 1/draftietflwigcurverepresentations01.txt 20190322 08:13:44.535014528 0700
+++ 2/draftietflwigcurverepresentations02.txt 20190322 08:13:44.635017005 0700
@@ 1,18 +1,18 @@
lwig R. Struik
InternetDraft Struik Security Consultancy
Intended status: Informational November 06, 2018
Expires: May 10, 2019
+Intended status: Informational March 11, 2019
+Expires: September 12, 2019
Alternative Elliptic Curve Representations
 draftietflwigcurverepresentations01
+ draftietflwigcurverepresentations02
Abstract
This document specifies how to represent Montgomery curves and
(twisted) Edwards curves as curves in shortWeierstrass form and
illustrates how this can be used to carry out elliptic curve
computations using existing implementations of, e.g., ECDSA and ECDH
using NIST prime curves.
Requirements Language
@@ 30,92 +30,115 @@
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InternetDrafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
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material or to cite them other than as "work in progress."
 This InternetDraft will expire on May 10, 2019.
+ This InternetDraft will expire on September 12, 2019.
Copyright Notice
 Copyright (c) 2018 IETF Trust and the persons identified as the
+ Copyright (c) 2019 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
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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
+ 2. Specification of Wei25519 . . . . . . . . . . . . . . . . . . 4
3. Use of Representation Switches . . . . . . . . . . . . . . . 4
4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1. Implementation of X25519 . . . . . . . . . . . . . . . . 5
 4.2. Implementation of Ed25519 . . . . . . . . . . . . . . . . 5
 4.3. Specification of ECDSASHA256Wei25519 . . . . . . . . . 5
+ 4.2. Implementation of Ed25519 . . . . . . . . . . . . . . . . 6
+ 4.3. Specification of ECDSA25519 . . . . . . . . . . . . . . . 6
4.4. Other Uses . . . . . . . . . . . . . . . . . . . . . . . 6
 5. Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
 6. Security Considerations . . . . . . . . . . . . . . . . . . . 7
+ 5. Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+ 6. Security Considerations . . . . . . . . . . . . . . . . . . . 8
7. Privacy Considerations . . . . . . . . . . . . . . . . . . . 8
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 8
 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 8
 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 8
 10.1. Normative References . . . . . . . . . . . . . . . . . . 8
 10.2. Informative References . . . . . . . . . . . . . . . . . 9
+ 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9
+ 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 9
+ 10.1. Normative References . . . . . . . . . . . . . . . . . . 9
+ 10.2. Informative References . . . . . . . . . . . . . . . . . 10
Appendix A. Some (nonBinary) Elliptic Curves . . . . . . . . . 10
A.1. Curves in shortWeierstrass Form . . . . . . . . . . . . 10
 A.2. Montgomery Curves . . . . . . . . . . . . . . . . . . . . 10
 A.3. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 10
 Appendix B. Elliptic Curve Nomenclature . . . . . . . . . . . . 11
 Appendix C. Elliptic Curve Group Operations . . . . . . . . . . 11
 C.1. Group Law for Weierstrass Curves . . . . . . . . . . . . 11
 C.2. Group Law for Montgomery Curves . . . . . . . . . . . . . 12
 C.3. Group Law for Twisted Edwards Curves . . . . . . . . . . 13
 Appendix D. Relationship Between Curve Models . . . . . . . . . 14
 D.1. Mapping between twisted Edwards Curves and Montgomery
 Curves . . . . . . . . . . . . . . . . . . . . . . . . . 14
 D.2. Mapping between Montgomery Curves and Weierstrass Curves 14
 D.3. Mapping between twisted Edwards Curves and Weierstrass
 Curves . . . . . . . . . . . . . . . . . . . . . . . . . 15
 Appendix E. Curve25519 and Cousins . . . . . . . . . . . . . . . 15
 E.1. Curve Definition and Alternative Representations . . . . 15
 E.2. Switching between Alternative Representations . . . . . . 16
 E.3. Domain Parameters . . . . . . . . . . . . . . . . . . . . 17
 Appendix F. Further Mappings . . . . . . . . . . . . . . . . . . 19
 F.1. Isomorphic Mapping between Weierstrass Curves . . . . . . 19
 F.2. Isogenous Mapping between Weierstrass Curves . . . . . . 20

 Appendix G. Further Cousins of Curve25519 . . . . . . . . . . . 21
 G.1. Further Alternative Representations . . . . . . . . . . . 21
 G.2. Further Switching . . . . . . . . . . . . . . . . . . . . 22
 G.3. Further Domain Parameters . . . . . . . . . . . . . . . . 23
 Appendix H. Isogeny Details . . . . . . . . . . . . . . . . . . 24
 H.1. Isogeny Parameters . . . . . . . . . . . . . . . . . . . 24
 H.1.1. Coefficients of u(x) . . . . . . . . . . . . . . . . 24
 H.1.2. Coefficients of v(x) . . . . . . . . . . . . . . . . 26
 H.1.3. Coefficients of w(x) . . . . . . . . . . . . . . . . 29
 H.2. Dual Isogeny Parameters . . . . . . . . . . . . . . . . . 30
 H.2.1. Coefficients of u'(x) . . . . . . . . . . . . . . . . 30
 H.2.2. Coefficients of v'(x) . . . . . . . . . . . . . . . . 32
 H.2.3. Coefficients of w'(x) . . . . . . . . . . . . . . . . 35
 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 36
+ A.2. Montgomery Curves . . . . . . . . . . . . . . . . . . . . 11
+ A.3. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 11
+ Appendix B. Elliptic Curve Nomenclature and Finite Fields . . . 11
+ B.1. Elliptic Curve Nomenclature . . . . . . . . . . . . . . . 11
+ B.2. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 13
+ Appendix C. Elliptic Curve Group Operations . . . . . . . . . . 14
+ C.1. Group Law for Weierstrass Curves . . . . . . . . . . . . 14
+ C.2. Group Law for Montgomery Curves . . . . . . . . . . . . . 15
+ C.3. Group Law for Twisted Edwards Curves . . . . . . . . . . 15
+ Appendix D. Relationship Between Curve Models . . . . . . . . . 16
+ D.1. Mapping between Twisted Edwards Curves and Montgomery
+ Curves . . . . . . . . . . . . . . . . . . . . . . . . . 16
+ D.2. Mapping between Montgomery Curves and Weierstrass Curves 17
+ D.3. Mapping between Twisted Edwards Curves and Weierstrass
+ Curves . . . . . . . . . . . . . . . . . . . . . . . . . 18
+ Appendix E. Curve25519 and Cousins . . . . . . . . . . . . . . . 18
+ E.1. Curve Definition and Alternative Representations . . . . 18
+ E.2. Switching between Alternative Representations . . . . . . 19
+ E.3. Domain Parameters . . . . . . . . . . . . . . . . . . . . 20
+ Appendix F. Further Mappings . . . . . . . . . . . . . . . . . . 22
+ F.1. Isomorphic Mapping between Twisted Edwards Curves . . . . 22
+ F.2. Isomorphic Mapping between Montgomery Curves . . . . . . 23
+ F.3. Isomorphic Mapping between Weierstrass Curves . . . . . . 24
+ F.4. Isogenous Mapping between Weierstrass Curves . . . . . . 25
+ Appendix G. Further Cousins of Curve25519 . . . . . . . . . . . 26
+ G.1. Further Alternative Representations . . . . . . . . . . . 26
+ G.2. Further Switching . . . . . . . . . . . . . . . . . . . . 26
+ G.3. Further Domain Parameters . . . . . . . . . . . . . . . . 27
+ Appendix H. Isogeny Details . . . . . . . . . . . . . . . . . . 29
+ H.1. Isogeny Parameters . . . . . . . . . . . . . . . . . . . 29
+ H.1.1. Coefficients of u(x) . . . . . . . . . . . . . . . . 29
+ H.1.2. Coefficients of v(x) . . . . . . . . . . . . . . . . 31
+ H.1.3. Coefficients of w(x) . . . . . . . . . . . . . . . . 34
+ H.2. Dual Isogeny Parameters . . . . . . . . . . . . . . . . . 35
+ H.2.1. Coefficients of u'(x) . . . . . . . . . . . . . . . . 35
+ H.2.2. Coefficients of v'(x) . . . . . . . . . . . . . . . . 37
+ H.2.3. Coefficients of w'(x) . . . . . . . . . . . . . . . . 40
+ Appendix I. Point Compression . . . . . . . . . . . . . . . . . 41
+ I.1. Point Compression for Weierstrass Curves . . . . . . . . 42
+ I.2. Point Compression for Montgomery Curves . . . . . . . . . 42
+ I.3. Point Compression for Twisted Edwards Curves . . . . . . 43
+ Appendix J. Data Conversions . . . . . . . . . . . . . . . . . . 43
+ J.1. Conversion between Bit Strings and Integers . . . . . . . 43
+ J.2. Conversion between Octet Strings and Integers (OS2I,
+ I2OS) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
+ J.3. Conversion between Octet Strings and Bit Strings (BS2OS,
+ OS2BS) . . . . . . . . . . . . . . . . . . . . . . . . . 44
+ J.4. Conversion between Field Elements and Octet Strings
+ (FE2OS, OS2FE) . . . . . . . . . . . . . . . . . . . . . 44
+ J.5. Ordering Conventions . . . . . . . . . . . . . . . . . . 45
+ Appendix K. Representations for Curve25519 Family Members . . . 46
+ K.1. Wei25519 . . . . . . . . . . . . . . . . . . . . . . . . 46
+ Appendix L. Auxiliary Functions . . . . . . . . . . . . . . . . 46
+ L.1. Square Roots in GF(q) . . . . . . . . . . . . . . . . . . 46
+ L.1.1. Square Roots in GF(q), where q = 3 (mod 4) . . . . . 47
+ L.1.2. Square Roots in GF(q), where q = 5 (mod 8) . . . . . 47
+ L.2. Inversion . . . . . . . . . . . . . . . . . . . . . . . . 47
+ Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 47
1. Fostering Code Reuse with New Elliptic Curves
It is wellknown 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 document specifies an
alternative representation of points of Curve25519, a socalled
Montgomery curve, and of points of Edwards25519, a socalled twisted
@@ 133,27 +156,28 @@
elliptic curve arithmetic for curves in "short" Weierstrass form (see
Appendix C.1).
2. Specification of Wei25519
For the specification of Wei25519 and its relationship to Curve25519
and Edwards25519, see Appendix E. 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 shortWeierstrass curves, such as the NIST curve P256, to
+ implements shortWeierstrass curves, such as the NIST curve P256, to
also implement the CFRG curves Curve25519 and Edwards25519. We also
cater to reusing of existing code where some domain parameters may
have been hardcoded, thereby widening the scope of applicability. To
this end, we specify the shortWeierstrass curves Wei25519.2 and
Wei25519.3, with hardcoded domain parameter a=2 and a=3 (mod p),
 respectively; see Appendix G.
+ respectively; see Appendix G. (Here, p is the characteristic of the
+ field over which these curves are defined.)
3. Use of Representation Switches
The curves Curve25519, Edwards25519, and Wei25519, as specified in
Appendix E.3, are all isomorphic, with the transformations of
Appendix E.2. These transformations map the specified base point of
each of these curves to the specified base point of each of the other
curves. Consequently, a publickey pair (k,R:=k*G) for any one of
these curves corresponds, via these isomorphic mappings, to the
publickey pair (k, R':=k*G') for each of these other curves (where G
@@ 216,44 +240,44 @@
RFC 8032 [RFC8032] specifies Ed25519, a "full" Schnorr signature
scheme, with instantiation by the twisted Edwards curve Edwards25519.
One can implement the computation of the ephemeral key pair for
Ed25519 using an existing Montgomery curve implementation by (1)
generating a publicprivate key pair (k, R':=k*G') for Curve25519;
(2) representing this publicprivate key as the pair (k, R:=k*G) for
Ed25519. As before, the representation change can be implemented via
a simple wrapper. Note that the Montgomery ladder specified in
Section 5 of RFC7748 [RFC7748] does not provide sufficient
 information to reconstruct R' (since it does not compute the
 ycoordinate of R'). However, this deficiency can be remedied by
+ information to reconstruct R':=(u, v) (since it does not compute the
+ vcoordinate of R'). However, this deficiency can be remedied by
using a slightly modified version of the Montgomery ladder that
 includes reconstruction of the ycoordinate of R':=k*G' at the end of
 hereof (which uses the vcoordinate Gv of the base point of
 Curve25519 as well). For details, see Appendix D.2.
+ includes reconstruction of the vcoordinate of R':=k*G' at the end of
+ hereof (which uses the vcoordinate of the base point of Curve25519
+ as well). For details, see Appendix C.1.
4.3. Specification of ECDSASHA256Wei25519
+4.3. Specification of ECDSA25519
FIPS Pub 1864 [FIPS1864] specifies the signature scheme ECDSA and
can be instantiated not just with the NIST prime curves, but also
with other Weierstrass curves (that satisfy additional cryptographic
criteria). In particular, one can instantiate this scheme with the
Weierstrass curve Wei25519 and the hash function SHA256, where an
implementation may generate a publicprivate key pair for Wei25519 by
(1) internally carrying out these computations on the Montgomery
curve Curve25519, the twisted Edwards curve Edwards25519, or even the
Weierstrass curve Wei25519.3 (with hardcoded a=3 domain parameter);
(2) representing the result as a key pair for the curve Wei25519.
Note that, in either case, one can implement these schemes with the
same representation conventions as used with existing NIST
specifications, including bit/byteordering, compression functions,
 and thelike. This allows implementations of ECDSA with the hash
 function SHA256 and with the NIST curve P256 or with the curve
+ and thelike. This allows generic implementations of ECDSA with the
+ hash function SHA256 and with the NIST curve P256 or with the curve
Wei25519 specified in this draft to use the same implementation
(instantiated with, respectively, the NIST P256 elliptic curve
domain parameters or with the domain parameters of curve Wei25519
specified in Appendix E).
4.4. 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,
@@ 264,65 +288,71 @@
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.
5. Caveats
The examples above illustrate how specifying the Weierstrass curve
 Wei25519 may facilitate reuse of existing code and may simplify
 standards development. However, the following caveats apply:
+ Wei25519 (or any curve in shortWeierstrass format, for that matter)
+ may facilitate reuse of existing code and may simplify standards
+ development. However, the following caveats apply:
 1. Unfriendly wire formats. The transformations between alternative
 curve representations can be implemented at negligible relative
+ 1. Wire format. The transformations between alternative curve
+ representations can be implemented at negligible relative
incremental cost if the curve points are represented as affine
points. If a point is represented in compressed format,
conversion usually requires a costly point decompression step.
This is the case in [RFC7748], where the inputs to the cofactor
DiffieHellman scheme X25519, as well as its output, are
 represented in xcoordinateonly format;
+ represented in ucoordinateonly format. This is also the case
+ in [RFC8032], where the EdDSA signature includes the ephemeral
+ signing key represented in compressed format (see Appendix I for
+ details);
 2. Unfriendly representation conventions. While elliptic curve
 computations are carriedout in a field GF(q) and, thereby,
 involve large integer arithmetic, these integers are represented
 as bit and bytestrings. Here, [RFC8032] uses least
 significantbyte (LSB)/leastsignificantbit (lsb) conventions,
 whereas [RFC7748] uses LSB/mostsignificantbit (msb)
 conventions, and where most other cryptographic specifications,
 including NIST SP80056a [SP80056a], FIPS Pub 1864
 [FIPS1864], and ANSI X9.622005 [ANSIX9.62] use MSB/msb
 conventions. Since each pair of conventions is different, this
 does necessitate bit/byte representation conversions;
+ 2. Representation conventions. While elliptic curve computations
+ are carriedout in a field GF(q) and, thereby, involve large
+ integer arithmetic, these integers are represented as bit and
+ bytestrings. Here, [RFC8032] uses leastsignificantbyte
+ (LSB)/leastsignificantbit (lsb) conventions, whereas [RFC7748]
+ uses LSB/mostsignificantbit (msb) conventions, and where most
+ other cryptographic specifications, including NIST SP80056a
+ [SP80056a], FIPS Pub 1864 [FIPS1864], and ANSI X9.622005
+ [ANSIX9.62] use MSB/msb conventions. Since each pair of
+ conventions is different (see Appendix J for details), this does
+ necessitate bit/byte representation conversions;
 3. Unfriendly domain parameters. All traditional NIST curves are
 Weierstrass curve with domain parameter a=3, while all Brainpool
 curves [RFC5639] are isomorphic to a Weierstrass curve of this
 form. Thus, one can expect there to be existing Weierstrass
+ 3. Domain parameters. All traditional NIST curves are Weierstrass
+ curves with domain parameter a=3, while all Brainpool curves
+ [RFC5639] are isomorphic to a Weierstrass curve of this form.
+ Thus, one can expect there to be existing Weierstrass
implementations with a hardcoded a=3 domain parameter
("Jacobianfriendly"). For those implementations, including the
curve Wei25519 as a potential vehicle for offering support for
the CFRG curves Curve25519 and Edwards25519 is not possible,
since not of the required form. Instead, one has to implement
 Curve25519.3 and include code that implements the isogeny and
 dual isogeny from and to Wei25519. This isogeny has degree l=47
 and requires roughly 9kB of storage for isogeny and dualisogeny
+ Wei25519.3 and include code that implements the isogeny and dual
+ isogeny from and to Wei25519. This isogeny has degree l=47 and
+ requires roughly 9kB of storage for isogeny and dualisogeny
computations (see the tables in Appendix H). Note that storage
 would have reduced to a single 32bye table if the curve would
 have been generated so as to be isomorphic to a Weierstrass curve
 with hardcoded a=3 parameter (this corresponds to l=1). Note:
 an example of such a curve is the Montgomery curve M_{A,B} over
 GF(p) with p=2^25519, A=1410290, and B=1 or (if one wants the
 base point to still have ucoordinate u=9) A=3960846. In either
 case, the resulting curve has the same cryptographic properties
 as Curve25519, while being more "Jacobianfriendly".
+ would have reduced to a single 64byte table if only the curve
+ would have been generated so as to be isomorphic to a Weierstrass
+ curve with hardcoded a=3 parameter (this corresponds to l=1).
+ Note: an example of such a curve is the Montgomery curve M_{A,B}
+ over GF(p) with p=2^25519, B=1, and A=1410290 or (if one wants
+ the base point to still have ucoordinate u=9) A=3960846. In
+ either case, the resulting curve has the same cryptographic
+ properties as Curve25519 and the same performance (since A is a
+ 3byte integer as is the case with the domain parameter A=486662
+ used with Curve25519), while being "Jacobianfriendly" by design.
6. Security Considerations
The different representations of elliptic curve points discussed in
this document are all obtained using a publicly known transformation,
which is either an isomorphism or a lowdegree isogeny. It is well
known that an isomorphism maps elliptic curve points to equivalent
mathematical objects and that the complexity of cryptographic
problems (such as the discrete logarithm problem) of curves related
via a lowdegree isogeny are tightly related. Thus, the use of these
@@ 339,30 +369,30 @@
for over 15 years.
7. Privacy Considerations
The transformations between different curve models described in this
document are publicly known and, therefore, do not affect privacy
provisions.
8. 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 4.
+ An object identifier is requested for Wei25519 and ECDSA25519, using
+ the representation conventions in this document.
9. Acknowledgements
Thanks to Nikolas Rosener for discussions surrounding implementation
details of the techniques described in this document and to Phillip
HallamBaker for triggering inclusion of verbiage on the use of
 Montgomery ladders with recovery of the ycoordinate.
+ Montgomery ladders with recovery of the ycoordinate. Thanks to
+ Stanislav Smyshlyaev for his careful review.
10. References
10.1. Normative References
[ANSIX9.62]
ANSI X9.622005, "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,
@@ 417,382 +447,514 @@
2003.
[GECC] D. Hankerson, A.J. Menezes, S.A. Vanstone, "Guide to
Elliptic Curve Cryptography", New York: SpringerVerlag,
2004.
[HWECC] W.P. Liu, "How to Use the Kinets LTC ECC HW to Accelerate
Curve25519 (version 7)", NXP,
https://community.nxp.com/docs/DOC330199, April 2017.
+ [Ladder] P.L. Montgomery, "Speeding the Pollard and Elliptic Curve
+ Methods of Factorization", Mathematics of
+ Computation, Vol. 48, 1987.
+
+ [tEdFormulas]
+ H. Hisil, K.K.H. Wong, G. Carter, E. Dawson, "Twisted
+ Edwards Curves Revisited", ASIACRYPT 2008, Lecture Notes
+ in Computer Science, Vol. 5350, New York: SpringerVerlag,
+ 2008.
+
Appendix A. Some (nonBinary) Elliptic Curves
A.1. Curves in shortWeierstrass 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,
+ 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
+ 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 socalled affine points), together with the special
point O (the socalled "point at infinity").This set forms a group
under addition, via the socalled "secantandtangent" rule, where
the point at infinity serves as the identity element. See
Appendix C.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 socalled affine points),
 together with the special point O (the socalled "point at
 infinity").This set forms a group under addition, via the socalled
 "secantandtangent" rule, where the point at infinity serves as the
 identity element. See Appendix C.2 for details of the group
 operation.
+ and where A is unequal to (+/)2 and where B is 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 socalled "secantandtangent" rule, where the point at
+ infinity serves as the identity element. See Appendix C.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 socalled 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
+ whereas d is not, where the point O:=(0, 1) serves as the identity
element. (Note that the identity element satisfies the defining
equation.) See Appendix C.3 for details of the group operation.
An Edwards curve is a twisted Edwards curve with a=1.
Appendix B. Elliptic Curve Nomenclature
+Appendix B. Elliptic Curve Nomenclature and Finite Fields
+
+B.1. Elliptic Curve Nomenclature
Each curve defined in Appendix A forms a commutative group under
 addition. In Appendix C we specify the group laws, which depend on
 the curve model in question. For completeness, we here include some
 common elliptic curve nomenclature and basic properties (primarily so
 as to keep this document selfcontained). These notions are mainly
 used in Appendix E and Appendix G and not essential for our
 exposition. This section can be skipped at first reading.
+ addition (denoted by '+'). In Appendix C we specify the group laws,
+ which depend on the curve model in question. For completeness, we
+ here include some common elliptic curve nomenclature and basic
+ properties (primarily so as to keep this document selfcontained).
+ These notions are mainly used in Appendix E and Appendix G and not
+ essential for our exposition. This section can be skipped at first
+ reading.
 Any point P of a curve is a generator of the cyclic subgroup
 (P):={k*P  k = 0, 1, 2,...} of the curve. If (P) has cardinality l,
 then l is called the order of P. The order of a curve is the
 cardinality of the set of its points. A curve is cyclic if it is
 generated by some point of this curve. All curves of prime order are
 cyclic, while all curves of order E = h*n, where n is a large prime
 number and where h is a small number (the socalled cofactor), have
 a large cyclic subgroup of prime order n. In this case, a generator
 of order n is called a base point, commonly denoted by G. A point of
 order dividing h is said to be in the small subgroup. For curves of
 prime order, this small subgroup is the singleton set, consisting of
 only the identity element.
+ Any point P of a curve E is a generator of the cyclic subgroup
+ (P):={k*P  k = 0, 1, 2,...} of the curve. (Here, k*P denotes the
+ sum of k copies of P, where 0*P is the identity element O of the
+ curve.) If (P) has cardinality l, then l is called the order of P.
+ The order of curve E is the cardinality of the set of its points,
+ commonly denoted by E. A curve is cyclic if it is generated by some
+ point of this curve. All curves of prime order are cyclic, while all
+ curves of order h*n, where n is a large prime number and where h is a
+ small number (the socalled cofactor), have a large cyclic subgroup
+ of prime order n. In this case, a generator of order n is called a
+ base point, commonly denoted by G. A point of order dividing h is
+ said to be in the small subgroup. For curves of prime order, this
+ small subgroup is the singleton set, consisting of only the identity
+ element O. If a point is not in the small subgroup, it has order at
+ least n.
 If R is a point on a curve E that is also contained in (P), there is
 a unique integer k in the interval [0, l1] so that R=kP, where l is
+ If R is a point of the curve that is also contained in (P), there is
+ a unique integer k in the interval [0, l1] so that R=k*P, where l is
the order of P. This number is called the discrete logarithm of R to
the base P. The discrete logarithm problem is the problem of finding
the discrete logarithm of R to the base P for any two points P and R
of the curve, if such a number exists.
 A publicprivate key pair is an ordered pair (k, R:=kG), where G is a
 fixed base point of the curve. Here, k (the private key) is an
 integer in the interval [0,n1], where G has order n.
+ If P is a fixed base point G of the curve, the pair (k, R:=k*G) is
+ called a publicprivate key pair, the integer k the private key, and
+ the point R the corresponding public key. The private key k can be
+ represented as an integer in the interval [0,n1], where G has order
+ n.
 A quadratic twist of a curve E defined over a field GF(q) is a curve
 E' related to E, with cardinality E+E'=2*(q+1). If E is a curve
 in one of the curve models specified in this document, a quadratic
 twist of this curve can be expressed using the same curve model,
 although (naturally) with different curve parameters.
+ In this document, a quadratic twist of a curve E defined over a field
+ GF(q) is a curve E' related to E, with cardinality E',
+ where E+E'=2*(q+1). If E is a curve in one of the curve models
+ specified in this document, a quadratic twist of this curve can be
+ expressed using the same curve model, although (naturally) with its
+ own curve parameters. Two curves E and E' defined over a field GF(q)
+ are said to be isogenous if these have the same order and are said to
+ be isomorphic if these have the same group structure. Note that
+ isomorphic curves have necessarily the same order and are, thus, a
+ special type of isogenous curves. Further details are out of scope.
+
+ Weierstrass curves can have prime order, whereas Montgomery curves
+ and twisted Edwards curves always have an order that is a multiple of
+ four (and, thereby, a small subgroup of cardinality four).
+
+ An ordered pair (x, y) whose coordinates are elements of GF(q) can be
+ associated with any ordered triple of the form [x*z: y*z: z], where z
+ is a nonzero element of GF(q), and can be uniquely recovered from
+ such a representation. The latter representation is commonly called
+ a representation in projective coordinates.
+
+ The group laws in Appendix C are mostly expressed in terms of affine
+ points, but can also be expressed in terms of the representation of
+ these points in projective coordinates, thereby allowing clearing of
+ denominators. The group laws may also involve nonaffine points
+ (such as the point at infinity O of a Weierstrass curve or of a
+ Montgomery curve). Those can also be represented in projective
+ coordinates. Further details are out of scope.
+
+B.2. Finite Fields
+
+ The field GF(q), where q is an odd prime power, is defined as
+ follows.
+
+ If p is a prime number, the field GF(p) consists of the integers in
+ the interval [0,p1] and two binary operations on this set: addition
+ and multiplication modulo p.
+
+ If q=p^m and m>0, the field GF(q) is defined in terms of an
+ irreducible polynomial f(z) in z of degree m with coefficients in
+ GF(p) (i.e., f(z) cannot be written as the product of two polynomials
+ in z of lower degree with coefficients in GF(p)): in this case, GF(q)
+ consists of the polynomials in z of degree smaller than m with
+ coefficients in GF(p) and two binary operations on this set:
+ polynomial addition and polynomial multiplication modulo the
+ irreducible polynomial f(z). By definition, each element x of GF(q)
+ is a polynomial in z of degree smaller than m and can, therefore, be
+ uniquely represented as a vector (x_{m1}, x_{m2}, ..., x_1, x_0) of
+ length m with coefficients in GF(p), where x_i is the coefficient of
+ z^i of polynomial x. Note that this representation depends on the
+ irreducible polynomial f(z) of the field GF(p^m) in question (which
+ is often fixed in practice). Note that GF(q) contains the prime
+ field GF(p) as a subset. If m=1, we always pick f(z):=z, so that the
+ definions of GF(p) and GF(p^1) above coincide. If m>1, then GF(q) is
+ called a (nontrivial) extension field over GF(p). The number p is
+ called the characteristic of GF(q).
+
+ A field element y is called a square in GF(q) if it can be expressed
+ as y:=x^2 for some x in GF(q); it is called a nonsquare in GF(q)
+ otherwise. If y is a square in GF(q), we denote by sqrt(y) one of
+ its square roots (the other one being sqrt(y)). For methods for
+ computing square roots and inverses in GF(q)  if these exist  see
+ Appendix L.1 and Appendix L.2, respectively.
+
+ NOTE: The curves in Appendix E and Appendix G are all defined over a
+ prime field GF(p), thereby reducing all operations to simple modular
+ integer arithmetic. Strictly speaking we could, therefore, have
+ refrained from introducing extension fields. Nevertheless, we
+ included the more general exposition, so as to accommodate potential
+ introduction of new curves that are defined over a (nontrivial)
+ extension field at some point in the future. This includes curves
+ proposed for postquantum isogenybased schemes, which are defined
+ over a quadratic extension field (i.e., where q:=p^2), and elliptic
+ curves used with pairingbased cryptography. The exposition in
+ either case is almost the same and now automatically yields, e.g.,
+ data conversion routines for any finite field object (see
+ Appendix J). Readers not interested in this, could simply view all
+ fields as prime fields.
Appendix C. Elliptic Curve Group Operations
C.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.
+ 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
+ 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).
+ X + X1 + X2 = lambda^2 and Y + Y1 = lambda*(X1  X), where
 Let P:= (x1, y1) be an affine point of the Weierstrass curve W_{a,b}
 and let Q:=2*P, where Q is not the identity element. Then Q:= (x,
 y), where
+ lambda:= (Y2  Y1)/(X2  X1).
 x + 2*x1 = lambda^2 and y + y1 = lambda*(x1  x), where
 lambda=(3*x1^2 + a)/(2*y1).
+ Let P:=(X1, Y1) be an affine point of the Weierstrass curve W_{a,b}
+ and let Q:=2*P, where Q is not the identity element. Then Q:=(X, Y),
+ where
 From the group law above it follows that if P=(x, y), P1=k*P=(x1,
 y1), and P2=(k+1)*P=(x2, y2) are affine points of the Weierstrass
 curve W_{a,b} and if y is nonzero, then the ycoordinate of P1 can be
 expressed in terms of the xcoordinates of P, P1, and P2, and the
 ycoordinate of P, as
+ X + 2*X1 = lambda^2 and Y + Y1 = lambda*(X1  X), where
 y1=((x*x1+a)*(x+x1)+2*bx2*(xx1)^2)/(2*y).
+ lambda:=(3*X1^2 + a)/(2*Y1).
 This property allows recovery of the ycoordinate of a point P1=k*P
+ From the group laws above it follows that if P=(X, Y), P1=k*P=(X1,
+ Y1), and P2=(k+1)*P=(X2, Y2) are distinct affine points of the
+ Weierstrass curve W_{a,b} and if Y is nonzero, then the Ycoordinate
+ of P1 can be expressed in terms of the Xcoordinates of P, P1, and
+ P2, and the Ycoordinate of P, as
+
+ Y1=((X*X1+a)*(X+X1)+2*bX2*(XX1)^2)/(2*Y).
+
+ This property allows recovery of the Ycoordinate of a point P1=k*P
that is computed via the socalled Montgomery ladder, where P is an
 affine point with nonzero ycoordinate. Further details are out of
 scope.
+ affine point with nonzero Ycoordinate (i.e., it does not have order
+ two). Further details are out of scope.
C.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.
+ For each affine point P:=(u, v) of the Montgomery curve M_{A,B}, the
+ point P is the point (u, v) and one has P + (P) = O.
 Let P1:=(x1, y1) and P2:=(x2, y2) be distinct affine points of the
+ Let P1:=(u1, v1) and P2:=(u2, v2) 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
+ identity element. Then Q:=(u, v), where
 x + x1 + x2 = B*lambda^2  A and y + y1 = lambda*(x1  x), where
 lambda=(y2  y1)/(x2  x1).
+ u + u1 + u2 = B*lambda^2  A and v + v1 = lambda*(u1  u), where
 Let P:= (x1, y1) be an affine point of the Montgomery curve M_{A,B}
 and let Q:=2*P, 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*B*y1).
+ lambda:=(v2  v1)/(u2  u1).
 From the group law above it follows that if P=(x, y), P1=k*P=(x1,
 y1), and P2=(k+1)*P=(x2, y2) are affine points of the Montgomery
 curve M_{A,B} and if y is nonzero, then the ycoordinate of P1 can be
 expressed in terms of the xcoordinates of P, P1, and P2, and the
 ycoordinate of P, as
+ Let P:=(u1, v1) be an affine point of the Montgomery curve M_{A,B}
+ and let Q:=2*P, where Q is not the identity element. Then Q:=(u, v),
+ where
 y1=((x*x1+1)*(x+x1+2*A)2*Ax2*(xx1)^2)/(2*B*y).
+ u + 2*u1 = B*lambda^2  A and v + v1 = lambda*(u1  u), where
 This property allows recovery of the ycoordinate of a point P1=k*P
+ lambda:=(3*u1^2 + 2*A*u1+1)/(2*B*v1).
+
+ From the group laws above it follows that if P=(u, v), P1=k*P=(u1,
+ v1), and P2=(k+1)*P=(u2, v2) are distinct affine points of the
+ Montgomery curve M_{A,B} and if v is nonzero, then the vcoordinate
+ of P1 can be expressed in terms of the ucoordinates of P, P1, and
+ P2, and the vcoordinate of P, as
+
+ v1=((u*u1+1)*(u+u1+2*A)2*Au2*(uu1)^2)/(2*B*v).
+
+ This property allows recovery of the vcoordinate of a point P1=k*P
that is computed via the socalled Montgomery ladder, where P is an
 affine point with nonzero ycoordinate. Further details are out of
 scope.
+ affine point with nonzero vcoordinate (i.e., it does not have order
+ one or two). Further details are out of scope.
C.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.
+ 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).
+ 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:=2*P. 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).
+ x = (2*x1*y1)/(1 + d*x1^2*y1^2) and
 Note that one can use the formulae for point addition for
 implementing 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)).
+ y = (y1^2  a*x1^2)/(1  d*x1^2*y1^2).
+
+ Note that one can use the formulae for point addition for 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)).
+
+ From the group laws above (subject to our Note above) it follows that
+ if P=(x, y), P1=k*P=(x1, y1), and P2=(k+1)*P=(x2, y2) are affine
+ points of the twisted Edwards curve E_{a,d} and if x is nonzero, then
+ the xcoordinate of P1 can be expressed in terms of the ycoordinates
+ of P, P1, and P2, and the xcoordinate of P, as
+
+ x1=(y*y1y2)/(x*(ad*y*y1*y2)).
+
+ This property allows recovery of the xcoordinate of a point P1=k*P
+ that is computed via the socalled Montgomery ladder, where P is an
+ affine point with nonzero xcoordinate (i.e., it does not have order
+ one or two). Further details are out of scope.
Appendix D. Relationship Between Curve Models
The nonbinary curves specified in Appendix A are expressed in
different curve models, viz. as curves in shortWeierstrass form, as
Montgomery curves, or as twisted Edwards curves. These curve models
are related, as follows.
D.1. Mapping between twisted Edwards Curves and Montgomery Curves
+D.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:=(A2)/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)/(ad) and where
B:=4/(ad). 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
toone correspondence, which  in fact  is an isomorphism between
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, (u1)/(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)/(1y), (1+y)/((1y)*x)) of M_{A,B}.
+ other point (x, y) of E_{a,d} is mapped to the point
+ (u,v):=((1+y)/(1y),(1+y)/((1y)*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 viceversa,
and translating the result back to the original curve, thereby
potentially allowing code reuse.
D.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:=(3A^2)/(3*B^2) and
b:=(2*A^39*A)/(27*B^3). This defines a onetoone 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.
+ 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 socalled "NIST curves", this inverse mapping is not
+ defined.
+
+ If the Weierstrass curve W_{a,b} has a point (alpha,0) of order two
+ and c:=a+3*(alpha)^2 is a square in GF(q), one can map points of this
+ curve to points of the Montgomery curve M_{A,B}, where A:=3*alpha/
+ gamma and B:=1/gamma and where gamma is any square root of c. In
+ this case, the mapping from W_{a,b} to M_{A,B} is defined by mapping
+ the point at infinity O of W_{a,b} to the point at infinity O of
+ M_{A,B}, while mapping each other point (X,Y) of W_{a,b} to the point
+ (u,v):=((Xalpha)/gamma,Y/gamma) of M_{A,B}. As before, this defines
+ a onetoone correspondence, which  in fact  is an isomorphism
+ between W_{a,b} and M_{A,B}. It is easy to see that the mapping from
+ W_{a,b} to M_{A,B} and that from M_{A,B} to W_{a,b} (if defined) are
+ each other's inverse.
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
shortWeierstrass form and translating the result back to the
original curve, thereby potentially allowing code reuse.
Note that implementations for elliptic curves with shortWeierstrass
form that hardcode 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} resulting from an isomorphic
mapping cannot always be expressed as a Weierstrass curve with a=3
via a coordinate transformation. For more details, see Appendix F.
D.3. Mapping between twisted Edwards Curves and Weierstrass Curves
+D.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 D.1 and the one between Montgomery and
Weierstrass curves of Appendix D.2. Obviously, one can use function
 composition (now using the respective inverses) to realize the
 inverse of this mapping.
+ composition (now using the respective inverses  if these exist) to
+ realize the inverse of this mapping.
Appendix E. Curve25519 and Cousins
E.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^24 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, p1].
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 E.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 C.3 apply.
+ the point (Gx, Gy), where parameters are as specified in
+ Appendix E.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 C.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 E.3. This
+ (GX, GY), where parameters are as specified in Appendix E.3. This
curve is denoted as Wei25519.
E.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
+ Each affine point (u, v) of Curve25519 corresponds to the point (X,
+ Y):=(u + A/3, v) of Wei25519, while the point at infinity of
Curve25519 corresponds to the point at infinity of Wei25519. (Here,
 we used the mapping of Appendix D.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
 that this mapping involves a simple shift of the first coordinate and
 can be implemented via integeronly 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
+ we used the mappings of Appendix D.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 that this mapping involves a simple shift of the first
+ coordinate and can be implemented via integeronly 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)
+ (=0x2aaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
+ aaaaaaaa aaad2451).
+
+ (Note that, depending on the implementation details of the field
+ arithmetic, one may have to shift the result by +p or p if this
+ integer is not in the interval [0,p1].)
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, (u1)/(u+1)) of Edwards25519, where c is the element of GF(p)
defined by
 c sqrt((A+2))
+ c sqrt((A+2)/B)
51042569399160536130206135233146329284152202253034631822681833788
666877215207

(=0x70d9120b 9f5ff944 2d84f723 fc03b081 3a5e2c2e b482e57d
 3391fb55 00ba81e7)
+ 3391fb55 00ba81e7).
 (Here, we used the mapping of Appendix D.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)/((1y)*x)).
+ (Here, we used the mapping of Appendix D.1 and normalized this using
+ the mapping of Appendix F.1 (where the element s of that appendix is
+ set to c above).) 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)/((1y)*x)) of Curve25519.
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
+ 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)/(1y)+A/3,
+ Edwards25519 corresponds to the point (X, Y):=((1+y)/(1y)+A/3,
c*(1+y)/((1y)*x)) of Wei25519, where c was defined before. (Here,
we used the mapping of Appendix D.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
 Edwards25519 and having each other point (x, y) of Wei25519
 correspond to the point (c*(3*xA)/(3*y), (3*xA3)/(3*xA+3)).
+ Edwards25519 and having each other point (X, Y) of Wei25519
+ correspond to the point (c*(3*XA)/(3*Y), (3*XA3)/(3*XA+3)) of
+ Edwards25519.
 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.
+ Note that these mappings can be easily realized if points are
+ represented in projective coordinates, using a few field
+ multiplications only, thus allowing switching between alternative
+ curve representations with negligible relative incremental cost.
E.3. Domain Parameters
The parameters of the Montgomery curve and the corresponding
isomorphic curves in twisted Edwards curve and shortWeierstrass 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
+ Edwards25519 are as specified in [RFC7748]; 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
@@ 820,21 +982,21 @@
Gv 14781619447589544791020593568409986887264606134616475288964881837
755586237401
(=0x20ae19a1 b8a086b4 e01edd2c 7748d14c 923d4d7e 6d7c61b2
29e9c5a2 7eced3d9)
Twisted Edwards curvespecific parameters (for Edwards25519):
a 1 (0x01)
 d 121665/121666
+ d 121665/121666 =  (A2)/(A+2)
(=370957059346694393431380835087545651895421138798432190163887855
33085940283555)
(=0x52036cee 2b6ffe73 8cc74079 7779e898 00700a4d 4141d8ab
75eb4dca 135978a3)
Gx 15112221349535400772501151409588531511454012693041857206046113283
949847762202
@@ 856,209 +1017,275 @@
(=0x2aaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
aaaaaa98 4914a144)
b 55751746669818908907645289078257140818241103727901012315294400837
956729358436
(=0x7b425ed0 97b425ed 097b425e d097b425 ed097b42 5ed097b4
260b5e9c 7710c864)
 Gx' 19298681539552699237261830834781317975544997444273427339909597334
+ GX 19298681539552699237261830834781317975544997444273427339909597334
652188435546
(=0x2aaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
aaaaaaaa aaad245a)
 Gy' 14781619447589544791020593568409986887264606134616475288964881837
+ GY 14781619447589544791020593568409986887264606134616475288964881837
755586237401
(=0x20ae19a1 b8a086b4 e01edd2c 7748d14c 923d4d7e 6d7c61b2
29e9c5a2 7eced3d9)
Appendix F. Further Mappings
The nonbinary curves specified in Appendix A are expressed in
different curve models, viz. as curves in shortWeierstrass 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 parameters to a value that allows a more efficient
 implementation of the addition formulae.
+ Montgomery curves, or as twisted Edwards curves. In Appendix D we
+ already described relationships between these various curve models.
+ Further mappings exist between elliptic curves within the same curve
+ model. These can be exploited to force some of the domain parameters
+ to specific values that allow for a more efficient implementation of
+ the addition formulae.
F.1. Isomorphic Mapping between Weierstrass Curves
+F.1. Isomorphic Mapping between Twisted Edwards Curves
+
+ One can map points of the twisted Edwards curve E_{a,d} to points of
+ the twisted Edwards curve E_{a',d'}, where a:=a'*s^2 and d:=d'*s^2
+ for some nonzero element s of GF(q). This defines a onetoone
+ correspondence, which  in fact  is an isomorphism between E_{a,d}
+ and E_{a',d'}.
+
+ The mapping from E_{a,d} to E_{a',d'} is defined by mapping the point
+ (x,y) of E_{a,d} to the point (x', y'):=(s*x, y) of E_{a',d'}. The
+ inverse mapping from E_{a',d'} to E_{a,d} is defined by mapping the
+ point (x', y') of E_{a',d'} to the point (x, y):=(x'/s, y') of
+ E_{a,d}.
+
+ Implementations may take advantage of this mapping to carry out
+ elliptic curve group operations originally defined for a twisted
+ Edwards curve with generic domain parameters a and d on a
+ corresponding isomorphic twisted Edwards curve with domain parameters
+ a' and d' that have a more special form, which are known to allow for
+ more efficient implementations of addition laws. In particular, it
+ is known that such efficiency improvements exist if a':=1 (see
+ [tEdFormulas]).
+
+F.2. Isomorphic Mapping between Montgomery Curves
+
+ One can map points of the Montgomery curve M_{A,B} to points of the
+ Montgomery curve M_{A',B'}, where A:=A' and B:=B'*s^2 for some
+ nonzero element s of GF(q). This defines a onetoone
+ correspondence, which  in fact  is an isomorphism between M_{A,B}
+ and M_{A',B'}.
+
+ The mapping from M_{A,B} to M_{A',B'} is defined by mapping the point
+ at infinity O of M_{A,B} to the point at infinity O of M_{A',B'},
+ while mapping each other point (u,v) of M_{A,B} to the point (u',
+ v'):=(u, s*v) of M_{A',B'}. The inverse mapping from M_{A',B'} to
+ M_{A,B} is defined by mapping the point at infinity O of M_{A',B'} to
+ the point at infinity O of M_{A,B}, while mapping each other point
+ (u',v') of M_{A',B'} to the point (u,v):=(u',v'/s) of M_{A,B}.
+
+ One can also map points of the Montgomery curve M_{A,B} to points of
+ the Montgomery curve M_{A',B'}, where A':=A and B':=B. This
+ defines a onetoone correspondence, which  in fact  is an
+ isomorphism between M_{A,B} and M_{A',B'}.
+
+ In this case, the mapping from M_{A,B} to M_{A',B'} is defined by
+ mapping the point at infinity O of M_{A,B} to the point at infinity O
+ of M_{A',B'}, while mapping each other point (u,v) of M_{A,B} to the
+ point (u',v'):=(u,v) of M_{A',B'}. The inverse mapping from
+ M_{A',B'} to M_{A,B} is defined by mapping the point at infinity O of
+ M_{A',B'} to the point at infinity O of M_{A,B}, while mapping each
+ other point (u',v') of M_{A',B'} to the point (u,v):=(u',v') of
+ M_{A,B}.
+
+ Implementations may take advantage of this mapping to carry out
+ elliptic curve groups operations originally defined for a Montgomery
+ curve with generic domain parameters A and B on a corresponding
+ isomorphic Montgomery curve with domain parameters A' and B' that
+ have a more special form, which is known to allow for more efficient
+ implementations of addition laws. In particular, it is known that
+ such efficiency improvements exist if B' assumes a small absolute
+ value, such as B':=(+/)1. (see [Ladder]).
+
+F.3. 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'*s^4 and b:=b'*s^6 for some
 nonzero value s of the finite field GF(q). This defines a onetoone
+ Weierstrass curve W_{a',b'}, where a':=a*s^4 and b':=b*s^6 for some
+ nonzero element s of GF(q). This defines a onetoone
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.
+ and W_{a',b'}.
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*s^2, y*s^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/s^2, y/s^3) of W_{a,b}.
+ while mapping each other point (X,Y) of W_{a,b} to the point
+ (X',Y'):=(X*s^2, Y*s^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'/s^2,Y'/s^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 socalled Jacobian coordinates with a
 particular projective version of the addition laws of Appendix C.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.
+ curve with generic domain parameters a and b on a corresponding
+ isomorphic Weierstrass curve with domain parameter a' and b' that
+ have a more 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), where p is the characteristic of
+ GF(q), and one uses socalled Jacobian coordinates with a particular
+ projective version of the addition laws of Appendix C.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.
Note that implementations for elliptic curves with shortWeierstrass
form that hardcode the domain parameter a to a= 3 cannot always be
used this way, since the curve W_{a,b} cannot 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)
(see Section 3.1.5 of [GECC]). However, even in this case, one can
still express the curve W_{a,b} as a Weierstrass curve with a small
domain parameter value a', thereby still allowing a more efficient
implementation than with a general domain parameter value a.
F.2. Isogenous Mapping between Weierstrass Curves
+F.4. Isogenous 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 F.1). Instead, the mapping is
 a socalled isogeny (or homomorphism). Since most elliptic curve
 operations process points of prime order or use socalled "cofactor
 multiplication", in practice the resulting mapping has similar
 properties as an isomorphism. 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.
+ of the Weierstrass curve W_{a',b'}, where a':=3 (mod p) and where p
+ is the characteristic of GF(q), even if a'/a is not a fourth power in
+ GF(q). In that case, this mappping cannot be an isomorphism (see
+ Appendix F.3). Instead, the mapping is a socalled isogeny (or
+ homomorphism). Since most elliptic curve operations process points
+ of prime order or use socalled "cofactor multiplication", in
+ practice the resulting mapping has similar properties as an
+ isomorphism. 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.
In this case, 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'):=(u(x)/w(x)^2, y*v(x)/w(x)^3) of W_{a',b'}. Here,
 u(x), v(x), and w(x) are polynomials that depend on the isogeny in
+ of W_{a',b'}, while mapping each other point (X,Y) of W_{a,b} to the
+ point (X',Y'):=(u(X)/w(X)^2,Y*v(X)/w(X)^3) of W_{a',b'}. Here, u(X),
+ v(X), and w(X) are polynomials in X that depend on the isogeny in
question. The inverse mapping from W_{a',b'} to W_{a,b} is again an
isogeny and 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):=(u'(x')/w'(x')^2,
 y'*v'(x')/w'(x')^3) of W_{a,b}, where  again  u'(x'), v'(x'), and
 w'(x') are polynomials that depend on the isogeny in question. These
 mappings have the property that their composition is not the identity
 mapping (as is the case with the isomorphic mappings discussed in
 Appendix F.1), but rather a fixed multiple hereof: if this multiple
 is l then the isogeny is called an isogeny of degree l (or lisogeny)
 and u, v, and w (and, similarly, u', v', and w') are polynomials of
 degrees l, 3(l1)/2, and (l1)/2, respectively. Note that an
 isomorphism is simply an isogeny of degree l=1. Details of how to
 determine isogenies are outside scope of this document (for this,
 contact the author of this document).
+ (X', Y') of W_{a',b'} to the point
+ (X,Y):=(u'(X')/w'(X')^2,Y'*v'(X')/w'(X')^3) of W_{a,b}, where 
+ again  u'(X'), v'(X'), and w'(X') are polynomials in X' that depend
+ on the isogeny in question. These mappings have the property that
+ their composition is not the identity mapping (as was the case with
+ the isomorphic mappings discussed in Appendix F.3), but rather a
+ fixed multiple hereof: if this multiple is l then the isogeny is
+ called an isogeny of degree l (or lisogeny) and u, v, and w (and,
+ similarly, u', v', and w') are polynomials of degrees l, 3*(l1)/2,
+ and (l1)/2, respectively. Note that an isomorphism is simply an
+ isogeny of degree l=1. Details of how to determine isogenies are
+ outside scope of this document (for this, contact the author of this
+ document).
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 isogenous
Weierstrass curve with domain parameter a'=3 (mod p), where one can
use socalled Jacobian coordinates with a particular projective
version of the addition laws of Appendix C.1. Since all traditional
NIST curves have domain parameter a=3, while all Brainpool curves
[RFC5639] are isomorphic to a Weierstrass curve of this form, this
allows taking advantage of existing implementations for these curves
that may have a hardcoded a=3 (mod p) domain parameter, provided one
switches back and forth to this curve form using the isogenous
mapping in question.
 Note that isogenous mappings can be easily realized in projective
 coordinates and involves roughly 3*l finite field multiplications,
 thus allowing switching between alternative representations at
 relative low incremental cost compared to that of elliptic curve
 scalar multiplications (provided the isogeny has low degree l).
 Note, however, that this does require storage of the polynomial
 coefficients of the isogeny and dual isogeny involved. This
 illustrates that lowdegree isogenies are to be preferred, since an
 lisogeny (usually) requires storing roughly 6*l elements of GF(q).
 While there are many isogenies, we therefore only consider those with
 the desired property with lowest possible degree.
+ Note that isogenous mappings can be easily realized using
+ representations in projective coordinates and involves roughly 3*l
+ finite field multiplications, thus allowing switching between
+ alternative representations at relatively low incremental cost
+ compared to that of elliptic curve scalar multiplications (provided
+ the isogeny has low degree l). Note, however, that this does require
+ storage of the polynomial coefficients of the isogeny and dual
+ isogeny involved. This illustrates that lowdegree isogenies are to
+ be preferred, since an lisogeny (usually) requires storing roughly
+ 6*l elements of GF(q). While there are many isogenies, we therefore
+ only consider those with the desired property with lowest possible
+ degree.
Appendix G. Further Cousins of Curve25519
G.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'), and isogenous to the Weierstrass curve Wei25519.3
 defined over GF(p), with as base point the pair (G2x', G2y'), where
 parameters are as specified in Appendix G.3 and where the related
 mappings are as specified in Appendix G.2.
+ Wei25519.2 defined over GF(p), with as base point the pair (G2X,G2Y),
+ and isogenous to the Weierstrass curve Wei25519.3 defined over
+ GF(p), with as base point the pair (G3X, G3Y), where parameters are
+ as specified in Appendix G.3 and where the related mappings are as
+ specified in Appendix G.2.
G.2. Further Switching
 Each affine point (x,y) of Wei25519 corresponds to the point
 (x',y'):=(x*s^2,y*s^3) of Wei25519.2, where s is the element of GF(p)
+ Each affine point (X, Y) of Wei25519 corresponds to the point (X',
+ Y'):=(X*s^2,Y*s^3) of Wei25519.2, where s is the element of GF(p)
defined by
s 20343593038935618591794247374137143598394058341193943326473831977
39407761440
(=0x047f6814 6d568b44 7e4552ea a5ed633d 02d62964 a2b0a120
5e7941e9 375de020),
while the point at infinity of Wei25519 corresponds to the point at
 infinity of Wei25519.2. (Here, we used the mapping of Appendix F.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'/s^2,y'/s^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
+ infinity of Wei25519.2. (Here, we used the mapping of Appendix F.3.)
+ Under this mapping, the base point (GX, GY) of Wei25519 corresponds
+ to the base point (G2X,G2Y) of Wei25519.2. The inverse mapping maps
+ the affine point (X', Y') of Wei25519.2 to (X,Y):=(X'/s^2,Y'/s^3) of
+ Wei25519, while mapping the point at infinity O of Wei25519.2 to the
+ point at infinity O of Wei25519. Note that this mapping (and its
inverse) involves a modular multiplication of both coordinates with
fixed constants s^2 and s^3 (respectively, 1/s^2 and 1/s^3), which
can be precomputed.
 Each affine point (x,y) of Wei25519 corresponds to the point
 (x',y'):=(x1/t^2,y1/t^3) of Wei25519.3, where
 (x1,y1)=(u(x)/w(x)^2,y*v(x)/w(x)^3), where u, v, and w are the
+ Each affine point (X,Y) of Wei25519 corresponds to the point
+ (X',Y'):=(X1*t^2,Y1*t^3) of Wei25519.3, where
+ (X1,Y1)=(u(X)/w(X)^2,Y*v(X)/w(X)^3), where u, v, and w are the
polynomials with coefficients in GF(p) as defined in Appendix H.1 and
where t is the element of GF(p) defined by
 t 26012855558634277483276064234565597076862996895623795164528458073
 435568115620
+ t 35728133398289175649586938605660542688691615699169662967154525084
+ 644181596229
 (=0x3982c126 59ad1749 ab8bc495 bb1a9d64 c9deffc5 e7b8e601
 a5651992 07d48fa4),
+ (=0x4efd6829 88ff8526 e189f712 5999550c e9ef729b ed1a7015
+ 73b1bab8 8bfcd845),
while the point at infinity of Wei25519 corresponds to the point at
infinity of Wei25519.3. (Here, we used the isogenous mapping of
 Appendix F.2.) Under this isogenous mapping, the base point
 (Gx',Gy') of Wei25519 corresponds to the base point (G2x',G2y') of
 Wei25519.3. The dual isogeny maps the affine point (x',y') of
 Wei25519.3 to to (x,y):=(u'(x1)/w'(x1)^2,y1*v'(x1)/w'(x1)^3) of
 Wei25519, where (x1,y1)=(x'*t^2,y'*t^3) and where u', v', and w' are
 the polynomials with coefficients in GF(p) as defined in
 Appendix H.2, while mapping the point at infinity of Wei25519.3 to
 the point at infinity of Wei25519. Under this dual isogenous
 mapping, the base point (G2x',G2y') of Wei25519.3 corresponds to a
 multiple of the base point (Gx',Gy') of Wei25519, where this multiple
 is l=47 (the degree of the isogeny; see the description in
 Appendix F.1). Note that this isogenous map (and its dual) primarily
 involves the evaluation of three fixed polynomials involving the
 xcoordinate, which takes roughly 140 modular multiplications (or
 less than 510% relative incremental cost compared to the cost of an
 elliptic curve scalar multiplication).
+ Appendix F.4.) Under this isogenous mapping, the base point (GX,GY)
+ of Wei25519 corresponds to the base point (G3X,G3Y) of Wei25519.3.
+ The dual isogeny maps the affine point (X',Y') of Wei25519.3 to the
+ affine point (X,Y):=(u'(X1)/w'(X1)^2,Y1*v'(x1)/w'(X1)^3) of Wei25519,
+ where (X1,Y1)=(X'/t^2,Y'/t^3) and where u', v', and w' are the
+ polynomials with coefficients in GF(p) as defined in Appendix H.2,
+ while mapping the point at infinity O of Wei25519.3 to the point at
+ infinity O of Wei25519. Under this dual isogenous mapping, the base
+ point (G3X, G3Y) of Wei25519.3 corresponds to a multiple of the base
+ point (GX, GY) of Wei25519, where this multiple is l=47 (the degree
+ of the isogeny; see the description in Appendix F.3). Note that this
+ isogenous map (and its dual) primarily involves the evaluation of
+ three fixed polynomials involving the xcoordinate, which takes
+ roughly 140 modular multiplications (or less than 510% relative
+ incremental cost compared to the cost of an elliptic curve scalar
+ multiplication).
G.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 isogenous 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.
General parameters: same as for Wei25519 (see Appendix E.3)
@@ 1067,53 +1293,54 @@
a=2):
a 2 (=0x02)
b 12102640281269758552371076649779977768474709596484288167752775713
178787220689
(=0x1ac1da05 b55bc146 33bd39e4 7f94302e f19843dc f669916f
6a5dfd01 65538cd1)
 G1x' 107705531383684005184170201967961611367923681983263378231495026
 81097436401658
+ G2X 10770553138368400518417020196796161136792368198326337823149502681
+ 097436401658
(=0x17cfeac3 78aed661 318e8634 582275b6 d9ad4def 072ea193
5ee3c4e8 7a940ffa)
 G1y' 544305758615084056530986689844575286168071033325025775211614397
 7388639873869
+ G2Y 54430575861508405653098668984457528616807103332502577521161439773
+ 88639873869
(=0x0c08a952 c55dfad6 2c4f13f1 a8f68dca dc5c331d 297a37b6
f0d7fdcc 51e16b4d)
Weierstrass curvespecific parameters (for Wei25519.3, i.e., with
a=3):
a 3
+
(=0x7fffffff ffffffff ffffffff ffffffff ffffffff ffffffff
ffffffff ffffffea)
b 29689592517550930188872794512874050362622433571298029721775200646
451501277098
(=0x41a3b6bf c668778e be2954a4 b1df36d1 485ecef1 ea614295
796e1022 40891faa)
 G2x' 538371792299408724349427232574807773704511272123391981336972078
 46219400243292
+ G3X 53837179229940872434942723257480777370451127212339198133697207846
+ 219400243292
(=0x7706c37b 5a84128a 3884a5d7 1811f1b5 5da3230f fb17a8ab
0b32e48d 31a6685c)
 G2y' 695480730911001844144020555292799703925148674228551417730708041
 8460388229929
+ G3Y 69548073091100184414402055529279970392514867422855141773070804184
+ 60388229929
(=0x0f60480c 7a5c0e11 40340adc 79d6a2bf 0cb57ad0 49d025dc
38d80c77 985f0329)
Appendix H. Isogeny Details
The isogeny and dual isogeny are both isogenies with degree l=47.
Both are specified by a triple of polynomials u, v, and w (resp. u',
v', and w') of degree 47, 69, and 23, respectively, with coefficients
in GF(p). The coeffients of each of these polynomials are specified
@@ 1687,16 +1914,327 @@
19 0x6868305b4f40654460aad63af3cb9151ab67c775eaac5e5df90d3aea58dee141
20 0x16bc90219a36063a22889db810730a8b719c267d538cd28fa7c0d04f124c8580
21 0x3628f9cf1fbe3eb559854e3b1c06a4cd6a26906b4e2d2e70616a493bba2dc574
22 0x64abcc6759f1ce1ab57d41e17c2633f717064e35a7233a6682f8cf8e9538afec
23 0x1
+Appendix I. Point Compression
+
+ Point compression allows a shorter representation of affine points of
+ an elliptic curve by exploiting algebraic relationships between the
+ coordinate values based on the defining equation of the curve in
+ question. Point decompression refers to the reverse process, where
+ one tries and recover the affine point from its compressed
+ representation and information on the domain parameters of the curve.
+ Consequently, point compression followed by point decompression is
+ the identity map.
+
+ The description below makes use of an auxiliary function (the parity
+ function), which we first define for prime fields GF(p) and then
+ extend to all fields GF(q), where q is an odd prime power. We assume
+ each finite field to be unambiguously defined.
+
+ Let y be a nonzero element of GF(q). If q:=p is an odd prime number,
+ y and py can be uniquely represented as integers in the interval
+ [1,p1] and have odd sum p. Consequently, one can distinguish y from
+ y via the parity of this representation, i.e., via par(y):=y (mod
+ 2). If q:=p^m, where p is an odd prime number and where m>0, both y
+ and y can be uniquely represented as vectors of length m, with
+ coefficients in GF(p) (see Appendix B.2). In this case, the leftmost
+ nonzero coordinate values of y and y are in the same position and
+ have representations in [1,p1] with different parity. As a result,
+ one can distinguish y from y via the parity of the representation of
+ this coordinate value. This extends the definition of the parity
+ function to any oddsize field GF(q), where one defines par(0):=0.
+
+I.1. Point Compression for Weierstrass Curves
+
+ If P:=(X, Y) is an affine point of the Weierstrass curve W_{a,b}
+ defined over the field GF(q), then so is P:=(X, Y). Since the
+ defining equation Y^2=X^2+a*X+b has at most two solutions with fixed
+ Xvalue, one can represent P by its Xcoordinate and one bit of
+ information that allows one to distinguish P from P, i.e., one can
+ represent P as the ordered pair compr(P):=(X, par(Y)). If P is a
+ point of order two, one can uniquely represent P by its Xcoordinate
+ alone, since Y=0 and has fixed parity. Conversely, given the ordered
+ pair (X, t), where X is an element of GF(q) and where t=0 or t=1, and
+ the domain parameters of the curve, one can use the defining equation
+ of the curve to try and determine candidate values for the
+ Ycoordinate given X, by solving the quadratic equation Y^2:=alpha,
+ where alpha:=X^3+a*X+b. If alpha is not a square in GF(q), this
+ equation does not have a solution in GF(q) and the ordered pair (X,
+ t) does not correspond to a point of this curve. Otherwise, there
+ are two solutions, viz. Y=sqrt(alpha) and Y. If alpha is a nonzero
+ element of GF(q), one can uniquely recover the Ycoordinate for which
+ par(Y):=t and, thereby, the point P:=(X, Y). This is also the case
+ if alpha=0 and t=0, in which case Y=0 and the point P has order two.
+ However, if alpha=0 and t=1, the ordered pair (X, t) does not
+ correspond to the outcome of the point compression function.
+
+I.2. Point Compression for Montgomery Curves
+
+ If P:=(u, v) is an affine point of the Montgomery curve M_{A,B}
+ defined over the field GF(q), then so is P:=(u, v). Since the
+ defining equation B*v^2=u^3+A*u^2+u has at most two solutions with
+ fixed uvalue, one can represent P by its ucoordinate and one bit of
+ information that allows one to distinguish P from P, i.e., one can
+ represent P as the ordered pair compr(P):=(u, par(v)). If P is a
+ point of order two, one can uniquely represent P by its ucoordinate
+ alone, since v=0 and has fixed parity. Conversely, given the ordered
+ pair (u, t), where u is an element of GF(q) and where t=0 or t=1, and
+ the domain parameters of the curve, one can use the defining equation
+ of the curve to try and determine candidate values for the
+ vcoordinate given u, by solving the quadratic equation v^2:=alpha,
+ where alpha:=(u^3+A*u^2+u)/B. If alpha is not a square in GF(q),
+ this equation does not have a solution in GF(q) and the ordered pair
+ (u, t) does not correspond to a point of this curve. Otherwise,
+ there are two solutions, viz. v=sqrt(alpha) and v. If alpha is a
+ nonzero element of GF(q), one can uniquely recover the vcoordinate
+ for which par(v):=t and, thereby, the affine point P:=(u, v). This
+ is also the case if alpha=0 and t=0, in which case v=0 and the point
+ P has order two. However, if alpha=0 and t=1, the ordered pair (u,
+ t) does not correspond to the outcome of the point compression
+ function.
+
+I.3. Point Compression for Twisted Edwards Curves
+
+ If P:=(x, y) is an affine point of the twisted Edwards curve E_{a,d}
+ defined over the field GF(q), then so is P:=(x, y). Since the
+ defining equation a*x^2+y^2=1+d*x^2*y^2 has at most two solutions
+ with fixed yvalue, one can represent P by its ycoordinate and one
+ bit of information that allows one to distinguish P from P, i.e.,
+ one can represent P as the ordered pair compr(P):=(par(x), y). If P
+ is a point of order one or two, one can uniquely represent P by its
+ ycoordinate alone, since x=0 and has fixed parity. Conversely,
+ given the ordered pair (t, y), where y is an element of GF(q) and
+ where t=0 or t=1, and the domain parameters of the curve, one can use
+ the defining equation of the curve to try and determine candidate
+ values for the xcoordinate given y, by solving the quadratic
+ equation x^2:=alpha, where alpha:=(1y^2)/(ad*y^2). If alpha is not
+ a square in GF(q), this equation does not have a solution in GF(q)
+ and the ordered pair (t, y) does not correspond to a point of this
+ curve. Otherwise, there are two solutions, viz. x=sqrt(alpha) and
+ x. If alpha is a nonzero element of GF(q), one can uniquely recover
+ the xcoordinate for which par(x):=t and, thereby, the affine point
+ P:=(x, y). This is also the case if alpha=0 and t=0, in which case
+ x=0 and the point P has order one or two. However, if alpha=0 and
+ t=1, the ordered pair (t, y) does not correspond to the outcome of
+ the point compression function.
+
+Appendix J. Data Conversions
+
+ The string over some alphabet S consisting of the symbols x_{l1},
+ x_{l2}, ..., x_1, x_0 (each in S), in this order, is denoted by
+ str(x_{l1}, x_{l2}, ..., x_1, x_0). The length of this string
+ (over S) is the number of symbols it contains (here: l). The empty
+ string is the (unique) string of length l=0.
+
+ An octet is an integer in the interval [0,256). An octet string is a
+ string, where the alphabet is the set of all octets. A binary string
+ (or bit string, for short) is a string, where the alphabet is the set
+ {0,1}. Note that the length of a string is defined in terms of the
+ underlying alphabet.
+
+J.1. Conversion between Bit Strings and Integers
+
+ There is a 11 correspondence between bit strings of length l and the
+ integers in the interval [0, 2^l), where the bit string
+ X:=str(x_{l1}, x_{l2}, ..., x_1, x_0) corresponds to the integer
+ x:=x_{l1}*2^{l1} + x_{l2}*2^{l2} + ... + x_1*2 + x_0*1. (If l=0,
+ the empty bit string corresponds to the integer zero.) Note that
+ while the mapping from bit strings to integers is uniquely defined,
+ the inverse mapping from integers to bit strings is not, since any
+ nonnegative integer smaller than 2^t can be represented as a bit
+ string of length at least t (due to leading zero coefficients in base
+ 2 representation). The latter representation is called tight if the
+ bit string representation has minimal length.
+
+J.2. Conversion between Octet Strings and Integers (OS2I, I2OS)
+
+ There is a 11 correspondence between octet strings of length l and
+ the integers in the interval [0, 256^l), where the octet string
+ X:=str(X_{l1}, X_{l2}, ..., X_1, X_0) corresponds to the integer
+ x:=X_{l1}*256^{l1} + X^{l2}*256^{l2} + ... + X_1*256 + X_0*1.
+ (If l=0, the empty string corresponds to the integer zero.) Note
+ that while the mapping from octet strings to integers is uniquely
+ defined, the inverse mapping from integers to octet strings is not,
+ since any nonnegative integer smaller than 256^t can be represented
+ as an octet string of length at least t (due to leading zero
+ coefficients in base 256 representation). The latter representation
+ is called tight if the octet string representation has minimal
+ length. This defines the mapping OS2I from octet strings to integers
+ and the mapping I2OS(x,l) from nonnegative integers smaller than
+ 256^l to octet strings of length l.
+
+J.3. Conversion between Octet Strings and Bit Strings (BS2OS, OS2BS)
+
+ There is a 11 correspondence between octet strings of length l and
+ and bit strings of length 8*l, where the octet string X:=str(X_{l1},
+ X_{l2}, ..., X_1, X_0) corresponds to the rightconcatenation of the
+ 8bit strings x_{l1}, x_{l2}, ..., x_1, x_0, where each octet X_i
+ corresponds to the 8bit string x_i according to the mapping of
+ Appendix J.1 above. Note that the mapping from octet strings to bit
+ strings is uniquely defined and so is the inverse mapping from bit
+ strings to octet strings, if one prepends each bit string with the
+ smallest number of 0 bits so as to result in a bit string of length
+ divisible by eight (i.e., one uses prepadding). This defines the
+ mapping OS2BS from octet strings to bit strings and the corresponding
+ mapping BS2OS from bit strings to octet strings.
+
+J.4. Conversion between Field Elements and Octet Strings (FE2OS, OS2FE)
+
+ There is a 11 correspondence between elements of a fixed finite
+ field GF(q), where q=p^m and m>0, and vectors of length m, with
+ coefficients in GF(p), where each element x of GF(q) is a vector
+ (x_{m1}, x_{m2}, ..., x_1, x_0) according to the conventions of
+ Appendix B.2. In this case, this field element can be uniquely
+ represented by the rightconcatenation of the octet strings X_{m1},
+ X_{m2}, ..., X_1, X_0, where each octet string X_i corresponds to
+ the integer x_i in the interval [0,p1] according to the mapping of
+ Appendix J.2 above. Note that both the mapping from field elements
+ to octet strings and the inverse mapping are only uniquely defined if
+ each octet string X_i has the same fixed size (e.g., the smallest
+ integer l so that 256^l >= p) and if all integers are reduced modulo
+ p. If so, the latter representation is called tight if l is minimal
+ so that 256^l >= p. This defines the mapping FE2OS(x,l) from field
+ elements to octet strings and the mapping OS2FE(X,l) from octet
+ strings to field elements, where the underlying field is implicit and
+ assumed to be known from context. In this case, the octet string has
+ length l*m.
+
+J.5. Ordering Conventions
+
+ One can consider various representation functions, depending on bit
+ ordering and octetordering conventions.
+
+ The description below makes use of an auxiliary function (the
+ reversion function), which we define both for bit strings and octet
+ strings. For a bit string [octet string] X:=str(x_{l1}, x_{l2},
+ ..., x_1, x_0), its reverse is the bit string [octet string]
+ X':=rev(X):=str(x_0, x_1, ..., x_{l2}, x_{l1}).
+
+ We now describe representations in mostsignificantbit first (msb)
+ or leastsignificantbit first (lsb) order and those in most
+ significantbyte first (MSB) or leastsignificantbyte first (LSB)
+ order.
+
+ One distinguishes the following octetstring representations of
+ integers and field elements:
+
+ 1. MSB, msb: represent field elements and integers as above,
+ yielding the octet string str(X_{l1}, X_{l2}, ..., X_1, X_0).
+
+ 2. MSB, lsb: reverse the bitorder of each octet, viewed as 8bit
+ string, yielding the octet string str((rev(X_{l1}),
+ rev(X_{l2}), ..., rev(X_1), rev(X_0)).
+
+ 3. LSB, lsb: reverse the octet string and bitorder of each octet,
+ yielding the octet string str(rev(X_{0}), rev(X_{1}), ...,
+ rev(X_{l2}), rev(X_{l1})).
+
+ 4. LSB, msb: reverse the octet string, yielding the octet string
+ str(X_{0}, X_{1}, ..., X_{l2}, X_{l1}).
+
+ Thus, the 2octet string "07e3" represents the integer 2019 (=0x07e3)
+ in MSB/msb order, the integer 57,543 (0xe0c7) in MSB/lsb order, the
+ integer 51,168 (0xc7e0) in LSB/lsb order, and the integer 58,119
+ (=0xe307) in LSB/msb order.
+
+ Note that, with the above data conversions, there is still some
+ ambiguity as to how to represent an integer or a field element as a
+ bit string or octet string (due to leading zeros). However, tight
+ representations (as defined above) are nonambiguous.
+
+Appendix K. Representations for Curve25519 Family Members
+
+K.1. Wei25519
+
+ The representation of integers, field elements, affine points, and
+ compressed points for the curve Wei25519 are as indicated below.
+ Representations are relative to the prime field GF(p), where
+ p=2^25519 is one of the general domain parameters of Appendix E.3.
+
+ Each field element z of GF(p) is represented as the octet string
+ FE2OS(z), where one uses one the MSB/msb conventions and tight
+ representation, as specified in Appendix J. In particular, each
+ element of GF(p) is represented as a 32byte octet string, which 
+ when viewed as a bit string  has the leftmost bit position set to 0.
+
+ Each affine point (X, Y) of Wei25519 is represented as the
+ rightconcatenation of the 32byte octet representations for the x
+ and ycoordinate of this point according to the conventions above,
+ i.e., it is represented as the 64byte octet string str(FE2OS(X),
+ FE2OS(Y)).
+
+ For each compressed point (X, t) of Wei25519, the parity bit t (which
+ is an element of the field GF(2)), is represented as a 1bit bit
+ string, whereas the xcoordinate X (which is an element of GF(p)), is
+ represented as a 32byte octet string FE2OS(X). The result is
+ "squeezed", by superimposing the 1bit representation of t on the
+ leftmost (unused) bitposition of the 32byte octet representation of
+ X.
+
+ Each integer in the interval [0,n1] is viewed as an element of the
+ prime field GF(n) and represented using MSB/msb conventions and a
+ tight representation. In particular, each element of GF(n) is
+ represented as a 32byte octet string, which  when viewed as a bit
+ string  has the lefmost three bit positions set to 0.
+
+Appendix L. Auxiliary Functions
+
+L.1. Square Roots in GF(q)
+
+ Square roots are easy to compute in GF(q) if q = 3 (mod 4) (see
+ Appendix L.1.1) or if q = 5 (mod 8) (see Appendix L.1.2). Details on
+ how to compute square roots for other values of q are out of scope.
+
+ If square roots are easy to compute in GF(q), then so are these in
+ GF(q^2).
+
+L.1.1. Square Roots in GF(q), where q = 3 (mod 4)
+
+ If y is a nonzero element of GF(q) and z:= y^{(q3)/4}, then y is a
+ square in GF(q) only if y*z^2=1. If y*z^2=1, z is a square root of
+ 1/y and y*z is a square root of y in GF(q).
+
+L.1.2. Square Roots in GF(q), where q = 5 (mod 8)
+
+ If y is a nonzero element of GF(q) and z:=y^{z5)/8}, then y is a
+ square in GF(q) only if y^2*z^4=1.
+
+ a. If y*z^2=+1, z is a square root of 1/y and y*z is a square root
+ of y in GF(q);
+
+ b. If y*z^2=1, i*z is a square root of 1/y and i*y*z is a square
+ root of y.
+
+ Here, i is an element of GF(q) for which i^2=1 (e.g.,
+ i:=2^{(q1)/4}). This field element can be precomputed.
+
+L.2. Inversion
+
+ If y is an integer and gcd(y,n)=1, one can efficiently compute 1/y
+ (mod n) via the extended Euclidean Algorithm (see Section 2.2.5 of
+ [GECC]). One can use this algorithm as well to compute the inverse
+ of a nonzero element y of a prime field GF(p), since gcd(y,p)=1.
+
+ The inverse of a nonzero element y of GF(q) can be computed as
+
+ 1/y:=y^{q2} (since y^{q1}=1).
+
+ Further details are out of scope. If inverses are easy to compute in
+ GF(q), then so are these in GF(q^2).
+
+ The inverses of two nonzero elements y1 and y2 of GF(q) can be
+ computed by first computing the inverse z of y1*y2 and by
+ subsequently computing y2*z=:1/y1 and y1*z=:1/y2.
+
Author's Address
Rene Struik
Struik Security Consultancy
Email: rstruik.ext@gmail.com