (for all curve models):
p 2^{255}-19
(=0x7fffffff ffffffff ffffffff ffffffff ffffffff ffffffff
ffffffff ffffffed)
h 8
n 72370055773322622139731865630429942408571163593799076060019509382
85454250989
(=2^{252} + 0x14def9de a2f79cd6 5812631a 5cf5d3ed)
h1 4
n1 14474011154664524427946373126085988481603263447650325797860494125
407373907997
(=2^{253} - 0x29bdf3bd 45ef39ac b024c634 b9eba7e3)
Montgomery curve-specific parameters (for Curve25519):
A 486662
B 1
Gu 9 (=0x9)
Gv 14781619447589544791020593568409986887264606134616475288964881837
755586237401
Struik Expires January 20, 2019 [Page 12]
Internet-Draft lwig-curve-representations July 2018
(=0x20ae19a1 b8a086b4 e01edd2c 7748d14c 923d4d7e 6d7c61b2
29e9c5a2 7eced3d9)
Twisted Edwards curve-specific parameters (for Edwards25519):
a -1 (-0x01)
d -121665/121666
(=370957059346694393431380835087545651895421138798432190163887855
33085940283555)
(=0x52036cee 2b6ffe73 8cc74079 7779e898 00700a4d 4141d8ab
75eb4dca 135978a3)
Gx 15112221349535400772501151409588531511454012693041857206046113283
949847762202
(=0x216936d3 cd6e53fe c0a4e231 fdd6dc5c 692cc760 9525a7b2
c9562d60 8f25d51a)
Gy 4/5
(=463168356949264781694283940034751631413079938662562256157830336
03165251855960)
(=0x66666666 66666666 66666666 66666666 66666666 66666666
66666666 66666658)
Weierstrass curve-specific parameters (for Wei25519):
a 19298681539552699237261830834781317975544997444273427339909597334
573241639236
(=0x2aaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
aaaaaa98 4914a144)
b 55751746669818908907645289078257140818241103727901012315294400837
956729358436
(=0x7b425ed0 97b425ed 097b425e d097b425 ed097b42 5ed097b4
260b5e9c 7710c864)
Gx' 19298681539552699237261830834781317975544997444273427339909597334
652188435546
(=0x2aaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
aaaaaaaa aaad245a)
Struik Expires January 20, 2019 [Page 13]
Internet-Draft lwig-curve-representations July 2018
Gy' 14781619447589544791020593568409986887264606134616475288964881837
755586237401
(=0x20ae19a1 b8a086b4 e01edd2c 7748d14c 923d4d7e 6d7c61b2
29e9c5a2 7eced3d9)
Appendix E. Further Mappings
The non-binary curves specified in Appendix A are expressed in
different curve models, viz. as curves in short-Weierstrass form, as
Montgomery curves, or as twisted Edwards curves. Within each curve
model, further mappings exist that induce a mapping between elliptic
curves within each curve model. This can be exploited to force some
of the domain parameter to a value that allows a more efficient
implementation of the addition formulae.
E.1. Isomorphic Mapping between Weierstrass Curves
One can map points of the Weierstrass curve W_{a,b} to points of the
Weierstrass curve W_{a',b'}, where a:=a'*u^4 and b:=b'*u^6 for some
nonzero value u of the finite field GF(q). This defines a one-to-one
correspondence, which - in fact - is an isomorphism between W_{a,b}
and W_{a',b'}, thereby showing that, e.g., the discrete logarithm
problem in either curve model is equally hard.
The mapping from W_{a,b} to W_{a',b'} is defined by mapping the point
at infinity O of W_{a,b} to the point at infinity O of W_{a',b'},
while mapping each other point (x, y) of W_{a,b} to the point (x',
y'):=(x*u^2, y*u^3) of W_{a',b'}. The inverse mapping from W_{a',b'}
to W_{a,b} is defined by mapping the point at infinity O of W_{a',b'}
to the point at infinity O of W_{a,b}, while mapping each other point
(x', y') of W_{a',b'} to the point (x, y):=(x/u^2, y/u^3) of W_{a,b}.
Implementations may take advantage of this mapping to carry out
elliptic curve group operations originally defined for a Weierstrass
curve with a generic domain parameter a on a corresponding isomorphic
Weierstrass curve with domain parameter a' that has a special form,
which is known to allow for more efficient implementations of
addition laws, and translating the result back to the original curve.
In particular, it is known that such efficiency improvements exist if
a'=-3 (mod p) and one uses so-called Jacobian coordinates with a
particular projective version of the addition laws of Appendix B.1.
While not all Weierstrass curves can be put into this form, all
traditional NIST curves have domain parameter a=-3, while all
Brainpool curves [RFC5639] are isomorphic to a Weierstrass curve of
this form. For details, we refer to [GECC].
Struik Expires January 20, 2019 [Page 14]
Internet-Draft lwig-curve-representations July 2018
Note that implementations for elliptic curves with short-Weierstrass
form that hard-code the domain parameter a to a= -3 (which value is
known to allow more efficient implementations) cannot always be used
this way, since the curve W_{a,b} may not always be expressed in
terms of a Weierstrass curve with a'=-3 via a coordinate
transformation: this only holds if a'/a is a fourth power in GF(q).
However, even in this case, one can still express the curve W_{a,b}
in terms of a Weierstrass curve with small a' domain parameter,
thereby still allowing a more efficient implementation than with a
general a value.
E.2. Isogeneous Mapping between Weierstrass Curves
One can still map points of the Weierstrass curve W_{a,b} to points
of the Weierstrass curve W_{a',b'}, where a':=-3 (mod p), even if
a'/a is not a fourth power in GF(q). In that case, this mappping
cannot be an isomorphism (see Appendix E.1) and, thereby, does not
define a one-to-one correspondence. Instead, the mapping is a so-
called isogeny (or homomorphism). Since most elliptic curve
operations process points of prime order or use so-called "co-factor
multiplication", in practice the resulting mapping has similar
properties. In particular, one can still take advantage of this
mapping to carry out elliptic curve group operations originally
defined for a Weierstrass curve with domain parameter a unequal to -3
(mod p) on a corresponding isogenous Weierstrass curve with domain
parameter a'=-3 (mod p) and translating the result back to the
original curve. Details of this mapping are outside scope of this
document.
Appendix F. Further Cousins of Curve25519
F.1. Further Alternative Representations
The Weierstrass curve Wei25519 is isomorphic to the Weierstrass curve
Wei25519.2 defined over GF(p), with as base point the pair (G1x,G1y),
where parameters are as specified in Appendix F.3.
F.2. Further Switching
Each affine point (x,y) of Wei25519 corresponds to the point
(x,y):=(x*u^2,y*u^3) of Wei25519.2, where u is the element of GF(p)
defined by
u 47731687248873559672555216906496754195083410699918207029391079363
6321486119
(=0x10e26dacae93602704c7e6cff9efe595764cb5c9e04931f6fdeefc657d4e5
27),
Struik Expires January 20, 2019 [Page 15]
Internet-Draft lwig-curve-representations July 2018
while the point at infinity of Wei25519 corresponds to the point at
infinity of Wei25519.2. (Here, we used the mapping of Appendix E.1.)
Under this mapping, the base point (Gx',Gy') of Wei25519 corresponds
to the base point (G1x',G1y') of Wei25519.2. The inverse mapping
maps the affine point (x,y) of Wei25519.2 to (x,y):=(x/u^2,y/u^3) of
Wei25519, while mapping the point at infinity of Wei25519.2 to the
point at infinity of Wei25519. Note that this mapping (and its
inverse) involves a multiplication of both coordinates with fixed
constants u^2 and u^3 (respectively, 1/u^2 and 1/u^3), which can be
precomputed.
F.3. Further Domain Parameters
The parameters of the Weierstrass curve with a=2 that is isomorphic
with Wei25519 and the parameters of the Weierstrass curve with a=-3
that is isogeneous with Wei25519 are as indicated below. Both domain
parameter sets can be exploited directly to derive more efficient
point addition formulae, should an implementation facilitate this.
Weierstrass curve-specific parameters (with a=2):
a 2 (=0x2)
b 45793404337388339159414415854563976158160282736335993851976016290
777777599260
(=0x653e25fa 4aa43eb9 cc42c61b 806bcfd1 0e67bc23 09966e90
95a202fe 9aac731c)
G1x' 218726072268944427441327971914352883414836203960572472224621495
35754145422686
(=0x305b74fc 935f1dad d440a88e 781f0a81 09d6a68d 98c6081a
660528e2 0746dd5e)
G1y' 139436179034864291344077235766386796155987755307479919871866321
47013341290929
(=0x1ed3cedc e78b6b19 5d1c361c e1d4ef00 5b5b102c 99083780
bf830f7e a89021b1)
Weierstrass curve-specific parameters (with a=-3):
[NOTE: parameters indicated with TBD still to be completed, pending
completion of Sage calculations.]
a -3
Struik Expires January 20, 2019 [Page 16]
Internet-Draft lwig-curve-representations July 2018
(=0x7fffffff ffffffff ffffffff ffffffff ffffffff ffffffff
ffffffff ffffffea)
b [TBD]
(=0x[TBD])
G2x' [TBD]
(=0x[TBD])
G2y' [TBD]
(=0x[TBD])
Author's Address
Rene Struik
Struik Security Consultancy
Email: rstruik.ext@gmail.com
Struik Expires January 20, 2019 [Page 17]